Research ArticleMultiframe Superresolution Reconstruction Based onSelf-Learning Method

Shao-Shuo Mu12 Ye Zhang1 Ping Jia1 Xun Yang1 and Xiao-Feng Qiu3

1Key Laboratory of Airborne Optical Imaging and Measurement Changchun Institute of Optics Fine Mechanics and PhysicsChinese Academy of Sciences Changchun 130033 China2Graduate University of Chinese Academy of Sciences Beijing 100039 China3Northeast Normal University Changchun 130033 China

Correspondence should be addressed to Shao-Shuo Mu 860600894qqcom

Received 28 August 2014 Revised 11 November 2014 Accepted 11 November 2014

Academic Editor Daniel Zaldivar

Copyright copy 2015 Shao-Shuo Mu et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

One category of the superresolution algorithms widely used in practical applications is dictionary-based superresolutionalgorithms which constructs a single high-resolution (HR) and high-clarity image from multiple low-resolution (LR) imagesDespite the fact that general dictionary-based superresolution algorithms obtain redundant dictionaries from numerous HR-LRimagesHR image distortion is unavoidable To solve this problem this paper proposes amultiframe superresolution reconstructionbased on self-learning methods First multiple images from the same scene are selected to be both input and training images andlarger-scale images which are also involved in the training set are constructed from the learning dictionaryThen different larger-scale images are constructed via repetition of the first step and the initial HR sets whose scale closely approximates that of the targetHR image are finally obtained Lastly initial HR images are fused into one target HR image under the NLM idea while the IBP ideais adopted to meet the global constraint The simulation results demonstrate that the proposed algorithm produces more accuratereconstructions than those produced by other general superresolution algorithms while in real scene experiments the proposedalgorithm can run well and create clearer HR images from input images captured by cameras

1 Introduction

In modern engineering high-resolution (HR) images arealways desirable which provides important information formaking important decisions in various practical applicationsfor example biometrics airborne detection system medicaldiagnosis high definition television (HDTV) and remotesurveillance One way to obtain HR images is to increasesensor pixel resolution which however leads to increasedcost in sensors A promising alternative is the superresolutiontechnique which reconstructs one HR image from one ormore low-resolution (LR) images shot from the same sceneproviding sufficient relevant information

In recent years superresolution techniques have beenextensively studied and put into practice CEVA Companyfirstly combined superresolution techniques with CEVA-MM3101 low-power imaging and vision platform successfullywhich used LR sensors to construct HR image and have been

applied to camera-enabled mobile devices French companySPOT applied superresolution techniques to SPOT-5 satellitewhich can obtain clearer ground scene images and identifytargets more accurately compared with SPOT-4 which didnot adopt superresolution techniques So superresolutiontechnique has been a hot spot to improve imagesrsquo resolutionwithout changing the optical system of the camera

Superresolution techniques have three categories in gen-eral namely interpolation-based reconstruction-based anddictionary-based algorithms Interpolation-based algorithmsare simple but ineffective in which the unknown image pixelvalues are calculated from the known pixels using interpo-lation formula [1ndash4] such as bilinear bicubic and Lanczosinterpolations Reconstruction-based algorithms have pro-cess inverse to imaging system solving These algorithmsuse prior image knowledge and recover high-frequencyinformation via established imaging models [5ndash10]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 181864 12 pageshttpdxdoiorg1011552015181864

2 Mathematical Problems in Engineering

Most recently dictionary-based algorithms have beenmore widely applied due to the development in compressedsensing [11ndash15] This category of algorithms requires a num-ber of external training images using similarity redundancystructure of natural images to restoremissing high-frequencyinformation in input images Roweis and Saul [16] proposeda nonlinear dimension reduction method namely locallylinear embedding (LLE) which has been widely applied toimage processing Based on [16] Chang et al [17] appliedLLE technology to superresolution reconstruction In thismethod the single input LR image is divided into blocks withimage shift then each image block searches for 119896 nearestblocks in all LR training image blocks whereby the optimalweights are obtained finally a HR image is constructed froma linear combination of the 119896HR training blocks correspond-ing to the 119896 LR training blocks References [18ndash20] improvedthe algorithm in [17] and achieved enhanced reconstructionresults Yang et al [21] proposed a sparse coding methodfor dictionary learning Compared with previous algorithmstheir algorithm relies on a larger number of databases whileobtaining missing high-frequency information more effec-tively In [22] their sparse coding is improved to strengthenthe sparse similarity between LR and HR image blocks Thismethod is simpler and more effective compared with [21]He et al [23] adopted a beta process dictionary learningapproach to make the HR and LR dictionaries more accurateand consistent gaining better results in comparison with[22] Jeong et al [24] proposed a superresolution methodrobust to noise which involves two phases the learningphase and the reconstruction phase In the learning phasethe training images are classified and different dictionariesare constructed according to noise quantity and in thereconstruction phase the input images with various noisequantities adopt appropriate dictionaries to reconstruct HRimages

All the above reconstruction methods require a largenumber of external training images When external imagescannot be accessed it is feasible to start from the point ofsimilarity redundancy structure of input images In [25ndash28]different self-learning methods are proposed Glasner et al[25] realized superresolution reconstruction without exter-nal images by adopting the gradual magnification schemeChen et al [26] used high-frequency and low-frequencycomponents of input images to train dictionary image blocksThen they used the Approximate Nearest Neighbor Searchto reconstruct images and recovered the missing high-frequency information Zhang et al [27] proposed a self-similarity redundant algorithm which selects original LRimages and corresponding degraded images as the trainingimages and uses LLE algorithm to obtain one HR image In[28] Ramakanth and Babu proposed single image learningdictionary and adopted the approximate nearest neighborfields (ANNF) method to find similarities between inputLR blocks and LR dictionary This method reconstructs HRimages more effectively and more efficiently compared withother dictionary-based methods

Other studies [25ndash28] have shown that natural imageshave self-similarity information on different scales and high-frequency information in small-scale images can be obtained

from large-scale images So this paper proposes a multiple-frame superresolution algorithm based on self-learning andthis algorithm requires no external training image First self-similarity information of multiple input images is used tocreate larger-scale training images through a step by step pro-cess Then an over-complete dictionary is studied by sparserepresentations Finally the reconstructed images are furtherimproved by exerting the global constraint Compared with[25ndash27] the proposed algorithm possesses several uniquefeatures as follows

(1) Larger-scale images are created from input imagesBoth input images and larger-scale images areinvolved in the training set Sparse representationis adopted to learn the mapping relation of thedictionary

(2) One larger-scale image is reconstructed from oneinput image apply this reconstruction to all 119899 inputimages Through gradual magnification 119899 initial esti-mates are obtained each corresponding to one of theinput images with a scale closely approximating thatof the target HR image Then the NLM method isemployed to fuse all estimates into one target HRimage

(3) In order to restore more high-frequency partsdegraded images of input images are used as LRtraining set and high-frequency parts of input imagesare used as HR training set

Our theoretical analysis demonstrates that the proposedalgorithm is feasible and the simulation results show that theproposed algorithm performs better than general dictionary-based superresolution algorithms In real scene experimentscarried out on a camera system the proposed algorithmobtains clearer real HR images

This paper is organized as follows Section 2 gives adiscussion on image sparse models Section 3 presents theframework of our proposed algorithm as well as its imple-mentation followed by experimental results in Section 4Finally Section 5 gives a conclusion of this paper

2 Image Sparse Model

Superresolution reconstruction is a reverse evaluation prob-lem Dictionary-based superresolution reconstruction trainsthe corresponding LR dictionary and HR dictionary bymapping the relationship between LR images andHR imagesThen the relationship between LR dictionary and HR dictio-nary is solved and aHR image is reconstructed with recoveryof the missing high-frequency information Imaging modelbetween LR image and HR image is as shown in (1) when thereal scene is captured through the optical system we obtaina group of LR image scenes due to the impact of optical blurand downsampling Consider

119884 = 119878119867119883 (1)

119884 is a LR image captured by the camera119883 is the real sceneimage 119867 is the optical blur matrix 119878 is the downsamplingmatrix

Mathematical Problems in Engineering 3

LR training images

HR training images

LR

HR

Optimal

LR image blocks

HR image blocks

LR input imageImage block

y

HR reconstructionblock x

Fuse

HR image

dictionary Dl

dictionary Dh

coefficient a+

Figure 1 Simple diagram of general dictionary-based reconstruction algorithm

Superresolution reconstruction is a highly ill-conditionproblem Given one LR input image 119884 many HR images 119883

are suitable for formula (1) With the development of com-pressed sensing dictionary-based superresolution methodshave been appliedmore widely in image restoration areasWecan divide image into blocks 119910 of the same dimension everyimage block is represented by both sparse representationldquo119886rdquo and sparse matrix 119863 such as 119910 = 119863 times 119886 where 119863

is a dictionary matrix and ldquo119886rdquo is sparse coefficients Thebasic process of dictionary-based reconstruction is shown inFigure 1

First we require lots of training images and train thecorresponding LR dictionary and HR dictionary by learningLR training images and HR training images Second theoptimal sparse coefficients ldquo119886rdquo of LR input image block aresolved according to the equation min(119910 minus 119863119897 times 119886) Thenwe get the corresponding HR image block 119909 via solving theformula 119863119897 times 119886 and fuse all HR image blocks into one HRimage119883

3 The Proposed Method

Whenone LR image is given to reconstruct oneHR image themain problemof dictionary-basedmethods is how to find lotsof HR-LR training images and utilize self-similarity informa-tion of external images But it will cause the reconstructedHRimage distortion given the absence of the input imagesrsquo fre-quency information in the selectedHR-LR training set In thispaper we propose a multiple superresolution reconstructionmethod based on self-learning dictionaryWe collectmultipleimages of the same scene as both input images and trainingimage set and larger-scale images which are constructedfrom last training set are also involved in the training setconstantly After that we get the initial reconstruction imageswhose scale closely approximates target HR image Then weuse NLM method to fuse the initial reconstruction imagesinto one HR image Lastly IBP idea is adopted to satisfy theglobal reconstruction constraint Here four major parts aredetailed in the following In Section 31 we mainly discusshow to use input images to gradually magnify image andcreate a large number of different larger-scale images astraining set In Section 32 sparse representation is adoptedto jointly learn redundant dictionary In Section 33 weemploy NLM method to fuse initial reconstruction images

into one HR image and introduce the importance of globalreconstruction constraint In Section 34 theworking processof the proposed algorithm is introduced In Section 35 wediscuss how to simplify the operation

31 Self-LearningMethod As we have said above dictionary-based reconstruction algorithms need lots of training imagesto learn the overcomplete dictionary for constructing HRimage Literatures [25ndash28] point out that input image hasenough self-similarity information and the high-frequencyinformation of small-scale image can be obtained fromlarge-scale image They can use multiframe input imagesto reconstruct images accurately Therefore when externalimages cannot be accessed this paper proposes a self-learningpyramidmethodwhich constructs larger-scale images gradu-ally to join in training set And it is feasibleWe can fully learnall the redundant information from different scale imagesFirst assuming that the target HR image resolution is 119875

times that of the input LR image resolution we will enlargeinput images resolution 119901119894 times step by step as the followingformula

1199011 times 1199012 times sdot sdot sdot times 119901119899 times 119904 = 119875

119904 lt 119901119894 119894 = 1 119899

1199011 times 1199012 times sdot sdot sdot times 119901119899 le 119875

(2)

Assuming that the input images are 119873 times 119873 dimensionthere are two cases for the principle of 119901 as follows

(a) 1199011 = 1199012 = sdot sdot sdot = 119901119899 = 119891119892 119892119899 times 119908 = 119873 (119908 lt 119892 and 119891119892 all are integers)

(b) 1199011 = 11989111198921 119901119899 = 119891119899119892119899 119891119894119892119895 gt 1 (119891119894 119892119895 all areintegers 119904 gt 119901119894 and 119894 = 1 2 119899)

We discuss this paper based on condition (a)Then the degraded images 119884and

1are solved via the formula

1198841darr119901uarr119901 (darr119901 is 119901 times downsampling uarr119901 is 119901 times upsam-pling) which together constitute the HR-LR training set 1198761with high-frequency part of input images 1198841 as the formula(3) (The degraded images 119884

and

1are LR training set High-

frequency part (1198841minus119884and

1) of input images1198841 isHR training set)

This is the difference between this paper and [25ndash27] andour method can recover the edge high-frequency better Wetrain1198761 to get dictionary1198631 and construct the corresponding

4 Mathematical Problems in Engineering

HR trainingimages

Constructing dictionary

images

Constructing larger-scale images

HR-LR training images

Constructing larger-scale

HR-LR trainging images

images of this reconstructionOriginal LR

images set

Degraded images set

LR trainingimages

Images set

Degraded images set

LR trainingimages

HR trainingimages

Images set

Reconstruction HR image

Y1

p

Take Y1 as input LR

HR-

LR tr

aini

ng im

ages

Q1

Y2 Y2

p

HR-

LR tr

aini

ng im

ages

Q2

Take Y2 as input LR

images Y3

Constructinglarger-scaleimages Y3

Take Yn as input LRimages of thisreconstruction

Yn + 1

NLM idea fusing multiple imagesYn + 1 into one target HR image

D1

D2 Dn

+ + +middot middot middot

middot middot middot

middot middot middot

minusminus

= Y1uarrpdarr

Y1 minus

= Y2uarrpdarr

Y2 minus

Q2 = Q1 cup (Y2 minus ) cup

(Yn minusQn = Qnminus1 cup

Yand2

Yand2

Yand2

Yand1

Yand1

Yand2

Yandn ) cup Yand

n

Figure 2 Process of training images

larger-scale HR images 1198842 whose resolution is 119901 times that ofthe input imagesThen degraded images119884and

2are solvedwhich

together constitute new HR-LR training set 1198762 with high-frequency part of 1198842 and 1198761 again It is as shown in (3) thatwe reconstruct final HR images whose resolution is closest totarget HR images followed by 119899 repeated learning Processof training images is as shown in Figure 2 We enlarge everyinput image step by step just like pyramid Consider

1198761 = 119884and

1cup (1198841 minus 119884

and

1)

1198762 = 119884and

2cup (1198842 minus 119884

and

2) cup 1198761

119876119898 = 119884and

119898cup (119884119898 minus 119884

and

119898) cup 119876119898minus1

(3)

