blind multichannel reconstruction of high-resolution images using wavelet fusion

Blind multichannel reconstruction of high-resolutionimages using wavelet fusion

Said E. El-Khamy, Mohiy M. Hadhoud, Moawad I. Dessouky, Bassiouny M. Salam,and Fathi E. Abd El-Samie

We developed an approach to the blind multichannel reconstruction of high-resolution images. Thisapproach is based on breaking the image reconstruction problem into three consecutive steps: a blindmultichannel restoration, a wavelet-based image fusion, and a maximum entropy image interpolation.The blind restoration step depends on estimating the two-dimensional (2-D) greatest common divisor(GCD) between each observation and a combinational image generated by a weighted averaging processof the available observations. The purpose of generating this combinational image is to get a new imagewith a higher signal-to-noise ratio and a blurring operator that is a coprime with all the blurringoperators of the available observations. The 2-D GCD is then estimated between the new image and eachobservation, and thus the effect of noise on the estimation process is reduced. The multiple outputs of therestoration step are then applied to the image fusion step, which is based on wavelets. The objective ofthis step is to integrate the data obtained from each observation into a single image, which is theninterpolated to give an enhanced resolution image. A maximum entropy algorithm is derived and usedin interpolating the resulting image from the fusion step. Results show that the suggested blind imagereconstruction approach succeeds in estimating a high-resolution image from noisy blurred observationsin the case of relatively coprime unknown blurring operators. The required computation time of thesuggested approach is moderate. © 2005 Optical Society of America

OCIS codes: 100.0100, 100.6640, 100.3010.

1. Introduction

Image reconstruction is an important field of imageprocessing. The development of this field has been mo-tivated by the requirement to acquire high-resolutionimages from degraded observations obtained by mul-tiple sensors. By using suitable image reconstructionalgorithms, we can obtain a single high-resolution im-age either from several degraded still images or fromseveral degraded multiframes. There are numerousapplications of the reconstruction of high-resolutionimages such as remote sensing, medical imaging, sat-

ellite imaging, and high definition television (HDTV).In particular, image reconstruction has been the sub-ject of much interest in the fields of optics and opticalimaging. Many references have appeared in the opticsliterature on image reconstruction (restoration, recov-ery, deconvolution), blind deconvolution, and multi-frame reconstruction.1–7

Most algorithms that have been developed for high-resolution image reconstruction are iterative innature.8–19 These algorithms seek to reduce the com-putational complexity of the matrix inversion pro-cesses involved in the solution by using successiveapproximation methods for estimation of the high-resolution image. The frequency domain treatmentof the image reconstruction problem has been a pop-ular treatment in the early solutions. This is achievedby making use of the specific time-shift properties ofthe Fourier transform.7–10 The maximum a posteriori(MAP) estimation algorithm has also been imple-mented in this field.11–13 Some other set theoreticalapproaches have also been presented.10,14

In the general solution of the image reconstructionproblem, we deal with three degradation phenomena:a general geometric registration warp, blurring, andadditive noise. Most of the above-mentioned ap-

S. E. El-Khamy ([email protected]) is with the Department ofElectrical Engineering, Faculty of Engineering, Alexandria Univer-sity, Alexandria 21544, Egypt. M. M. Hadhoud is with the Depart-ment of Information Technology, Faculty of Computers andInformation, Menoufia University, 32511 Shebin Elkom,Egypt. M. I. Dessouky, B. M. Salam, and F. E. Abd El-Samie([email protected]) are with the Department of Electronicsand Electrical Communications, Faculty of Electronic Engineer-ing, Menoufia University, 32951 Menouf, Egypt.

Received 28 February 2005; revised manuscript received 29June 2005; accepted 1 July 2005.

0003-6935/05/347349-08$15.00/0© 2005 Optical Society of America

1 December 2005 � Vol. 44, No. 34 � APPLIED OPTICS 7349

proaches depend mainly on the assumption that thedegradation in each image or frame is known a priorior is to be estimated during the image reconstructionprocess. Few methods deal only with the problem ofblind image reconstruction.10,20 In blind methods, theblurring operators that affect each observation imageare unknown. The first step in blind image restora-tion is to register the images, which means estimat-ing the geometric registration warp between differentimages. This step can be considered as a preprocess-ing step, and we assume in our research that theregistration process has already been performed.20

When multiple distorted registered versions of thesame scene are available, it is possible to restore theoriginal image from these distorted versions withoutany prior knowledge of the distortion functions.21–25

The problem of reconstructing an image from two dis-torted observations in a high signal-to-noise ratio(SNR) environment has been investigated using a two-dimensional (2-D) greatest common divisor (GCD) al-gorithm that is sensitive to the presence of noise.25 Inthe z domain, the original image can be regarded as theGCD among the distorted observations if noise is ne-glected and the distortion filters are assumed to be offinite impulse response (FIR) and are relativelycoprime.25 Since any small variations in the point-spread function (PSF) of the imaging devices used incapturing the observations lead to blurring operatorsof different parameters, which are thus coprime in thez domain, the coprimeness assumption of blurring op-erators is a realistic assumption.

We develop a blind reconstruction approach ofhigh-resolution images that is based on breakingthe image reconstruction problem into three consec-utive steps: a blind multichannel GCD restoration,a wavelet-based image fusion, and a maximum en-tropy image interpolation. The blind restorationstep depends on estimating the 2-D GCD betweeneach observation and a combinational image gener-ated by a weighted averaging process of availableobservations. The 2-D GCD is then estimated be-tween the new image and each observation. Themultiple outputs of the restoration step are thenapplied to the wavelet-based image fusion step. Amaximum entropy algorithm is derived and used ininterpolating the resulting image from the fusionstep.