32 Learning Dictionary In the previous section we havediscussed the self-learning pyramid method Chen et al [26]and Zhang et al [27] use image blocks manifold to learndictionary They solve the optimal weight between inputimage block and several nearest local training blocks But itcannot effectively utilize global image information Yang etal [22] point out that sparse representation can utilize globalimage information better and construct universal dictionarywith a better reconstructed image So we propose jointlylearning dictionarymethod based on sparse representation inthis section As shown in Section 31 that 119876119898 is the trainingset we use images 119884119898 of 119876119898 as input images of next recon-struction 119909 and 119910 are HR training blocks and LR trainingblocks respectively which are represented by atomic linearcombination of matrix 119863 119909 = 119863ℎ times 120572 119910 = 119863119897 times 120573 (120572 and120573

are sparse coefficients 119863ℎ and 119863119897 are HR dictionary andLR dictionary resp) In this section the premise of learningdictionary is that sparse coefficients 120572 and 120573 are the same Itis as shown in (4) Consider

119910 = 119863119897 times 120572

119909 = 119863ℎ times 120572

(4)

By solving the sparse coefficients 120572 of LR dictionary 119863119897we can get HR image blocks according to 119909 = 119863ℎ times 120572 Thisis the basic idea of training dictionary Before reconstructingHR image we must learn dictionaries119863119897 and119863ℎ which mustsatisfy constraints of (4) firstly So we can convert (4) into thefollowing optimization problem as shown in (5) Consider

min 1205720 st 1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2le 120576

min 1205720 st 1003817100381710038171003817119910 minus 119863ℎ times 12057210038171003817100381710038172

2le 120576

(5)

As long as the sparse coefficients 120572 are sparse enough (5)is solved by solving 119871

1 norm problem as shown in (6) and(7) Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721 (6)

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721 (7)

In order to get HR dictionary and LR dictionary of thesame coefficient 120572 (6) and (7) can be converted into (8)Consider

argmin1

2119875 minus 119863 times 120572

2

2+ 120583 1205721 (8)

In (8)

119875 = (

119909

radic119873

119910

radic119872

) 119863 = (

119863ℎ

radic119873

119863119897

radic119872

) (9)

Mathematical Problems in Engineering 5

Dictionary D

Original LR

Degraded images set hellip

p interpolation

HR blockBlock

High frequency HR part

Fusion

Reconstruction HR image

High frequency image

images set Ym

Interpolation image Alm

+

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Ym minus

= Ymuarrpdarrp

Y1uarrp

Alm = Y1uarrp

minus

Yandm

Yandm

Figure 3 Process of constructing dictionary and reconstructing image (the upscale factor is 119901)

where 119873 is dimension of HR image blocks and 119872 isdimension of LR image blocks In order to solve (8) we setthe initial value of matrix 119863 and (8) can be converted into(10) Consider

min 1

2(119875 minus 119863 times 120572)

119879(119875 minus 119863 times 120572) + 120583 |120572| (10)

Then the transformation of (10) is as shown in (11)Consider

min 1

21198761015840120572119876 + 119901

1015840120572 + 120583 |120572| (11)

where 119876 = 119863119879119863 and 119901 = 119863

119879119910 After obtaining 120572 we can

solve (12) to get the value of matrix119863 Consider

argmin 119875 minus 119863 times 1205722

2 (12)

Therefore dictionaries 119863119897 and 119863ℎ are obtained Then weenlarge every input image of 119884119898 one by one as shown inFigure 3 First every input image is enlarged to 119860 119897119898 by inter-polation and the upscale factor is 119901 Second 119860 119897119898 is dividedinto blocks 119910 of the same size According to dictionaries 119863119897and 119863ℎ we solve (7) (8) and (10) to get 120572 Then HR image

blocks are obtained according to the formula 119909 = 119863ℎ times

120572 When all of the HR image blocks are gained they aremerged into one HR image 119860ℎ119898 according to the fixed order119860ℎ119898 + 119860 119897119898 is the larger-scale reconstructed image and alsois one of reconstruction images 119884119898+1 as shown in Figure 3Lastly we obtain the reconstruction images set 119884119898+1 by usingdictionary to construct every image in 119884119898 Then image sets119884119898+1 and 119876119898+1 will become the input images and trainingimages set of next reconstruction enlargement respectively119876119898+1 is as shown in (13) Consider

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898 (13)

33 Reconstruction Fusion In Sections 31 and 32 we useself-learning pyramid method to construct training imagesand get reconstruction set 119884119899+1 But the scale of reconstruc-tion images is 119901119899 times that of the original input imagesrsquo scalewhichmay not be the same scale with target HR image So weuse NLM idea to fuse all images of 119884119899+1 into one HR imagewhich is the same scale with target HR image Meanwhile itcan effectively eliminate fusion noise

Nonlocal mean (NLM) filter is a very effective denoisingfilter which is based on an assumption that each pixel is arepetition of surrounding pixels It can be used to obtain the

6 Mathematical Problems in Engineering

target pixel value with a weighted sum of the surroundingpixels Now NLM is widely used in the field of superreso-lution reconstruction Just as [9] when we use NLM idea toreconstructmultiframe images the smaller themagnificationis the smaller the reconstruction error is So when imagedimension of 119884119899+1 is closest to target HR image using NLMidea to process 119884119899+1 is feasible In this section we adopt (14)to fuse images of 119884119899+1 into target HR image1198830 Consider

119883 = argmin sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times10038171003817100381710038171003817119863119896119897119864119867

119896119897119883 minus 119864

119871

119896119897119910119905

10038171003817100381710038171003817

(14)

where120596(119896 119897 119894 119895 119905) is the weight coefficient usually calculatedby the following formula Consider

120596 (119896 119897 119894 119895 119905) = exp(minus

10038171003817100381710038171003817119875119896119897119910119905 minus 119875119894119895119910119905

10038171003817100381710038171003817

2

21205902) (15)

The (119896 119897) is a pixel of the target HR image 119910119905 repre-sents one of the image sets 119884119899+1 (119894 119895) is a pixel of image119910119905 119875119896119897119910119905 represents extracting image block from image 119910119905whose center is (119896 119897) After amplification and denoising ofreconstruction image we get the prime estimation image1198830Finally wemust add global reconstruction constraint tomeet119884 = 119878119867119883 as shown in (16) Consider

119883 = argmin 1003817100381710038171003817119910 minus 11987811986711988310038171003817100381710038172

2+ 120583

1003817100381710038171003817119883 minus 119883010038171003817100381710038172

2 (16)

Turn into the following formula (17) Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (17)

We can solve (17) to obtain the optimal estimated HRimage119883 via IBP idea

34 The Proposed Algorithm The process of the proposedalgorithm is as shown in the following writing based on theabove sections Suppose that we have four LR images of thesame sceneObjective Reconstruct HR Image from Multiple LR Images

(1) Initialization input four LR images 1198841 and solve thecorresponding degrade images by formula 1198841darr119901uarr119901construct the first training image set 1198761 set magni-fication 119901 and iterations 119899

(2) For119898 = 1 119899 consider the following

(a) HR image and LR image of the HR-LR trainingset 119876119898 are divided into image blocks 119909 and119910 respectively Training dictionaries 119863ℎ and119863119897 are solved according to section (b) and thefollowing formula Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721

(18)

(b) Take 119884119898 as new input LR images to be recon-structed And solve new HR images 119884119898+1 fromdictionaries 119863ℎ and 119863119897 Then we get fourreconstruction images119884119898+1 inwhich resolutionis 119901119898 times of the original input images

(c) Put high-frequency of 119884119898+1 and correspondingdegrade images into the HR training set and LRtraining set respectively So we get new trainingimage set according to section (a) and (3) Take119884119898+1 as new LR images to be reconstructedConsider

119884and

119898= ((119884119898) darr119901) uarr119901

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898

(19)

(3) End

(4) Adopt NLM idea to fuse images of 119884119899+1 into one thesame scale HR image1198830 with target image

(5) Add global reconstruction constraint to meet 119884 =

119878119867119883 Use IBP idea and gradient descent algorithm tosolve (16) and obtain the optimal reconstruction HRimage119883 Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (20)

(6) Output the final HR image 119883

35 The Discussion of Reconstruction Speed When weapply the NLM method to adjust reconstruction image inSection 33 it will spend much operation time by solving (14)directly So in this section we will accelerate reconstructionspeed by simplifying section (c) in this section Set derivationof the (14) to be equal to zero We can get (21) Consider

119883 = ( sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times (119864119867

119896119897)119879

119863119879

119896119897119863119896119897119864119867

119896119897)

minus1

times [

[

sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) times (119864119867

119896119897)119879

119863119879

119896119897119864119867

119896119897119910119905

]

]

(21)

119864119871

119896119897represents extracting one pixel from the HR image

Equation (21) becomes (22) Consider

119883 =

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) 119910119905 (119894 119895)

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) (22)

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 4 The input LR images (a) is one of several Lena images (b) is one of several building images (c) is one of several flower images

As described above it will reduce operation time andavoid complicated matrix operations Meanwhile this algo-rithm can be parallelized to deal with such steps as 2(a)2(b) and (4) of Section 34Therefore we can take advantageof GPU to further improve the reconstruction speed Asshown in Section 42 we use Gforce 780MGPU to processthe captured images in camera system experimentation

Some blocks of the input LR image are smooth withfew details as shown in Figure 4 (the black box) Theseblocks contain little high-frequency information and have noeffect for reconstruction HR image So this section proposesinterpolation algorithm to enlarge these blocks and thealgorithm can run faster Smooth image block is defined bythe following formula

119860 = radic1

1199012(119911 minus 119911119871

119894)2 (23)

where 119911119871119894is the pixel value of image block and 119911 is the average

value of block pixelWe set a fixed value1198600 If119860 lt 1198600 we use

interpolation method to reconstruct the block

4 Experimental Results and Analysis

41 Simulation Experimentation In this section we perform2x magnification experiments on three test images fromFigure 4 to validate the effectiveness of the proposedmethodThese images cover a lot of contents including Lena flowerand building as shown in Figure 4 The compared groupincludes five other methods the interpolation-based method[4] the NLMmethod based on reconstruction [8] Glassnerrsquosmethod [25] Yangrsquos method [23] and Zhangrsquos method[27] For example the input LR Lena images are simulatedfrom a 256 times 256 normative HR image revolved translatedrandomly and downsampled by a factor of two The imageblock size is set to 5 times 5 with overlap of 4 pixels betweenadjacent blocks And upscale factor 119875 is 2 The gradualenlargement 119901 is set to 125 119899 is 3 and 119904 is 128125 Althoughthe proposed method has a certain similarity in the idea ofdictionary-based reconstruction [23] it takes the followingunique features

(1) This paper proposes self-learning pyramid method toconstruct larger-scale image step by step And it does

not need to search lots of external HR-LR images as atraining set

(2) The reconstructed result depends on the selectedtraining images It will cause to the reconstructedHR image distortion given the absence of the inputimagesrsquo frequency information in the selected HR-LR training set This method puts different scalereconstructed image into the training set and it canprovide much real similarity information to ensurethe authenticity of the reconstruction HR image

(3) Because the reconstruction results have fusion noiseandmaynot be the same scalewith targetHR image inSection 33 we use NLM idea to meet both cases andmerge them into one HR image which contains moretarget image information Lastly we use IBP idea tomeet global reconstruction constraint

Figures 5 6 and 7 are the reconstruction results of thetesting images Lena flower and building

We prove the superiority of the proposed algorithm fromthe visual quality and objective quality of the reconstructedHR image And peak signal to noise ratio (PSNR) structuralsimilarity (SSIM) mean squared error (MSE) and meanabsolute error (MAE) are employed to evaluate the objectivequality of the reconstructed HR image Definitions are asfollows

Mean squared error is as follows

MSE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

(119892 (119894 119895) minus 119891 (119894 119895))2 (24)

Peak signal to noise ratio is as follows

PSNR = 10 log10

2552

MSE (25)

Mean absolute error is as follows

MAE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

1003816100381610038161003816119892 (119894 119895) minus 1198921003816100381610038161003816 (26)

The 119892(119894 119895) and 119891(119894 119895) are pixels of the reconstructed HRimage and the original HR image respectively The119872 and119873

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Most recently dictionary-based algorithms have beenmore widely applied due to the development in compressedsensing [11ndash15] This category of algorithms requires a num-ber of external training images using similarity redundancystructure of natural images to restoremissing high-frequencyinformation in input images Roweis and Saul [16] proposeda nonlinear dimension reduction method namely locallylinear embedding (LLE) which has been widely applied toimage processing Based on [16] Chang et al [17] appliedLLE technology to superresolution reconstruction In thismethod the single input LR image is divided into blocks withimage shift then each image block searches for 119896 nearestblocks in all LR training image blocks whereby the optimalweights are obtained finally a HR image is constructed froma linear combination of the 119896HR training blocks correspond-ing to the 119896 LR training blocks References [18ndash20] improvedthe algorithm in [17] and achieved enhanced reconstructionresults Yang et al [21] proposed a sparse coding methodfor dictionary learning Compared with previous algorithmstheir algorithm relies on a larger number of databases whileobtaining missing high-frequency information more effec-tively In [22] their sparse coding is improved to strengthenthe sparse similarity between LR and HR image blocks Thismethod is simpler and more effective compared with [21]He et al [23] adopted a beta process dictionary learningapproach to make the HR and LR dictionaries more accurateand consistent gaining better results in comparison with[22] Jeong et al [24] proposed a superresolution methodrobust to noise which involves two phases the learningphase and the reconstruction phase In the learning phasethe training images are classified and different dictionariesare constructed according to noise quantity and in thereconstruction phase the input images with various noisequantities adopt appropriate dictionaries to reconstruct HRimages

All the above reconstruction methods require a largenumber of external training images When external imagescannot be accessed it is feasible to start from the point ofsimilarity redundancy structure of input images In [25ndash28]different self-learning methods are proposed Glasner et al[25] realized superresolution reconstruction without exter-nal images by adopting the gradual magnification schemeChen et al [26] used high-frequency and low-frequencycomponents of input images to train dictionary image blocksThen they used the Approximate Nearest Neighbor Searchto reconstruct images and recovered the missing high-frequency information Zhang et al [27] proposed a self-similarity redundant algorithm which selects original LRimages and corresponding degraded images as the trainingimages and uses LLE algorithm to obtain one HR image In[28] Ramakanth and Babu proposed single image learningdictionary and adopted the approximate nearest neighborfields (ANNF) method to find similarities between inputLR blocks and LR dictionary This method reconstructs HRimages more effectively and more efficiently compared withother dictionary-based methods

Other studies [25ndash28] have shown that natural imageshave self-similarity information on different scales and high-frequency information in small-scale images can be obtained

from large-scale images So this paper proposes a multiple-frame superresolution algorithm based on self-learning andthis algorithm requires no external training image First self-similarity information of multiple input images is used tocreate larger-scale training images through a step by step pro-cess Then an over-complete dictionary is studied by sparserepresentations Finally the reconstructed images are furtherimproved by exerting the global constraint Compared with[25ndash27] the proposed algorithm possesses several uniquefeatures as follows

(1) Larger-scale images are created from input imagesBoth input images and larger-scale images areinvolved in the training set Sparse representationis adopted to learn the mapping relation of thedictionary

(2) One larger-scale image is reconstructed from oneinput image apply this reconstruction to all 119899 inputimages Through gradual magnification 119899 initial esti-mates are obtained each corresponding to one of theinput images with a scale closely approximating thatof the target HR image Then the NLM method isemployed to fuse all estimates into one target HRimage

(3) In order to restore more high-frequency partsdegraded images of input images are used as LRtraining set and high-frequency parts of input imagesare used as HR training set

Our theoretical analysis demonstrates that the proposedalgorithm is feasible and the simulation results show that theproposed algorithm performs better than general dictionary-based superresolution algorithms In real scene experimentscarried out on a camera system the proposed algorithmobtains clearer real HR images

This paper is organized as follows Section 2 gives adiscussion on image sparse models Section 3 presents theframework of our proposed algorithm as well as its imple-mentation followed by experimental results in Section 4Finally Section 5 gives a conclusion of this paper

2 Image Sparse Model

Superresolution reconstruction is a reverse evaluation prob-lem Dictionary-based superresolution reconstruction trainsthe corresponding LR dictionary and HR dictionary bymapping the relationship between LR images andHR imagesThen the relationship between LR dictionary and HR dictio-nary is solved and aHR image is reconstructed with recoveryof the missing high-frequency information Imaging modelbetween LR image and HR image is as shown in (1) when thereal scene is captured through the optical system we obtaina group of LR image scenes due to the impact of optical blurand downsampling Consider

119884 = 119878119867119883 (1)

119884 is a LR image captured by the camera119883 is the real sceneimage 119867 is the optical blur matrix 119878 is the downsamplingmatrix

Mathematical Problems in Engineering 3

LR training images

HR training images

LR

HR

Optimal

LR image blocks

HR image blocks

LR input imageImage block

y

HR reconstructionblock x

Fuse

HR image

dictionary Dl

dictionary Dh

coefficient a+

Figure 1 Simple diagram of general dictionary-based reconstruction algorithm

Superresolution reconstruction is a highly ill-conditionproblem Given one LR input image 119884 many HR images 119883

are suitable for formula (1) With the development of com-pressed sensing dictionary-based superresolution methodshave been appliedmore widely in image restoration areasWecan divide image into blocks 119910 of the same dimension everyimage block is represented by both sparse representationldquo119886rdquo and sparse matrix 119863 such as 119910 = 119863 times 119886 where 119863

is a dictionary matrix and ldquo119886rdquo is sparse coefficients Thebasic process of dictionary-based reconstruction is shown inFigure 1

First we require lots of training images and train thecorresponding LR dictionary and HR dictionary by learningLR training images and HR training images Second theoptimal sparse coefficients ldquo119886rdquo of LR input image block aresolved according to the equation min(119910 minus 119863119897 times 119886) Thenwe get the corresponding HR image block 119909 via solving theformula 119863119897 times 119886 and fuse all HR image blocks into one HRimage119883

3 The Proposed Method

Whenone LR image is given to reconstruct oneHR image themain problemof dictionary-basedmethods is how to find lotsof HR-LR training images and utilize self-similarity informa-tion of external images But it will cause the reconstructedHRimage distortion given the absence of the input imagesrsquo fre-quency information in the selectedHR-LR training set In thispaper we propose a multiple superresolution reconstructionmethod based on self-learning dictionaryWe collectmultipleimages of the same scene as both input images and trainingimage set and larger-scale images which are constructedfrom last training set are also involved in the training setconstantly After that we get the initial reconstruction imageswhose scale closely approximates target HR image Then weuse NLM method to fuse the initial reconstruction imagesinto one HR image Lastly IBP idea is adopted to satisfy theglobal reconstruction constraint Here four major parts aredetailed in the following In Section 31 we mainly discusshow to use input images to gradually magnify image andcreate a large number of different larger-scale images astraining set In Section 32 sparse representation is adoptedto jointly learn redundant dictionary In Section 33 weemploy NLM method to fuse initial reconstruction images

into one HR image and introduce the importance of globalreconstruction constraint In Section 34 theworking processof the proposed algorithm is introduced In Section 35 wediscuss how to simplify the operation

31 Self-LearningMethod As we have said above dictionary-based reconstruction algorithms need lots of training imagesto learn the overcomplete dictionary for constructing HRimage Literatures [25ndash28] point out that input image hasenough self-similarity information and the high-frequencyinformation of small-scale image can be obtained fromlarge-scale image They can use multiframe input imagesto reconstruct images accurately Therefore when externalimages cannot be accessed this paper proposes a self-learningpyramidmethodwhich constructs larger-scale images gradu-ally to join in training set And it is feasibleWe can fully learnall the redundant information from different scale imagesFirst assuming that the target HR image resolution is 119875

times that of the input LR image resolution we will enlargeinput images resolution 119901119894 times step by step as the followingformula

1199011 times 1199012 times sdot sdot sdot times 119901119899 times 119904 = 119875

119904 lt 119901119894 119894 = 1 119899

1199011 times 1199012 times sdot sdot sdot times 119901119899 le 119875

(2)

Assuming that the input images are 119873 times 119873 dimensionthere are two cases for the principle of 119901 as follows

(a) 1199011 = 1199012 = sdot sdot sdot = 119901119899 = 119891119892 119892119899 times 119908 = 119873 (119908 lt 119892 and 119891119892 all are integers)

(b) 1199011 = 11989111198921 119901119899 = 119891119899119892119899 119891119894119892119895 gt 1 (119891119894 119892119895 all areintegers 119904 gt 119901119894 and 119894 = 1 2 119899)

We discuss this paper based on condition (a)Then the degraded images 119884and

1are solved via the formula

1198841darr119901uarr119901 (darr119901 is 119901 times downsampling uarr119901 is 119901 times upsam-pling) which together constitute the HR-LR training set 1198761with high-frequency part of input images 1198841 as the formula(3) (The degraded images 119884

and

1are LR training set High-

frequency part (1198841minus119884and

1) of input images1198841 isHR training set)

This is the difference between this paper and [25ndash27] andour method can recover the edge high-frequency better Wetrain1198761 to get dictionary1198631 and construct the corresponding

4 Mathematical Problems in Engineering

HR trainingimages

Constructing dictionary

images

Constructing larger-scale images

HR-LR training images

Constructing larger-scale

HR-LR trainging images

images of this reconstructionOriginal LR

images set

Degraded images set

LR trainingimages

Images set

Degraded images set

LR trainingimages

HR trainingimages

Images set

Reconstruction HR image

Y1

p

Take Y1 as input LR

HR-

LR tr

aini

ng im

ages

Q1

Y2 Y2

p

HR-

LR tr

aini

ng im

ages

Q2

Take Y2 as input LR

images Y3

Constructinglarger-scaleimages Y3

Take Yn as input LRimages of thisreconstruction

Yn + 1

NLM idea fusing multiple imagesYn + 1 into one target HR image

D1

D2 Dn

+ + +middot middot middot

middot middot middot

middot middot middot

minusminus

= Y1uarrpdarr

Y1 minus

= Y2uarrpdarr

Y2 minus

Q2 = Q1 cup (Y2 minus ) cup

(Yn minusQn = Qnminus1 cup

Yand2

Yand2

Yand2

Yand1

Yand1

Yand2

Yandn ) cup Yand

n

Figure 2 Process of training images

larger-scale HR images 1198842 whose resolution is 119901 times that ofthe input imagesThen degraded images119884and

2are solvedwhich

together constitute new HR-LR training set 1198762 with high-frequency part of 1198842 and 1198761 again It is as shown in (3) thatwe reconstruct final HR images whose resolution is closest totarget HR images followed by 119899 repeated learning Processof training images is as shown in Figure 2 We enlarge everyinput image step by step just like pyramid Consider

1198761 = 119884and

1cup (1198841 minus 119884

and

1)

1198762 = 119884and

2cup (1198842 minus 119884

and

2) cup 1198761

119876119898 = 119884and

119898cup (119884119898 minus 119884

and

119898) cup 119876119898minus1

(3)

32 Learning Dictionary In the previous section we havediscussed the self-learning pyramid method Chen et al [26]and Zhang et al [27] use image blocks manifold to learndictionary They solve the optimal weight between inputimage block and several nearest local training blocks But itcannot effectively utilize global image information Yang etal [22] point out that sparse representation can utilize globalimage information better and construct universal dictionarywith a better reconstructed image So we propose jointlylearning dictionarymethod based on sparse representation inthis section As shown in Section 31 that 119876119898 is the trainingset we use images 119884119898 of 119876119898 as input images of next recon-struction 119909 and 119910 are HR training blocks and LR trainingblocks respectively which are represented by atomic linearcombination of matrix 119863 119909 = 119863ℎ times 120572 119910 = 119863119897 times 120573 (120572 and120573

are sparse coefficients 119863ℎ and 119863119897 are HR dictionary andLR dictionary resp) In this section the premise of learningdictionary is that sparse coefficients 120572 and 120573 are the same Itis as shown in (4) Consider

119910 = 119863119897 times 120572

119909 = 119863ℎ times 120572

(4)

By solving the sparse coefficients 120572 of LR dictionary 119863119897we can get HR image blocks according to 119909 = 119863ℎ times 120572 Thisis the basic idea of training dictionary Before reconstructingHR image we must learn dictionaries119863119897 and119863ℎ which mustsatisfy constraints of (4) firstly So we can convert (4) into thefollowing optimization problem as shown in (5) Consider

min 1205720 st 1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2le 120576

min 1205720 st 1003817100381710038171003817119910 minus 119863ℎ times 12057210038171003817100381710038172

2le 120576

(5)

As long as the sparse coefficients 120572 are sparse enough (5)is solved by solving 119871

1 norm problem as shown in (6) and(7) Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721 (6)

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721 (7)

In order to get HR dictionary and LR dictionary of thesame coefficient 120572 (6) and (7) can be converted into (8)Consider

argmin1

2119875 minus 119863 times 120572

2

2+ 120583 1205721 (8)

In (8)

119875 = (

119909

radic119873

119910

radic119872

) 119863 = (

119863ℎ

radic119873

119863119897

radic119872

) (9)

Mathematical Problems in Engineering 5

Dictionary D

Original LR

Degraded images set hellip

p interpolation

HR blockBlock

High frequency HR part

Fusion

Reconstruction HR image

High frequency image

images set Ym

Interpolation image Alm

+

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Ym minus

= Ymuarrpdarrp

Y1uarrp

Alm = Y1uarrp

minus

Yandm

Yandm

Figure 3 Process of constructing dictionary and reconstructing image (the upscale factor is 119901)

where 119873 is dimension of HR image blocks and 119872 isdimension of LR image blocks In order to solve (8) we setthe initial value of matrix 119863 and (8) can be converted into(10) Consider

min 1

2(119875 minus 119863 times 120572)

119879(119875 minus 119863 times 120572) + 120583 |120572| (10)

Then the transformation of (10) is as shown in (11)Consider

min 1

21198761015840120572119876 + 119901

1015840120572 + 120583 |120572| (11)

where 119876 = 119863119879119863 and 119901 = 119863

119879119910 After obtaining 120572 we can

solve (12) to get the value of matrix119863 Consider

argmin 119875 minus 119863 times 1205722

2 (12)

Therefore dictionaries 119863119897 and 119863ℎ are obtained Then weenlarge every input image of 119884119898 one by one as shown inFigure 3 First every input image is enlarged to 119860 119897119898 by inter-polation and the upscale factor is 119901 Second 119860 119897119898 is dividedinto blocks 119910 of the same size According to dictionaries 119863119897and 119863ℎ we solve (7) (8) and (10) to get 120572 Then HR image

blocks are obtained according to the formula 119909 = 119863ℎ times

120572 When all of the HR image blocks are gained they aremerged into one HR image 119860ℎ119898 according to the fixed order119860ℎ119898 + 119860 119897119898 is the larger-scale reconstructed image and alsois one of reconstruction images 119884119898+1 as shown in Figure 3Lastly we obtain the reconstruction images set 119884119898+1 by usingdictionary to construct every image in 119884119898 Then image sets119884119898+1 and 119876119898+1 will become the input images and trainingimages set of next reconstruction enlargement respectively119876119898+1 is as shown in (13) Consider

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898 (13)

33 Reconstruction Fusion In Sections 31 and 32 we useself-learning pyramid method to construct training imagesand get reconstruction set 119884119899+1 But the scale of reconstruc-tion images is 119901119899 times that of the original input imagesrsquo scalewhichmay not be the same scale with target HR image So weuse NLM idea to fuse all images of 119884119899+1 into one HR imagewhich is the same scale with target HR image Meanwhile itcan effectively eliminate fusion noise

Nonlocal mean (NLM) filter is a very effective denoisingfilter which is based on an assumption that each pixel is arepetition of surrounding pixels It can be used to obtain the

6 Mathematical Problems in Engineering

target pixel value with a weighted sum of the surroundingpixels Now NLM is widely used in the field of superreso-lution reconstruction Just as [9] when we use NLM idea toreconstructmultiframe images the smaller themagnificationis the smaller the reconstruction error is So when imagedimension of 119884119899+1 is closest to target HR image using NLMidea to process 119884119899+1 is feasible In this section we adopt (14)to fuse images of 119884119899+1 into target HR image1198830 Consider

119883 = argmin sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times10038171003817100381710038171003817119863119896119897119864119867

119896119897119883 minus 119864

119871

119896119897119910119905

10038171003817100381710038171003817

(14)

where120596(119896 119897 119894 119895 119905) is the weight coefficient usually calculatedby the following formula Consider

120596 (119896 119897 119894 119895 119905) = exp(minus

10038171003817100381710038171003817119875119896119897119910119905 minus 119875119894119895119910119905

10038171003817100381710038171003817

2

21205902) (15)

The (119896 119897) is a pixel of the target HR image 119910119905 repre-sents one of the image sets 119884119899+1 (119894 119895) is a pixel of image119910119905 119875119896119897119910119905 represents extracting image block from image 119910119905whose center is (119896 119897) After amplification and denoising ofreconstruction image we get the prime estimation image1198830Finally wemust add global reconstruction constraint tomeet119884 = 119878119867119883 as shown in (16) Consider

119883 = argmin 1003817100381710038171003817119910 minus 11987811986711988310038171003817100381710038172

2+ 120583

1003817100381710038171003817119883 minus 119883010038171003817100381710038172

2 (16)

Turn into the following formula (17) Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (17)

We can solve (17) to obtain the optimal estimated HRimage119883 via IBP idea

34 The Proposed Algorithm The process of the proposedalgorithm is as shown in the following writing based on theabove sections Suppose that we have four LR images of thesame sceneObjective Reconstruct HR Image from Multiple LR Images

(1) Initialization input four LR images 1198841 and solve thecorresponding degrade images by formula 1198841darr119901uarr119901construct the first training image set 1198761 set magni-fication 119901 and iterations 119899

(2) For119898 = 1 119899 consider the following

(a) HR image and LR image of the HR-LR trainingset 119876119898 are divided into image blocks 119909 and119910 respectively Training dictionaries 119863ℎ and119863119897 are solved according to section (b) and thefollowing formula Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721

(18)

(b) Take 119884119898 as new input LR images to be recon-structed And solve new HR images 119884119898+1 fromdictionaries 119863ℎ and 119863119897 Then we get fourreconstruction images119884119898+1 inwhich resolutionis 119901119898 times of the original input images

(c) Put high-frequency of 119884119898+1 and correspondingdegrade images into the HR training set and LRtraining set respectively So we get new trainingimage set according to section (a) and (3) Take119884119898+1 as new LR images to be reconstructedConsider

119884and

119898= ((119884119898) darr119901) uarr119901

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898

(19)

(3) End

(4) Adopt NLM idea to fuse images of 119884119899+1 into one thesame scale HR image1198830 with target image

(5) Add global reconstruction constraint to meet 119884 =

119878119867119883 Use IBP idea and gradient descent algorithm tosolve (16) and obtain the optimal reconstruction HRimage119883 Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (20)

(6) Output the final HR image 119883

35 The Discussion of Reconstruction Speed When weapply the NLM method to adjust reconstruction image inSection 33 it will spend much operation time by solving (14)directly So in this section we will accelerate reconstructionspeed by simplifying section (c) in this section Set derivationof the (14) to be equal to zero We can get (21) Consider

119883 = ( sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times (119864119867

119896119897)119879

119863119879

119896119897119863119896119897119864119867

119896119897)

minus1

times [

[

sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) times (119864119867

119896119897)119879

119863119879

119896119897119864119867

119896119897119910119905

]

]

(21)

119864119871

119896119897represents extracting one pixel from the HR image

Equation (21) becomes (22) Consider

119883 =

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) 119910119905 (119894 119895)

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) (22)

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 4 The input LR images (a) is one of several Lena images (b) is one of several building images (c) is one of several flower images

As described above it will reduce operation time andavoid complicated matrix operations Meanwhile this algo-rithm can be parallelized to deal with such steps as 2(a)2(b) and (4) of Section 34Therefore we can take advantageof GPU to further improve the reconstruction speed Asshown in Section 42 we use Gforce 780MGPU to processthe captured images in camera system experimentation

Some blocks of the input LR image are smooth withfew details as shown in Figure 4 (the black box) Theseblocks contain little high-frequency information and have noeffect for reconstruction HR image So this section proposesinterpolation algorithm to enlarge these blocks and thealgorithm can run faster Smooth image block is defined bythe following formula

119860 = radic1

1199012(119911 minus 119911119871

119894)2 (23)

where 119911119871119894is the pixel value of image block and 119911 is the average

value of block pixelWe set a fixed value1198600 If119860 lt 1198600 we use

interpolation method to reconstruct the block

4 Experimental Results and Analysis

41 Simulation Experimentation In this section we perform2x magnification experiments on three test images fromFigure 4 to validate the effectiveness of the proposedmethodThese images cover a lot of contents including Lena flowerand building as shown in Figure 4 The compared groupincludes five other methods the interpolation-based method[4] the NLMmethod based on reconstruction [8] Glassnerrsquosmethod [25] Yangrsquos method [23] and Zhangrsquos method[27] For example the input LR Lena images are simulatedfrom a 256 times 256 normative HR image revolved translatedrandomly and downsampled by a factor of two The imageblock size is set to 5 times 5 with overlap of 4 pixels betweenadjacent blocks And upscale factor 119875 is 2 The gradualenlargement 119901 is set to 125 119899 is 3 and 119904 is 128125 Althoughthe proposed method has a certain similarity in the idea ofdictionary-based reconstruction [23] it takes the followingunique features

(1) This paper proposes self-learning pyramid method toconstruct larger-scale image step by step And it does

not need to search lots of external HR-LR images as atraining set

(2) The reconstructed result depends on the selectedtraining images It will cause to the reconstructedHR image distortion given the absence of the inputimagesrsquo frequency information in the selected HR-LR training set This method puts different scalereconstructed image into the training set and it canprovide much real similarity information to ensurethe authenticity of the reconstruction HR image

(3) Because the reconstruction results have fusion noiseandmaynot be the same scalewith targetHR image inSection 33 we use NLM idea to meet both cases andmerge them into one HR image which contains moretarget image information Lastly we use IBP idea tomeet global reconstruction constraint

Figures 5 6 and 7 are the reconstruction results of thetesting images Lena flower and building

We prove the superiority of the proposed algorithm fromthe visual quality and objective quality of the reconstructedHR image And peak signal to noise ratio (PSNR) structuralsimilarity (SSIM) mean squared error (MSE) and meanabsolute error (MAE) are employed to evaluate the objectivequality of the reconstructed HR image Definitions are asfollows

Mean squared error is as follows

MSE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

(119892 (119894 119895) minus 119891 (119894 119895))2 (24)

Peak signal to noise ratio is as follows

PSNR = 10 log10

2552

MSE (25)

Mean absolute error is as follows

MAE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

1003816100381610038161003816119892 (119894 119895) minus 1198921003816100381610038161003816 (26)

The 119892(119894 119895) and 119891(119894 119895) are pixels of the reconstructed HRimage and the original HR image respectively The119872 and119873

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

LR training images

HR training images

LR

HR

Optimal

LR image blocks

HR image blocks

LR input imageImage block

y

HR reconstructionblock x

Fuse

HR image

dictionary Dl

dictionary Dh

coefficient a+

Figure 1 Simple diagram of general dictionary-based reconstruction algorithm

Superresolution reconstruction is a highly ill-conditionproblem Given one LR input image 119884 many HR images 119883

are suitable for formula (1) With the development of com-pressed sensing dictionary-based superresolution methodshave been appliedmore widely in image restoration areasWecan divide image into blocks 119910 of the same dimension everyimage block is represented by both sparse representationldquo119886rdquo and sparse matrix 119863 such as 119910 = 119863 times 119886 where 119863

is a dictionary matrix and ldquo119886rdquo is sparse coefficients Thebasic process of dictionary-based reconstruction is shown inFigure 1

First we require lots of training images and train thecorresponding LR dictionary and HR dictionary by learningLR training images and HR training images Second theoptimal sparse coefficients ldquo119886rdquo of LR input image block aresolved according to the equation min(119910 minus 119863119897 times 119886) Thenwe get the corresponding HR image block 119909 via solving theformula 119863119897 times 119886 and fuse all HR image blocks into one HRimage119883

3 The Proposed Method

Whenone LR image is given to reconstruct oneHR image themain problemof dictionary-basedmethods is how to find lotsof HR-LR training images and utilize self-similarity informa-tion of external images But it will cause the reconstructedHRimage distortion given the absence of the input imagesrsquo fre-quency information in the selectedHR-LR training set In thispaper we propose a multiple superresolution reconstructionmethod based on self-learning dictionaryWe collectmultipleimages of the same scene as both input images and trainingimage set and larger-scale images which are constructedfrom last training set are also involved in the training setconstantly After that we get the initial reconstruction imageswhose scale closely approximates target HR image Then weuse NLM method to fuse the initial reconstruction imagesinto one HR image Lastly IBP idea is adopted to satisfy theglobal reconstruction constraint Here four major parts aredetailed in the following In Section 31 we mainly discusshow to use input images to gradually magnify image andcreate a large number of different larger-scale images astraining set In Section 32 sparse representation is adoptedto jointly learn redundant dictionary In Section 33 weemploy NLM method to fuse initial reconstruction images

into one HR image and introduce the importance of globalreconstruction constraint In Section 34 theworking processof the proposed algorithm is introduced In Section 35 wediscuss how to simplify the operation

31 Self-LearningMethod As we have said above dictionary-based reconstruction algorithms need lots of training imagesto learn the overcomplete dictionary for constructing HRimage Literatures [25ndash28] point out that input image hasenough self-similarity information and the high-frequencyinformation of small-scale image can be obtained fromlarge-scale image They can use multiframe input imagesto reconstruct images accurately Therefore when externalimages cannot be accessed this paper proposes a self-learningpyramidmethodwhich constructs larger-scale images gradu-ally to join in training set And it is feasibleWe can fully learnall the redundant information from different scale imagesFirst assuming that the target HR image resolution is 119875

times that of the input LR image resolution we will enlargeinput images resolution 119901119894 times step by step as the followingformula

1199011 times 1199012 times sdot sdot sdot times 119901119899 times 119904 = 119875

119904 lt 119901119894 119894 = 1 119899

1199011 times 1199012 times sdot sdot sdot times 119901119899 le 119875

(2)

Assuming that the input images are 119873 times 119873 dimensionthere are two cases for the principle of 119901 as follows

(a) 1199011 = 1199012 = sdot sdot sdot = 119901119899 = 119891119892 119892119899 times 119908 = 119873 (119908 lt 119892 and 119891119892 all are integers)

(b) 1199011 = 11989111198921 119901119899 = 119891119899119892119899 119891119894119892119895 gt 1 (119891119894 119892119895 all areintegers 119904 gt 119901119894 and 119894 = 1 2 119899)

We discuss this paper based on condition (a)Then the degraded images 119884and

1are solved via the formula

1198841darr119901uarr119901 (darr119901 is 119901 times downsampling uarr119901 is 119901 times upsam-pling) which together constitute the HR-LR training set 1198761with high-frequency part of input images 1198841 as the formula(3) (The degraded images 119884

and

1are LR training set High-

frequency part (1198841minus119884and

1) of input images1198841 isHR training set)

This is the difference between this paper and [25ndash27] andour method can recover the edge high-frequency better Wetrain1198761 to get dictionary1198631 and construct the corresponding

4 Mathematical Problems in Engineering

HR trainingimages

Constructing dictionary

images

Constructing larger-scale images

HR-LR training images

Constructing larger-scale

HR-LR trainging images

images of this reconstructionOriginal LR

images set

Degraded images set

LR trainingimages

Images set

Degraded images set

LR trainingimages

HR trainingimages

Images set

Reconstruction HR image

Y1

p

Take Y1 as input LR

HR-

LR tr

aini

ng im

ages

Q1

Y2 Y2

p

HR-

LR tr

aini

ng im

ages

Q2

Take Y2 as input LR

images Y3

Constructinglarger-scaleimages Y3

Take Yn as input LRimages of thisreconstruction

Yn + 1

NLM idea fusing multiple imagesYn + 1 into one target HR image

D1

D2 Dn

+ + +middot middot middot

middot middot middot

middot middot middot

minusminus

= Y1uarrpdarr

Y1 minus

= Y2uarrpdarr

Y2 minus

Q2 = Q1 cup (Y2 minus ) cup

(Yn minusQn = Qnminus1 cup

Yand2

Yand2

Yand2

Yand1

Yand1

Yand2

Yandn ) cup Yand

n

Figure 2 Process of training images

larger-scale HR images 1198842 whose resolution is 119901 times that ofthe input imagesThen degraded images119884and

2are solvedwhich

together constitute new HR-LR training set 1198762 with high-frequency part of 1198842 and 1198761 again It is as shown in (3) thatwe reconstruct final HR images whose resolution is closest totarget HR images followed by 119899 repeated learning Processof training images is as shown in Figure 2 We enlarge everyinput image step by step just like pyramid Consider

1198761 = 119884and

1cup (1198841 minus 119884

and

1)

1198762 = 119884and

2cup (1198842 minus 119884

and

2) cup 1198761

119876119898 = 119884and

119898cup (119884119898 minus 119884

and

119898) cup 119876119898minus1

(3)

32 Learning Dictionary In the previous section we havediscussed the self-learning pyramid method Chen et al [26]and Zhang et al [27] use image blocks manifold to learndictionary They solve the optimal weight between inputimage block and several nearest local training blocks But itcannot effectively utilize global image information Yang etal [22] point out that sparse representation can utilize globalimage information better and construct universal dictionarywith a better reconstructed image So we propose jointlylearning dictionarymethod based on sparse representation inthis section As shown in Section 31 that 119876119898 is the trainingset we use images 119884119898 of 119876119898 as input images of next recon-struction 119909 and 119910 are HR training blocks and LR trainingblocks respectively which are represented by atomic linearcombination of matrix 119863 119909 = 119863ℎ times 120572 119910 = 119863119897 times 120573 (120572 and120573

are sparse coefficients 119863ℎ and 119863119897 are HR dictionary andLR dictionary resp) In this section the premise of learningdictionary is that sparse coefficients 120572 and 120573 are the same Itis as shown in (4) Consider

119910 = 119863119897 times 120572

119909 = 119863ℎ times 120572

(4)

By solving the sparse coefficients 120572 of LR dictionary 119863119897we can get HR image blocks according to 119909 = 119863ℎ times 120572 Thisis the basic idea of training dictionary Before reconstructingHR image we must learn dictionaries119863119897 and119863ℎ which mustsatisfy constraints of (4) firstly So we can convert (4) into thefollowing optimization problem as shown in (5) Consider

min 1205720 st 1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2le 120576

min 1205720 st 1003817100381710038171003817119910 minus 119863ℎ times 12057210038171003817100381710038172

2le 120576

(5)

As long as the sparse coefficients 120572 are sparse enough (5)is solved by solving 119871

1 norm problem as shown in (6) and(7) Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721 (6)

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721 (7)

In order to get HR dictionary and LR dictionary of thesame coefficient 120572 (6) and (7) can be converted into (8)Consider

argmin1

2119875 minus 119863 times 120572

2

2+ 120583 1205721 (8)

In (8)

119875 = (

119909

radic119873

119910

radic119872

) 119863 = (

119863ℎ

radic119873

119863119897

radic119872

) (9)

Mathematical Problems in Engineering 5

Dictionary D

Original LR

Degraded images set hellip

p interpolation

HR blockBlock

High frequency HR part

Fusion

Reconstruction HR image

High frequency image

images set Ym

Interpolation image Alm

+

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Ym minus

= Ymuarrpdarrp

Y1uarrp

Alm = Y1uarrp

minus

Yandm

Yandm

Figure 3 Process of constructing dictionary and reconstructing image (the upscale factor is 119901)

where 119873 is dimension of HR image blocks and 119872 isdimension of LR image blocks In order to solve (8) we setthe initial value of matrix 119863 and (8) can be converted into(10) Consider

min 1

2(119875 minus 119863 times 120572)

119879(119875 minus 119863 times 120572) + 120583 |120572| (10)

Then the transformation of (10) is as shown in (11)Consider

min 1

21198761015840120572119876 + 119901

1015840120572 + 120583 |120572| (11)

where 119876 = 119863119879119863 and 119901 = 119863

119879119910 After obtaining 120572 we can

solve (12) to get the value of matrix119863 Consider

argmin 119875 minus 119863 times 1205722

2 (12)

Therefore dictionaries 119863119897 and 119863ℎ are obtained Then weenlarge every input image of 119884119898 one by one as shown inFigure 3 First every input image is enlarged to 119860 119897119898 by inter-polation and the upscale factor is 119901 Second 119860 119897119898 is dividedinto blocks 119910 of the same size According to dictionaries 119863119897and 119863ℎ we solve (7) (8) and (10) to get 120572 Then HR image

blocks are obtained according to the formula 119909 = 119863ℎ times

120572 When all of the HR image blocks are gained they aremerged into one HR image 119860ℎ119898 according to the fixed order119860ℎ119898 + 119860 119897119898 is the larger-scale reconstructed image and alsois one of reconstruction images 119884119898+1 as shown in Figure 3Lastly we obtain the reconstruction images set 119884119898+1 by usingdictionary to construct every image in 119884119898 Then image sets119884119898+1 and 119876119898+1 will become the input images and trainingimages set of next reconstruction enlargement respectively119876119898+1 is as shown in (13) Consider

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898 (13)

33 Reconstruction Fusion In Sections 31 and 32 we useself-learning pyramid method to construct training imagesand get reconstruction set 119884119899+1 But the scale of reconstruc-tion images is 119901119899 times that of the original input imagesrsquo scalewhichmay not be the same scale with target HR image So weuse NLM idea to fuse all images of 119884119899+1 into one HR imagewhich is the same scale with target HR image Meanwhile itcan effectively eliminate fusion noise

Nonlocal mean (NLM) filter is a very effective denoisingfilter which is based on an assumption that each pixel is arepetition of surrounding pixels It can be used to obtain the

6 Mathematical Problems in Engineering

target pixel value with a weighted sum of the surroundingpixels Now NLM is widely used in the field of superreso-lution reconstruction Just as [9] when we use NLM idea toreconstructmultiframe images the smaller themagnificationis the smaller the reconstruction error is So when imagedimension of 119884119899+1 is closest to target HR image using NLMidea to process 119884119899+1 is feasible In this section we adopt (14)to fuse images of 119884119899+1 into target HR image1198830 Consider

119883 = argmin sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times10038171003817100381710038171003817119863119896119897119864119867

119896119897119883 minus 119864

119871

119896119897119910119905

10038171003817100381710038171003817

(14)

where120596(119896 119897 119894 119895 119905) is the weight coefficient usually calculatedby the following formula Consider

120596 (119896 119897 119894 119895 119905) = exp(minus

10038171003817100381710038171003817119875119896119897119910119905 minus 119875119894119895119910119905

10038171003817100381710038171003817

2

21205902) (15)

The (119896 119897) is a pixel of the target HR image 119910119905 repre-sents one of the image sets 119884119899+1 (119894 119895) is a pixel of image119910119905 119875119896119897119910119905 represents extracting image block from image 119910119905whose center is (119896 119897) After amplification and denoising ofreconstruction image we get the prime estimation image1198830Finally wemust add global reconstruction constraint tomeet119884 = 119878119867119883 as shown in (16) Consider

119883 = argmin 1003817100381710038171003817119910 minus 11987811986711988310038171003817100381710038172

2+ 120583

1003817100381710038171003817119883 minus 119883010038171003817100381710038172

2 (16)

Turn into the following formula (17) Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (17)

We can solve (17) to obtain the optimal estimated HRimage119883 via IBP idea

34 The Proposed Algorithm The process of the proposedalgorithm is as shown in the following writing based on theabove sections Suppose that we have four LR images of thesame sceneObjective Reconstruct HR Image from Multiple LR Images

(1) Initialization input four LR images 1198841 and solve thecorresponding degrade images by formula 1198841darr119901uarr119901construct the first training image set 1198761 set magni-fication 119901 and iterations 119899

(2) For119898 = 1 119899 consider the following

(a) HR image and LR image of the HR-LR trainingset 119876119898 are divided into image blocks 119909 and119910 respectively Training dictionaries 119863ℎ and119863119897 are solved according to section (b) and thefollowing formula Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721

(18)

(b) Take 119884119898 as new input LR images to be recon-structed And solve new HR images 119884119898+1 fromdictionaries 119863ℎ and 119863119897 Then we get fourreconstruction images119884119898+1 inwhich resolutionis 119901119898 times of the original input images

(c) Put high-frequency of 119884119898+1 and correspondingdegrade images into the HR training set and LRtraining set respectively So we get new trainingimage set according to section (a) and (3) Take119884119898+1 as new LR images to be reconstructedConsider

119884and

119898= ((119884119898) darr119901) uarr119901

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898

(19)

(3) End

(4) Adopt NLM idea to fuse images of 119884119899+1 into one thesame scale HR image1198830 with target image

(5) Add global reconstruction constraint to meet 119884 =

119878119867119883 Use IBP idea and gradient descent algorithm tosolve (16) and obtain the optimal reconstruction HRimage119883 Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (20)

(6) Output the final HR image 119883

35 The Discussion of Reconstruction Speed When weapply the NLM method to adjust reconstruction image inSection 33 it will spend much operation time by solving (14)directly So in this section we will accelerate reconstructionspeed by simplifying section (c) in this section Set derivationof the (14) to be equal to zero We can get (21) Consider

119883 = ( sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times (119864119867

119896119897)119879

119863119879

119896119897119863119896119897119864119867

119896119897)

minus1

times [

[

sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) times (119864119867

119896119897)119879

119863119879

119896119897119864119867

119896119897119910119905

]

]

(21)

119864119871

119896119897represents extracting one pixel from the HR image

Equation (21) becomes (22) Consider

119883 =

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) 119910119905 (119894 119895)

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) (22)

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 4 The input LR images (a) is one of several Lena images (b) is one of several building images (c) is one of several flower images

As described above it will reduce operation time andavoid complicated matrix operations Meanwhile this algo-rithm can be parallelized to deal with such steps as 2(a)2(b) and (4) of Section 34Therefore we can take advantageof GPU to further improve the reconstruction speed Asshown in Section 42 we use Gforce 780MGPU to processthe captured images in camera system experimentation

Some blocks of the input LR image are smooth withfew details as shown in Figure 4 (the black box) Theseblocks contain little high-frequency information and have noeffect for reconstruction HR image So this section proposesinterpolation algorithm to enlarge these blocks and thealgorithm can run faster Smooth image block is defined bythe following formula

119860 = radic1

1199012(119911 minus 119911119871

119894)2 (23)

where 119911119871119894is the pixel value of image block and 119911 is the average

value of block pixelWe set a fixed value1198600 If119860 lt 1198600 we use

interpolation method to reconstruct the block

4 Experimental Results and Analysis

41 Simulation Experimentation In this section we perform2x magnification experiments on three test images fromFigure 4 to validate the effectiveness of the proposedmethodThese images cover a lot of contents including Lena flowerand building as shown in Figure 4 The compared groupincludes five other methods the interpolation-based method[4] the NLMmethod based on reconstruction [8] Glassnerrsquosmethod [25] Yangrsquos method [23] and Zhangrsquos method[27] For example the input LR Lena images are simulatedfrom a 256 times 256 normative HR image revolved translatedrandomly and downsampled by a factor of two The imageblock size is set to 5 times 5 with overlap of 4 pixels betweenadjacent blocks And upscale factor 119875 is 2 The gradualenlargement 119901 is set to 125 119899 is 3 and 119904 is 128125 Althoughthe proposed method has a certain similarity in the idea ofdictionary-based reconstruction [23] it takes the followingunique features

(1) This paper proposes self-learning pyramid method toconstruct larger-scale image step by step And it does

not need to search lots of external HR-LR images as atraining set

(2) The reconstructed result depends on the selectedtraining images It will cause to the reconstructedHR image distortion given the absence of the inputimagesrsquo frequency information in the selected HR-LR training set This method puts different scalereconstructed image into the training set and it canprovide much real similarity information to ensurethe authenticity of the reconstruction HR image

(3) Because the reconstruction results have fusion noiseandmaynot be the same scalewith targetHR image inSection 33 we use NLM idea to meet both cases andmerge them into one HR image which contains moretarget image information Lastly we use IBP idea tomeet global reconstruction constraint

Figures 5 6 and 7 are the reconstruction results of thetesting images Lena flower and building

We prove the superiority of the proposed algorithm fromthe visual quality and objective quality of the reconstructedHR image And peak signal to noise ratio (PSNR) structuralsimilarity (SSIM) mean squared error (MSE) and meanabsolute error (MAE) are employed to evaluate the objectivequality of the reconstructed HR image Definitions are asfollows

Mean squared error is as follows

MSE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

(119892 (119894 119895) minus 119891 (119894 119895))2 (24)

Peak signal to noise ratio is as follows

PSNR = 10 log10

2552

MSE (25)

Mean absolute error is as follows

MAE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

1003816100381610038161003816119892 (119894 119895) minus 1198921003816100381610038161003816 (26)

The 119892(119894 119895) and 119891(119894 119895) are pixels of the reconstructed HRimage and the original HR image respectively The119872 and119873

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

HR trainingimages

Constructing dictionary

images

Constructing larger-scale images

HR-LR training images

Constructing larger-scale

HR-LR trainging images

images of this reconstructionOriginal LR

images set

Degraded images set

LR trainingimages

Images set

Degraded images set

LR trainingimages

HR trainingimages

Images set

Reconstruction HR image

Y1

p

Take Y1 as input LR

HR-

LR tr

aini

ng im

ages

Q1

Y2 Y2

p

HR-

LR tr

aini

ng im

ages

Q2

Take Y2 as input LR

images Y3

Constructinglarger-scaleimages Y3

Take Yn as input LRimages of thisreconstruction

Yn + 1

NLM idea fusing multiple imagesYn + 1 into one target HR image

D1

D2 Dn

+ + +middot middot middot

middot middot middot

middot middot middot

minusminus

= Y1uarrpdarr

Y1 minus

= Y2uarrpdarr

Y2 minus

Q2 = Q1 cup (Y2 minus ) cup

(Yn minusQn = Qnminus1 cup

Yand2

Yand2

Yand2

Yand1

Yand1

Yand2

Yandn ) cup Yand

n

Figure 2 Process of training images

larger-scale HR images 1198842 whose resolution is 119901 times that ofthe input imagesThen degraded images119884and

2are solvedwhich

together constitute new HR-LR training set 1198762 with high-frequency part of 1198842 and 1198761 again It is as shown in (3) thatwe reconstruct final HR images whose resolution is closest totarget HR images followed by 119899 repeated learning Processof training images is as shown in Figure 2 We enlarge everyinput image step by step just like pyramid Consider

1198761 = 119884and

1cup (1198841 minus 119884

and

1)

1198762 = 119884and

2cup (1198842 minus 119884

and

2) cup 1198761

119876119898 = 119884and

119898cup (119884119898 minus 119884

and

119898) cup 119876119898minus1

(3)

32 Learning Dictionary In the previous section we havediscussed the self-learning pyramid method Chen et al [26]and Zhang et al [27] use image blocks manifold to learndictionary They solve the optimal weight between inputimage block and several nearest local training blocks But itcannot effectively utilize global image information Yang etal [22] point out that sparse representation can utilize globalimage information better and construct universal dictionarywith a better reconstructed image So we propose jointlylearning dictionarymethod based on sparse representation inthis section As shown in Section 31 that 119876119898 is the trainingset we use images 119884119898 of 119876119898 as input images of next recon-struction 119909 and 119910 are HR training blocks and LR trainingblocks respectively which are represented by atomic linearcombination of matrix 119863 119909 = 119863ℎ times 120572 119910 = 119863119897 times 120573 (120572 and120573

are sparse coefficients 119863ℎ and 119863119897 are HR dictionary andLR dictionary resp) In this section the premise of learningdictionary is that sparse coefficients 120572 and 120573 are the same Itis as shown in (4) Consider

119910 = 119863119897 times 120572

119909 = 119863ℎ times 120572

(4)

By solving the sparse coefficients 120572 of LR dictionary 119863119897we can get HR image blocks according to 119909 = 119863ℎ times 120572 Thisis the basic idea of training dictionary Before reconstructingHR image we must learn dictionaries119863119897 and119863ℎ which mustsatisfy constraints of (4) firstly So we can convert (4) into thefollowing optimization problem as shown in (5) Consider

min 1205720 st 1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2le 120576

min 1205720 st 1003817100381710038171003817119910 minus 119863ℎ times 12057210038171003817100381710038172

2le 120576

(5)

As long as the sparse coefficients 120572 are sparse enough (5)is solved by solving 119871

1 norm problem as shown in (6) and(7) Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721 (6)

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721 (7)

In order to get HR dictionary and LR dictionary of thesame coefficient 120572 (6) and (7) can be converted into (8)Consider

argmin1

2119875 minus 119863 times 120572

2

2+ 120583 1205721 (8)

In (8)

119875 = (

119909

radic119873

119910

radic119872

) 119863 = (

119863ℎ

radic119873

119863119897

radic119872

) (9)

Mathematical Problems in Engineering 5

Dictionary D

Original LR

Degraded images set hellip

p interpolation

HR blockBlock

High frequency HR part

Fusion

Reconstruction HR image

High frequency image

images set Ym

Interpolation image Alm

+

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Ym minus

= Ymuarrpdarrp

Y1uarrp

Alm = Y1uarrp

minus

Yandm

Yandm

Figure 3 Process of constructing dictionary and reconstructing image (the upscale factor is 119901)

where 119873 is dimension of HR image blocks and 119872 isdimension of LR image blocks In order to solve (8) we setthe initial value of matrix 119863 and (8) can be converted into(10) Consider

min 1

2(119875 minus 119863 times 120572)

119879(119875 minus 119863 times 120572) + 120583 |120572| (10)

Then the transformation of (10) is as shown in (11)Consider

min 1

21198761015840120572119876 + 119901

1015840120572 + 120583 |120572| (11)

where 119876 = 119863119879119863 and 119901 = 119863

119879119910 After obtaining 120572 we can

solve (12) to get the value of matrix119863 Consider

argmin 119875 minus 119863 times 1205722

2 (12)

Therefore dictionaries 119863119897 and 119863ℎ are obtained Then weenlarge every input image of 119884119898 one by one as shown inFigure 3 First every input image is enlarged to 119860 119897119898 by inter-polation and the upscale factor is 119901 Second 119860 119897119898 is dividedinto blocks 119910 of the same size According to dictionaries 119863119897and 119863ℎ we solve (7) (8) and (10) to get 120572 Then HR image

blocks are obtained according to the formula 119909 = 119863ℎ times

120572 When all of the HR image blocks are gained they aremerged into one HR image 119860ℎ119898 according to the fixed order119860ℎ119898 + 119860 119897119898 is the larger-scale reconstructed image and alsois one of reconstruction images 119884119898+1 as shown in Figure 3Lastly we obtain the reconstruction images set 119884119898+1 by usingdictionary to construct every image in 119884119898 Then image sets119884119898+1 and 119876119898+1 will become the input images and trainingimages set of next reconstruction enlargement respectively119876119898+1 is as shown in (13) Consider

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898 (13)

33 Reconstruction Fusion In Sections 31 and 32 we useself-learning pyramid method to construct training imagesand get reconstruction set 119884119899+1 But the scale of reconstruc-tion images is 119901119899 times that of the original input imagesrsquo scalewhichmay not be the same scale with target HR image So weuse NLM idea to fuse all images of 119884119899+1 into one HR imagewhich is the same scale with target HR image Meanwhile itcan effectively eliminate fusion noise

Nonlocal mean (NLM) filter is a very effective denoisingfilter which is based on an assumption that each pixel is arepetition of surrounding pixels It can be used to obtain the

6 Mathematical Problems in Engineering

target pixel value with a weighted sum of the surroundingpixels Now NLM is widely used in the field of superreso-lution reconstruction Just as [9] when we use NLM idea toreconstructmultiframe images the smaller themagnificationis the smaller the reconstruction error is So when imagedimension of 119884119899+1 is closest to target HR image using NLMidea to process 119884119899+1 is feasible In this section we adopt (14)to fuse images of 119884119899+1 into target HR image1198830 Consider

119883 = argmin sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times10038171003817100381710038171003817119863119896119897119864119867

119896119897119883 minus 119864

119871

119896119897119910119905

10038171003817100381710038171003817

(14)

where120596(119896 119897 119894 119895 119905) is the weight coefficient usually calculatedby the following formula Consider

120596 (119896 119897 119894 119895 119905) = exp(minus

10038171003817100381710038171003817119875119896119897119910119905 minus 119875119894119895119910119905

10038171003817100381710038171003817

2

21205902) (15)

The (119896 119897) is a pixel of the target HR image 119910119905 repre-sents one of the image sets 119884119899+1 (119894 119895) is a pixel of image119910119905 119875119896119897119910119905 represents extracting image block from image 119910119905whose center is (119896 119897) After amplification and denoising ofreconstruction image we get the prime estimation image1198830Finally wemust add global reconstruction constraint tomeet119884 = 119878119867119883 as shown in (16) Consider

119883 = argmin 1003817100381710038171003817119910 minus 11987811986711988310038171003817100381710038172

2+ 120583

1003817100381710038171003817119883 minus 119883010038171003817100381710038172

2 (16)

Turn into the following formula (17) Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (17)

We can solve (17) to obtain the optimal estimated HRimage119883 via IBP idea

34 The Proposed Algorithm The process of the proposedalgorithm is as shown in the following writing based on theabove sections Suppose that we have four LR images of thesame sceneObjective Reconstruct HR Image from Multiple LR Images

(1) Initialization input four LR images 1198841 and solve thecorresponding degrade images by formula 1198841darr119901uarr119901construct the first training image set 1198761 set magni-fication 119901 and iterations 119899

(2) For119898 = 1 119899 consider the following

(a) HR image and LR image of the HR-LR trainingset 119876119898 are divided into image blocks 119909 and119910 respectively Training dictionaries 119863ℎ and119863119897 are solved according to section (b) and thefollowing formula Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721

(18)

(b) Take 119884119898 as new input LR images to be recon-structed And solve new HR images 119884119898+1 fromdictionaries 119863ℎ and 119863119897 Then we get fourreconstruction images119884119898+1 inwhich resolutionis 119901119898 times of the original input images

(c) Put high-frequency of 119884119898+1 and correspondingdegrade images into the HR training set and LRtraining set respectively So we get new trainingimage set according to section (a) and (3) Take119884119898+1 as new LR images to be reconstructedConsider

119884and

119898= ((119884119898) darr119901) uarr119901

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898

(19)

(3) End

(4) Adopt NLM idea to fuse images of 119884119899+1 into one thesame scale HR image1198830 with target image

(5) Add global reconstruction constraint to meet 119884 =

119878119867119883 Use IBP idea and gradient descent algorithm tosolve (16) and obtain the optimal reconstruction HRimage119883 Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (20)

(6) Output the final HR image 119883

35 The Discussion of Reconstruction Speed When weapply the NLM method to adjust reconstruction image inSection 33 it will spend much operation time by solving (14)directly So in this section we will accelerate reconstructionspeed by simplifying section (c) in this section Set derivationof the (14) to be equal to zero We can get (21) Consider

119883 = ( sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times (119864119867

119896119897)119879

119863119879

119896119897119863119896119897119864119867

119896119897)

minus1

times [

[

sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) times (119864119867

119896119897)119879

119863119879

119896119897119864119867

119896119897119910119905

]

]

(21)

119864119871

119896119897represents extracting one pixel from the HR image

Equation (21) becomes (22) Consider

119883 =

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) 119910119905 (119894 119895)

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) (22)

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 4 The input LR images (a) is one of several Lena images (b) is one of several building images (c) is one of several flower images

As described above it will reduce operation time andavoid complicated matrix operations Meanwhile this algo-rithm can be parallelized to deal with such steps as 2(a)2(b) and (4) of Section 34Therefore we can take advantageof GPU to further improve the reconstruction speed Asshown in Section 42 we use Gforce 780MGPU to processthe captured images in camera system experimentation

Some blocks of the input LR image are smooth withfew details as shown in Figure 4 (the black box) Theseblocks contain little high-frequency information and have noeffect for reconstruction HR image So this section proposesinterpolation algorithm to enlarge these blocks and thealgorithm can run faster Smooth image block is defined bythe following formula

119860 = radic1

1199012(119911 minus 119911119871

119894)2 (23)

where 119911119871119894is the pixel value of image block and 119911 is the average

value of block pixelWe set a fixed value1198600 If119860 lt 1198600 we use

interpolation method to reconstruct the block

4 Experimental Results and Analysis

41 Simulation Experimentation In this section we perform2x magnification experiments on three test images fromFigure 4 to validate the effectiveness of the proposedmethodThese images cover a lot of contents including Lena flowerand building as shown in Figure 4 The compared groupincludes five other methods the interpolation-based method[4] the NLMmethod based on reconstruction [8] Glassnerrsquosmethod [25] Yangrsquos method [23] and Zhangrsquos method[27] For example the input LR Lena images are simulatedfrom a 256 times 256 normative HR image revolved translatedrandomly and downsampled by a factor of two The imageblock size is set to 5 times 5 with overlap of 4 pixels betweenadjacent blocks And upscale factor 119875 is 2 The gradualenlargement 119901 is set to 125 119899 is 3 and 119904 is 128125 Althoughthe proposed method has a certain similarity in the idea ofdictionary-based reconstruction [23] it takes the followingunique features

(1) This paper proposes self-learning pyramid method toconstruct larger-scale image step by step And it does

not need to search lots of external HR-LR images as atraining set

(2) The reconstructed result depends on the selectedtraining images It will cause to the reconstructedHR image distortion given the absence of the inputimagesrsquo frequency information in the selected HR-LR training set This method puts different scalereconstructed image into the training set and it canprovide much real similarity information to ensurethe authenticity of the reconstruction HR image

(3) Because the reconstruction results have fusion noiseandmaynot be the same scalewith targetHR image inSection 33 we use NLM idea to meet both cases andmerge them into one HR image which contains moretarget image information Lastly we use IBP idea tomeet global reconstruction constraint

Figures 5 6 and 7 are the reconstruction results of thetesting images Lena flower and building

We prove the superiority of the proposed algorithm fromthe visual quality and objective quality of the reconstructedHR image And peak signal to noise ratio (PSNR) structuralsimilarity (SSIM) mean squared error (MSE) and meanabsolute error (MAE) are employed to evaluate the objectivequality of the reconstructed HR image Definitions are asfollows

Mean squared error is as follows

MSE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

(119892 (119894 119895) minus 119891 (119894 119895))2 (24)

Peak signal to noise ratio is as follows

PSNR = 10 log10

2552

MSE (25)

Mean absolute error is as follows

MAE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

1003816100381610038161003816119892 (119894 119895) minus 1198921003816100381610038161003816 (26)

The 119892(119894 119895) and 119891(119894 119895) are pixels of the reconstructed HRimage and the original HR image respectively The119872 and119873

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Dictionary D

Original LR

Degraded images set hellip

p interpolation

HR blockBlock

High frequency HR part

Fusion

Reconstruction HR image

High frequency image

images set Ym

Interpolation image Alm

+

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Ym minus

= Ymuarrpdarrp

Y1uarrp

Alm = Y1uarrp

minus

Yandm

Yandm

Figure 3 Process of constructing dictionary and reconstructing image (the upscale factor is 119901)

where 119873 is dimension of HR image blocks and 119872 isdimension of LR image blocks In order to solve (8) we setthe initial value of matrix 119863 and (8) can be converted into(10) Consider

min 1

2(119875 minus 119863 times 120572)

119879(119875 minus 119863 times 120572) + 120583 |120572| (10)

Then the transformation of (10) is as shown in (11)Consider

min 1

21198761015840120572119876 + 119901

1015840120572 + 120583 |120572| (11)

where 119876 = 119863119879119863 and 119901 = 119863

119879119910 After obtaining 120572 we can

solve (12) to get the value of matrix119863 Consider

argmin 119875 minus 119863 times 1205722

2 (12)

Therefore dictionaries 119863119897 and 119863ℎ are obtained Then weenlarge every input image of 119884119898 one by one as shown inFigure 3 First every input image is enlarged to 119860 119897119898 by inter-polation and the upscale factor is 119901 Second 119860 119897119898 is dividedinto blocks 119910 of the same size According to dictionaries 119863119897and 119863ℎ we solve (7) (8) and (10) to get 120572 Then HR image

blocks are obtained according to the formula 119909 = 119863ℎ times

120572 When all of the HR image blocks are gained they aremerged into one HR image 119860ℎ119898 according to the fixed order119860ℎ119898 + 119860 119897119898 is the larger-scale reconstructed image and alsois one of reconstruction images 119884119898+1 as shown in Figure 3Lastly we obtain the reconstruction images set 119884119898+1 by usingdictionary to construct every image in 119884119898 Then image sets119884119898+1 and 119876119898+1 will become the input images and trainingimages set of next reconstruction enlargement respectively119876119898+1 is as shown in (13) Consider

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898 (13)

33 Reconstruction Fusion In Sections 31 and 32 we useself-learning pyramid method to construct training imagesand get reconstruction set 119884119899+1 But the scale of reconstruc-tion images is 119901119899 times that of the original input imagesrsquo scalewhichmay not be the same scale with target HR image So weuse NLM idea to fuse all images of 119884119899+1 into one HR imagewhich is the same scale with target HR image Meanwhile itcan effectively eliminate fusion noise

Nonlocal mean (NLM) filter is a very effective denoisingfilter which is based on an assumption that each pixel is arepetition of surrounding pixels It can be used to obtain the

6 Mathematical Problems in Engineering

target pixel value with a weighted sum of the surroundingpixels Now NLM is widely used in the field of superreso-lution reconstruction Just as [9] when we use NLM idea toreconstructmultiframe images the smaller themagnificationis the smaller the reconstruction error is So when imagedimension of 119884119899+1 is closest to target HR image using NLMidea to process 119884119899+1 is feasible In this section we adopt (14)to fuse images of 119884119899+1 into target HR image1198830 Consider

119883 = argmin sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times10038171003817100381710038171003817119863119896119897119864119867

119896119897119883 minus 119864

119871

119896119897119910119905

10038171003817100381710038171003817

(14)

where120596(119896 119897 119894 119895 119905) is the weight coefficient usually calculatedby the following formula Consider

120596 (119896 119897 119894 119895 119905) = exp(minus

10038171003817100381710038171003817119875119896119897119910119905 minus 119875119894119895119910119905

10038171003817100381710038171003817

2

21205902) (15)

The (119896 119897) is a pixel of the target HR image 119910119905 repre-sents one of the image sets 119884119899+1 (119894 119895) is a pixel of image119910119905 119875119896119897119910119905 represents extracting image block from image 119910119905whose center is (119896 119897) After amplification and denoising ofreconstruction image we get the prime estimation image1198830Finally wemust add global reconstruction constraint tomeet119884 = 119878119867119883 as shown in (16) Consider

119883 = argmin 1003817100381710038171003817119910 minus 11987811986711988310038171003817100381710038172

2+ 120583

1003817100381710038171003817119883 minus 119883010038171003817100381710038172

2 (16)

Turn into the following formula (17) Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (17)

We can solve (17) to obtain the optimal estimated HRimage119883 via IBP idea

34 The Proposed Algorithm The process of the proposedalgorithm is as shown in the following writing based on theabove sections Suppose that we have four LR images of thesame sceneObjective Reconstruct HR Image from Multiple LR Images

(1) Initialization input four LR images 1198841 and solve thecorresponding degrade images by formula 1198841darr119901uarr119901construct the first training image set 1198761 set magni-fication 119901 and iterations 119899

(2) For119898 = 1 119899 consider the following

(a) HR image and LR image of the HR-LR trainingset 119876119898 are divided into image blocks 119909 and119910 respectively Training dictionaries 119863ℎ and119863119897 are solved according to section (b) and thefollowing formula Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721

(18)

(b) Take 119884119898 as new input LR images to be recon-structed And solve new HR images 119884119898+1 fromdictionaries 119863ℎ and 119863119897 Then we get fourreconstruction images119884119898+1 inwhich resolutionis 119901119898 times of the original input images

(c) Put high-frequency of 119884119898+1 and correspondingdegrade images into the HR training set and LRtraining set respectively So we get new trainingimage set according to section (a) and (3) Take119884119898+1 as new LR images to be reconstructedConsider

119884and

119898= ((119884119898) darr119901) uarr119901

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898

(19)

(3) End

(4) Adopt NLM idea to fuse images of 119884119899+1 into one thesame scale HR image1198830 with target image

(5) Add global reconstruction constraint to meet 119884 =

119878119867119883 Use IBP idea and gradient descent algorithm tosolve (16) and obtain the optimal reconstruction HRimage119883 Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (20)

(6) Output the final HR image 119883

35 The Discussion of Reconstruction Speed When weapply the NLM method to adjust reconstruction image inSection 33 it will spend much operation time by solving (14)directly So in this section we will accelerate reconstructionspeed by simplifying section (c) in this section Set derivationof the (14) to be equal to zero We can get (21) Consider

119883 = ( sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times (119864119867

119896119897)119879

119863119879

119896119897119863119896119897119864119867

119896119897)

minus1

times [

[

sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) times (119864119867

119896119897)119879

119863119879

119896119897119864119867

119896119897119910119905

]

]

(21)

119864119871

119896119897represents extracting one pixel from the HR image

Equation (21) becomes (22) Consider

119883 =

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) 119910119905 (119894 119895)

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) (22)

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 4 The input LR images (a) is one of several Lena images (b) is one of several building images (c) is one of several flower images

As described above it will reduce operation time andavoid complicated matrix operations Meanwhile this algo-rithm can be parallelized to deal with such steps as 2(a)2(b) and (4) of Section 34Therefore we can take advantageof GPU to further improve the reconstruction speed Asshown in Section 42 we use Gforce 780MGPU to processthe captured images in camera system experimentation

Some blocks of the input LR image are smooth withfew details as shown in Figure 4 (the black box) Theseblocks contain little high-frequency information and have noeffect for reconstruction HR image So this section proposesinterpolation algorithm to enlarge these blocks and thealgorithm can run faster Smooth image block is defined bythe following formula

119860 = radic1

1199012(119911 minus 119911119871

119894)2 (23)

where 119911119871119894is the pixel value of image block and 119911 is the average

value of block pixelWe set a fixed value1198600 If119860 lt 1198600 we use

interpolation method to reconstruct the block

4 Experimental Results and Analysis

41 Simulation Experimentation In this section we perform2x magnification experiments on three test images fromFigure 4 to validate the effectiveness of the proposedmethodThese images cover a lot of contents including Lena flowerand building as shown in Figure 4 The compared groupincludes five other methods the interpolation-based method[4] the NLMmethod based on reconstruction [8] Glassnerrsquosmethod [25] Yangrsquos method [23] and Zhangrsquos method[27] For example the input LR Lena images are simulatedfrom a 256 times 256 normative HR image revolved translatedrandomly and downsampled by a factor of two The imageblock size is set to 5 times 5 with overlap of 4 pixels betweenadjacent blocks And upscale factor 119875 is 2 The gradualenlargement 119901 is set to 125 119899 is 3 and 119904 is 128125 Althoughthe proposed method has a certain similarity in the idea ofdictionary-based reconstruction [23] it takes the followingunique features

(1) This paper proposes self-learning pyramid method toconstruct larger-scale image step by step And it does

not need to search lots of external HR-LR images as atraining set

(2) The reconstructed result depends on the selectedtraining images It will cause to the reconstructedHR image distortion given the absence of the inputimagesrsquo frequency information in the selected HR-LR training set This method puts different scalereconstructed image into the training set and it canprovide much real similarity information to ensurethe authenticity of the reconstruction HR image

(3) Because the reconstruction results have fusion noiseandmaynot be the same scalewith targetHR image inSection 33 we use NLM idea to meet both cases andmerge them into one HR image which contains moretarget image information Lastly we use IBP idea tomeet global reconstruction constraint

Figures 5 6 and 7 are the reconstruction results of thetesting images Lena flower and building

We prove the superiority of the proposed algorithm fromthe visual quality and objective quality of the reconstructedHR image And peak signal to noise ratio (PSNR) structuralsimilarity (SSIM) mean squared error (MSE) and meanabsolute error (MAE) are employed to evaluate the objectivequality of the reconstructed HR image Definitions are asfollows

Mean squared error is as follows

MSE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

(119892 (119894 119895) minus 119891 (119894 119895))2 (24)

Peak signal to noise ratio is as follows

PSNR = 10 log10

2552

MSE (25)

Mean absolute error is as follows

MAE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

1003816100381610038161003816119892 (119894 119895) minus 1198921003816100381610038161003816 (26)

The 119892(119894 119895) and 119891(119894 119895) are pixels of the reconstructed HRimage and the original HR image respectively The119872 and119873

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

target pixel value with a weighted sum of the surroundingpixels Now NLM is widely used in the field of superreso-lution reconstruction Just as [9] when we use NLM idea toreconstructmultiframe images the smaller themagnificationis the smaller the reconstruction error is So when imagedimension of 119884119899+1 is closest to target HR image using NLMidea to process 119884119899+1 is feasible In this section we adopt (14)to fuse images of 119884119899+1 into target HR image1198830 Consider

119883 = argmin sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times10038171003817100381710038171003817119863119896119897119864119867

119896119897119883 minus 119864

119871

119896119897119910119905

10038171003817100381710038171003817

(14)

where120596(119896 119897 119894 119895 119905) is the weight coefficient usually calculatedby the following formula Consider

120596 (119896 119897 119894 119895 119905) = exp(minus

10038171003817100381710038171003817119875119896119897119910119905 minus 119875119894119895119910119905

10038171003817100381710038171003817

2

21205902) (15)

The (119896 119897) is a pixel of the target HR image 119910119905 repre-sents one of the image sets 119884119899+1 (119894 119895) is a pixel of image119910119905 119875119896119897119910119905 represents extracting image block from image 119910119905whose center is (119896 119897) After amplification and denoising ofreconstruction image we get the prime estimation image1198830Finally wemust add global reconstruction constraint tomeet119884 = 119878119867119883 as shown in (16) Consider

119883 = argmin 1003817100381710038171003817119910 minus 11987811986711988310038171003817100381710038172

2+ 120583

1003817100381710038171003817119883 minus 119883010038171003817100381710038172

2 (16)

Turn into the following formula (17) Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (17)

We can solve (17) to obtain the optimal estimated HRimage119883 via IBP idea

34 The Proposed Algorithm The process of the proposedalgorithm is as shown in the following writing based on theabove sections Suppose that we have four LR images of thesame sceneObjective Reconstruct HR Image from Multiple LR Images

(1) Initialization input four LR images 1198841 and solve thecorresponding degrade images by formula 1198841darr119901uarr119901construct the first training image set 1198761 set magni-fication 119901 and iterations 119899

(2) For119898 = 1 119899 consider the following

(a) HR image and LR image of the HR-LR trainingset 119876119898 are divided into image blocks 119909 and119910 respectively Training dictionaries 119863ℎ and119863119897 are solved according to section (b) and thefollowing formula Consider

argmin1

2

1003817100381710038171003817119909 minus 119863ℎ times 12057210038171003817100381710038172

2+ 120583 1205721

argmin1

2

1003817100381710038171003817119910 minus 119863119897 times 12057210038171003817100381710038172

2+ 120583 1205721

(18)

(b) Take 119884119898 as new input LR images to be recon-structed And solve new HR images 119884119898+1 fromdictionaries 119863ℎ and 119863119897 Then we get fourreconstruction images119884119898+1 inwhich resolutionis 119901119898 times of the original input images

(c) Put high-frequency of 119884119898+1 and correspondingdegrade images into the HR training set and LRtraining set respectively So we get new trainingimage set according to section (a) and (3) Take119884119898+1 as new LR images to be reconstructedConsider

119884and

119898= ((119884119898) darr119901) uarr119901

119876119898+1 = 119884and

119898+1cup (119884119898+1 minus 119884

and

119898+1) cup 119876119898

(19)

(3) End

(4) Adopt NLM idea to fuse images of 119884119899+1 into one thesame scale HR image1198830 with target image

(5) Add global reconstruction constraint to meet 119884 =

119878119867119883 Use IBP idea and gradient descent algorithm tosolve (16) and obtain the optimal reconstruction HRimage119883 Consider

119883 = 1198830 + 119878119867119879(119910 minus 119878119867119883) + 120583 (119883 minus 1198830) (20)

(6) Output the final HR image 119883

35 The Discussion of Reconstruction Speed When weapply the NLM method to adjust reconstruction image inSection 33 it will spend much operation time by solving (14)directly So in this section we will accelerate reconstructionspeed by simplifying section (c) in this section Set derivationof the (14) to be equal to zero We can get (21) Consider

119883 = ( sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905)

times (119864119867

119896119897)119879

119863119879

119896119897119863119896119897119864119867

119896119897)

minus1

times [

[

sum

(119896119897)isin120596

120588

sum

119905=1

sum

(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) times (119864119867

119896119897)119879

119863119879

119896119897119864119867

119896119897119910119905

]

]

(21)

119864119871

119896119897represents extracting one pixel from the HR image

Equation (21) becomes (22) Consider

119883 =

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) 119910119905 (119894 119895)

sum120588

119905=1sum(119894119895)isin119873119896119897

120596 (119896 119897 119894 119895 119905) (22)

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 4 The input LR images (a) is one of several Lena images (b) is one of several building images (c) is one of several flower images

As described above it will reduce operation time andavoid complicated matrix operations Meanwhile this algo-rithm can be parallelized to deal with such steps as 2(a)2(b) and (4) of Section 34Therefore we can take advantageof GPU to further improve the reconstruction speed Asshown in Section 42 we use Gforce 780MGPU to processthe captured images in camera system experimentation

Some blocks of the input LR image are smooth withfew details as shown in Figure 4 (the black box) Theseblocks contain little high-frequency information and have noeffect for reconstruction HR image So this section proposesinterpolation algorithm to enlarge these blocks and thealgorithm can run faster Smooth image block is defined bythe following formula

119860 = radic1

1199012(119911 minus 119911119871

119894)2 (23)

where 119911119871119894is the pixel value of image block and 119911 is the average

value of block pixelWe set a fixed value1198600 If119860 lt 1198600 we use

interpolation method to reconstruct the block

4 Experimental Results and Analysis

41 Simulation Experimentation In this section we perform2x magnification experiments on three test images fromFigure 4 to validate the effectiveness of the proposedmethodThese images cover a lot of contents including Lena flowerand building as shown in Figure 4 The compared groupincludes five other methods the interpolation-based method[4] the NLMmethod based on reconstruction [8] Glassnerrsquosmethod [25] Yangrsquos method [23] and Zhangrsquos method[27] For example the input LR Lena images are simulatedfrom a 256 times 256 normative HR image revolved translatedrandomly and downsampled by a factor of two The imageblock size is set to 5 times 5 with overlap of 4 pixels betweenadjacent blocks And upscale factor 119875 is 2 The gradualenlargement 119901 is set to 125 119899 is 3 and 119904 is 128125 Althoughthe proposed method has a certain similarity in the idea ofdictionary-based reconstruction [23] it takes the followingunique features

(1) This paper proposes self-learning pyramid method toconstruct larger-scale image step by step And it does

not need to search lots of external HR-LR images as atraining set

(2) The reconstructed result depends on the selectedtraining images It will cause to the reconstructedHR image distortion given the absence of the inputimagesrsquo frequency information in the selected HR-LR training set This method puts different scalereconstructed image into the training set and it canprovide much real similarity information to ensurethe authenticity of the reconstruction HR image

(3) Because the reconstruction results have fusion noiseandmaynot be the same scalewith targetHR image inSection 33 we use NLM idea to meet both cases andmerge them into one HR image which contains moretarget image information Lastly we use IBP idea tomeet global reconstruction constraint

Figures 5 6 and 7 are the reconstruction results of thetesting images Lena flower and building

We prove the superiority of the proposed algorithm fromthe visual quality and objective quality of the reconstructedHR image And peak signal to noise ratio (PSNR) structuralsimilarity (SSIM) mean squared error (MSE) and meanabsolute error (MAE) are employed to evaluate the objectivequality of the reconstructed HR image Definitions are asfollows

Mean squared error is as follows

MSE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

(119892 (119894 119895) minus 119891 (119894 119895))2 (24)

Peak signal to noise ratio is as follows

PSNR = 10 log10

2552

MSE (25)

Mean absolute error is as follows

MAE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

1003816100381610038161003816119892 (119894 119895) minus 1198921003816100381610038161003816 (26)

The 119892(119894 119895) and 119891(119894 119895) are pixels of the reconstructed HRimage and the original HR image respectively The119872 and119873

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 4 The input LR images (a) is one of several Lena images (b) is one of several building images (c) is one of several flower images

As described above it will reduce operation time andavoid complicated matrix operations Meanwhile this algo-rithm can be parallelized to deal with such steps as 2(a)2(b) and (4) of Section 34Therefore we can take advantageof GPU to further improve the reconstruction speed Asshown in Section 42 we use Gforce 780MGPU to processthe captured images in camera system experimentation

Some blocks of the input LR image are smooth withfew details as shown in Figure 4 (the black box) Theseblocks contain little high-frequency information and have noeffect for reconstruction HR image So this section proposesinterpolation algorithm to enlarge these blocks and thealgorithm can run faster Smooth image block is defined bythe following formula

119860 = radic1

1199012(119911 minus 119911119871

119894)2 (23)

where 119911119871119894is the pixel value of image block and 119911 is the average

value of block pixelWe set a fixed value1198600 If119860 lt 1198600 we use

interpolation method to reconstruct the block

4 Experimental Results and Analysis

41 Simulation Experimentation In this section we perform2x magnification experiments on three test images fromFigure 4 to validate the effectiveness of the proposedmethodThese images cover a lot of contents including Lena flowerand building as shown in Figure 4 The compared groupincludes five other methods the interpolation-based method[4] the NLMmethod based on reconstruction [8] Glassnerrsquosmethod [25] Yangrsquos method [23] and Zhangrsquos method[27] For example the input LR Lena images are simulatedfrom a 256 times 256 normative HR image revolved translatedrandomly and downsampled by a factor of two The imageblock size is set to 5 times 5 with overlap of 4 pixels betweenadjacent blocks And upscale factor 119875 is 2 The gradualenlargement 119901 is set to 125 119899 is 3 and 119904 is 128125 Althoughthe proposed method has a certain similarity in the idea ofdictionary-based reconstruction [23] it takes the followingunique features

(1) This paper proposes self-learning pyramid method toconstruct larger-scale image step by step And it does

not need to search lots of external HR-LR images as atraining set

(2) The reconstructed result depends on the selectedtraining images It will cause to the reconstructedHR image distortion given the absence of the inputimagesrsquo frequency information in the selected HR-LR training set This method puts different scalereconstructed image into the training set and it canprovide much real similarity information to ensurethe authenticity of the reconstruction HR image

(3) Because the reconstruction results have fusion noiseandmaynot be the same scalewith targetHR image inSection 33 we use NLM idea to meet both cases andmerge them into one HR image which contains moretarget image information Lastly we use IBP idea tomeet global reconstruction constraint

Figures 5 6 and 7 are the reconstruction results of thetesting images Lena flower and building

We prove the superiority of the proposed algorithm fromthe visual quality and objective quality of the reconstructedHR image And peak signal to noise ratio (PSNR) structuralsimilarity (SSIM) mean squared error (MSE) and meanabsolute error (MAE) are employed to evaluate the objectivequality of the reconstructed HR image Definitions are asfollows

Mean squared error is as follows

MSE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

(119892 (119894 119895) minus 119891 (119894 119895))2 (24)

Peak signal to noise ratio is as follows

PSNR = 10 log10

2552

MSE (25)

Mean absolute error is as follows

MAE =1

119872119873

119872

sum

119894=1

119873

sum

119895=1

1003816100381610038161003816119892 (119894 119895) minus 1198921003816100381610038161003816 (26)

The 119892(119894 119895) and 119891(119894 119895) are pixels of the reconstructed HRimage and the original HR image respectively The119872 and119873

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

(a) (b) (c) (d)

(e) (f) (g)

Figure 5 Comparison of reconstructed HR images of Lena (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

(a) (b) (c) (d)

(e) (f) (g)

Figure 6 Comparison of reconstructed HR images of building (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

are dimensions of reconstructedHR imageThe 119892 is themeanvalue of all pixels of reconstructed HR image When PSNR islarger and MAE is smaller the quality of the reconstructedimage is better SSIM is to show the reconstructed qualityby comparing image intensity image contrast and structural

similarity based on the visual system The larger the SSIMis the clearer the image is Values of PSNR SSIM andMAE are shown in Table 1 The first second and third rowsfor each test image indicate PSNR MAE and SSIM valuesrespectively Table 1 suggests that the proposed algorithm

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

(a) (b) (c) (d)

(e) (f) (g)

Figure 7 Comparison of reconstructed HR images of flower (a) Original HR image (b) Interpolation-based method results [4] (c) NLMresults [8] (d) Glassnerrsquos method [25] (e) Yangrsquos method [23] (f) Zhangrsquos method [27] (g) The proposed method results

can achieve better reconstructed result and recover moremissing high-frequency information and clearer edge detailsin comparison with other five methods

To further assess the visual quality obtained by differentmethods we compare reconstructed image details of differentmethods with original HR image as shown in Figures 5 6and 7Whenwe contrast eye and the brim of a hat in Figure 5the result of interpolation-based method is vaguer and NLMmethodrsquos reconstructed result appears distortion andmosaicOn the contrary the proposed method does not causedistortion and vague compared with interpolation-based andNLM cases Moreover reconstructed details of our methodare more delicate than other algorithms In Figures 6 and 7the proposed method can also produce better reconstructeddetails eaves and flower border both accurately and visuallyin comparison with other superresolution algorithms

42 Camera System Experimentation In this section weconduct research on real scene reconstruction to prove thesuperiority of the proposed algorithm It is the camerasystem assembled by ourselves which consists of a microdis-placement lens PI turntable and Gforce 780MGPU Bycontrolling the PI turntable we can continuously receivemultiple real images as shown in Figure 8(a) (one of multiplereal images) The images are taken to be reconstructed byour method and result is as shown in Figure 8(f) Figures8(b)ndash8(e) are the reconstruction results of the NLM methodbased on reconstruction [9] Glassnerrsquos method [25] Yangrsquosmethod and Zhangrsquos method [27] Signal noise ratio (SNR)average gradient (AG) information entropy (IE) and stan-dard deviation (SD) which do not require reference HR

image are employed to evaluate the objective quality of thereconstructed real HR image in real scene reconstruction ofcamera system Definitions are as follows Consider

SNR =119872

STDmax (27)

119872 represents image mean value The STDmax representsthe max value of local standard deviation Consider

IE =

119871minus1

sum

119894=0

119875119894log2119875119894 (28)

119871 is the total image gray level and 119875119894 represents the ratioof pixels in gray value 119894 to pixels in all image gray valuesConsider

AG =

119872minus1

sum

119894=1

119873minus1

sum

119895=1

radic((120597119892 (119909119894 119910119895)

120597119909119894

)

2

+ (120597119892 (119909119894 119910119895)

119910119895

)

2

) times1

2

times ((119872 minus 1) (119873 minus 1))minus1

SD =radic

sum119872

119894=1sum119873

119895=1(119892 (119909119894 119910119895) minus 119892)

2

119872 times 119873

(29)

The average gradient represents clarity Informationentropy can indicate image information abundance Standarddeviation represents discrete situation of image gray levelWhen SNR SD IE and AG are larger the reconstructed

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 1 The results of PSNR MAE and SSIM For every image there are three rows The first second and third rows for each test imageindicate PSNR MAE and SSIM values

Interpolation [4] NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed method

Lena273993 274644 306150 310123 315623 32601660320 58437 42831 39561 38601 3773508631 08643 09051 09168 09250 09538

Building312571 312962 371000 370549 369531 37699041998 40993 23300 23254 25365 2051008746 08753 09537 09524 09124 09653

Flower263318 257424 305731 312439 313826 32105480847 83127 5240 46710 46425 4437108176 07931 09031 09280 09315 09501

Average283294 281676 3276 331037 332993 34135461055 60852 39510 36508 36797 3420508517 08442 09206 09324 09229 09564

(a) (b) (c) (d)

(e) (f)

Figure 8 Real scene reconstruction (a) One of the captured images (b) NLM results [8] (c) Glassnerrsquos method [25] (d) Yangrsquos method [23](e) Zhangrsquos method [27] (f) The proposed method results

image is clearer and contains richer information Thereforethe numeric results of proposed method are better thanthose of other algorithms as shown in Table 2 Visually incomparison of contour outline of the reconstructed imagedetail the reconstructed image in Figure 8(f) is significantlyclearer than the other reconstructed image Consequently theproposed method can work out well in the camera systemexperimentation and be capable of reproducing plausibledetails and sharp edges with minimal artifacts

5 Conclusion

It is known that in superresolution reconstruction distortionof reconstructed images occurs when frequency information

of target images is not contained in training images Tosolve this problem this paper proposes a multiple super-resolution reconstruction algorithm based on self-learningdictionary First images on a series of scales created frommultiple input LR images are used as training imagesThen a learned overcomplete dictionary provides sufficientreal image information that ensures the authenticity ofthe reconstructed HR image In both simulation and realexperiments the proposed algorithm achieves better resultsthan that achieved by other algorithms recovering missinghigh-frequency information and edge detailsmore effectivelyFinally several ideas towards higher computation speed aresuggested

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 2 The results of SNR AG (average gradient) IE (information entropy) and SD (standard deviation)

Real scene NLMmethod [8] Glassnerrsquos method [25] Yangrsquos method [23] Zhangrsquos method [27] The proposed methodSNR 53115 67257 66978 67899 68049 71254AG 11059 11895 17399 17296 17327 18138IE 63191 63151 63536 63411 63609 64141SD 262563 262807 262999 263164 263043 263733

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the project of National ScienceFund for Distinguished Young Scholars of China (Grant no60902067) and the key Science-Technology Project of Jilinprovince (Grant no 11ZDGG001)

References

[1] P Thevenaz T Blu and M Unser Image Interpolation andResampling Academic Press New York NY USA 2000

[2] H S Hou and H C Andrews ldquoCubic spline for imageinterpolation and digital filteringrdquo IEEE Transactions on SignalProcess vol 26 no 6 pp 508ndash517 1978

[3] NADodgson ldquoQuadratic interpolation for image resamplingrdquoIEEE Transactions on Image Processing vol 6 no 9 pp 1322ndash1326 1997

[4] V Patrick S Sabine and V Martin ldquoA frequency domainapproach to registration of aliased images with application tosuper-resolutionrdquo EURASIP Journal on Applied Signal Process-ing vol 2006 Article ID 71459 14 pages 2006

[5] M Irani and S Peleg ldquoImproving resolution by image registra-tionrdquo Graphical Models and Image Processing vol 53 no 3 pp231ndash239 1991

[6] P Cheeseman B Kanefsky R Kraft J Stutz and R HansonldquoSuper-resolved surface reconstruction from multiple imagesrdquoin Maximum Entropy and Bayesian Methods Santa BarbaraCalifornia USA 1993 vol 62 of Fundamental Theories ofPhysics pp 293ndash308 Springer 1996

[7] R C Hardie K J Barnard and E E Armstrong ldquoJoint MAPregistration and high-resolution image estimation using asequence of under-sampled imagesrdquo IEEE Transactions onImage Processing vol 6 no 12 pp 1621ndash1633 1997

[8] S Farsiu M D Robinson M Elad and P Milanfar ldquoFast androbust multiframe super resolutionrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1327ndash1344 2004

[9] M Protter M Elad H Takeda and P Milanfar ldquoGeneralizingthe nonlocal-means to super-resolution reconstructionrdquo IEEETransactions on Image Processing vol 18 no 1 pp 36ndash51 2009

[10] J-D Li Y Zhang and P Jia ldquoSuper-resolution reconstructionusing nonlocal meansrdquoOptics and Precision Engineering vol 21no 6 pp 1576ndash1585 2013

[11] C Yang J Huang andM Yang ldquoExploiting self-similarities forsingle frame super-resolutionrdquo in Proceedings of the 10th AsianConference on Computer Vision (ACCV rsquo10) vol 3 pp 497ndash5102010

[12] K Zhang X Gao X Li andD Tao ldquoPartially supervised neigh-bor embedding for example-based image super-resolutionrdquoIEEE Journal on Selected Topics in Signal Processing vol 5 no 2pp 230ndash239 2011

[13] R Zeyde M Elad and M Protter ldquoOn single image scale-upusing sparse-representationsrdquo in Curves and Surfaces pp 711ndash730 Springer 2010

[14] Y-WTai S LiuM S Brown and S Lin ldquoSuper resolutionusingedge prior and single image detail synthesisrdquo in Proceedings ofthe IEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo10) pp 2400ndash2407 San FranciscoCalif USA June 2010

[15] J Sun J Zhu and M F Tappen ldquoContext-constrained halluci-nation for image super-resolutionrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo10) pp 231ndash238 IEEE San Francisco Calif USA June 2010

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[17] H ChangD Y Yeung andY Xiong ldquoSuper-resolution throughneighbor embeddingrdquo in Proceedings of the IEEE ComputerSociety Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp 275ndash282 2004

[18] Q Liu L Liu Y Wang and Z Zhang ldquoLocally linear embed-ding based example learning for pan-sharpeningrdquo in Proceed-ings of the 21st International Conference on Pattern Recognition(ICPR rsquo12) pp 1928ndash1931 Tsukuba Japan 2012

[19] Y Tao Z L Gan and X C Zhang ldquoNovel neighbor embeddingsuper resolutionmethod for compressed imagesrdquo inProceedingsof the 4th International Conference on Image Analysis and SignalProcessing (IASP rsquo12) pp 1ndash4 November 2012

[20] M Bevilacqua A Roumy C Guillemot and M-L AlberiMorel ldquoNeighbor embedding based single-image super-reso-lution using semi-nonnegative matrix factorizationrdquo in Pro-ceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo12) vol 1 pp 1289ndash1292Kyoto Japan March 2012

[21] J Yang JWright T Huang and YMa ldquoImage super-resolutionas sparse representation of raw image patchesrdquo in Proceedingsof the 26th IEEE Conference on Computer Vision and PatternRecognition pp 1ndash8 June 2008

[22] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010

[23] L He H Qi and R Zaretzki ldquoBeta process joint dictionarylearning for coupled feature spaces with application to singleimage super-resolutionrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo13) pp345ndash352 Portland Ore USA June 2013

[24] S-C Jeong Y-W Kang and B C Song ldquoSuper-resolution algo-rithm using noise level adaptive dictionaryrdquo in Proceedings ofthe 14th IEEE International Symposium on Consumer Electronics(ISCE rsquo10) June 2010

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[25] D Glasner S Bagon and M Irani ldquoSuper-resolution from asingle imagerdquo in Proceedings of the 12th International Conferenceon Computer Vision (ICCV rsquo09) pp 349ndash356 Kyoto JapanOctober 2009

[26] S Chen H Gong and C Li ldquoSuper resolution from a singleimage based on self similarityrdquo in Proceedings of the Interna-tional Conference on Computational and Information Sciences(ICCIS rsquo11) pp 91ndash94 Chengdu China October 2011

[27] K Zhang X Gao D Tao and X Li ldquoSingle image super-resolution with multiscale similarity learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems vol 24 no 10pp 1648ndash1659 2013

[28] S A Ramakanth and R V Babu ldquoSuper resolution using asingle image dictionaryrdquo inProceedings of the IEEE InternationalConference on Electronics Computing and CommunicationTechnologies (IEEE CONECCT rsquo14) pp 1ndash6 Bangalore IndiaJanuary 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of