The paper is organized as follows: The problem isformulated in Section 2; Section 3 reviews the 2-DGCD algorithm; Section 4 presents the suggested ap-proach to obtain the high-resolution image from mul-tiple observations; Section 5 discusses the wavelet-based image fusion process; in Section 6, thesuggested maximum entropy image interpolation al-gorithm is derived and its implementation is given;results are presented in Section 7; and concludingremarks are given in Section 8.

2. Problem Formulation

The image reconstruction problem to be solved is theestimation of a high-resolution image from severalimages degraded by both blurring and noise. An im-

age degraded by both blurring and noise can be mod-eled by the following equation25–28:

y�m, n� � x�m, n� � b�m, n� � v�m, n�, (1)

where x(m, n) is the original image; b(m, n) is thedegradation PSF; and v(m, n) is an additive zeromean white Gaussian noise.

Assuming that we have two degraded observationsof the same scene given by the following equation:

yk�m, n� � x�m, n� � bk�m, n� � vk�m, n�,

k � 1, 2, (2)

where b1�m, n� and b2�m, n� are coprime filters. Thecoprimeness assumption is justified, since any slightvariation of a group of similar blurring operators af-fecting the same observation will lead to a differentblurring operator that is coprime with the other op-erators.

In the z domain this equation translates to

Yk�z1, z2� � X�z1, z2�Bk�z1, z2� � Vk�z1, z2�,

k � 1, 2. (3)

If the noise vk�m, n� is neglected, then the termV�z1, z2� vanishes, and hence the above equation be-comes

Yk�z1, z2� � X�z1, z2�Bk�z1, z2�, k � 1, 2. (4)

From the above equation, if the two distortion trans-fer functions B1�z1, z2� and B2�z1, z2� are coprime, thenX�z1, z2� is the GCD of Y1�z1, z2� and Y2�z1, z2�. Thiscan be written mathematically as follows:if

GCD�B1�z1, z2�, B2�z1, z2�� 1, (5)

then

GCD�Y1�z1, z2�, Y2�z1, z2�� X�z1, z2�. (6)

The GCD between two 2-D functions using the 2-DGCD algorithm is revised in Section 3.

3. Two-Dimensional Greatest Common DivisorAlgorithm

In the GCD algorithm25 it is assumed that the twoblurred images y1�m, n� and y2�m, n� of the originalimage x�m, n� are both M � N matrices. Substitutingz1 � exp��j2�m�P�, m � 0, 1, . . . , P � 1 into bothY1�z1, z2� and Y2�z1, z2� for each m, yields two one-dimensional (1-D) polynomials:

Yk�exp��j(2�m�P)�, z2� � X�exp��j(2�m�P)�, z2�· Bk�exp��j(2�m�P)�,z2�, k � 1, 2. (7)

7350 APPLIED OPTICS � Vol. 44, No. 34 � 1 December 2005

Thus the 1-D GCD between these two polynomialsyields the scaled quantity c0 � �exp��j�2�m�P��X�exp��j�2�m�P��, z2�. For each value of m we furthersubstitute z2 � exp��j2�n�P�, n � 0, 1, . . . , P � 1 inthis GCD to form a matrix of discrete Fourier trans-form elements,

A(m, n)a(m) � X�exp��j(2�m�P)�� exp��j(2�n�P)��. (8)

Carrying out similar operations on columns, we get

L�m, n�l�n� � X�exp��j�2�m�P��

� exp��j�2�n�P�� (9)

where a�m� and l�n� are scalar quantities that mustbe determined. From Eqs. (8) and (9) we have25

A�m, n�a�m� � L�m, n�l�n� � 0. (10)

The evaluation of a�m� and l�n� can be made on aleast-squares basis. Thus the estimated Fouriertransform of the original image is then calculatedby25

X�e�j�2�m�P�, e�j�2�n�P�� 12 �A�m, n�a�m�

� L�m, n�l�n��. (11)

The inverse Fourier transform is then used to obtainan estimate of the original image.

4. Proposed Reconstruction Approach ofHigh-Resolution Images

Assume that we have K degraded observations of thesame scene given by the following equation:

yk�m, n� � x�m, n� � bk�m, n� � vk�m, n�,

k � 1, 2, . . . , K. (12)

The application of the 2-D GCD algorithm describedin Section 3 that involves only two observations at atime is not an efficient approach to restoration, be-cause it does not incorporate the information in allobservations into the restoration process. To do so, wesuggest generating a new observation image, yK�1,given by the following equation:

yK�1�m, n� � �k�1

K

wkyk�m, n�, (13)

where wk values are scalars chosen according to anestimation of the SNRs in each image, which is madebased on a noise variance estimation. Another re-striction on the values of wk is the normal-

ization condition as follows:

�k�1

K

wk � 1. (14)

Substituting Eq. (12) into Eq. (13), we obtain

yK�1�m, n� � �k�1

K

wk�x�m, n� � bk�m, n� � vk�m, n��.

(15)

Thus

yK�1�m, n� � x�m, n� � �k�1

K

wkbk�m, n��

k�1

K

wkvk�m, n�. (16)

This equation can be written in the form

yK�1�m, n� � x�m, n� � bK�1�m, n� � vK�1�m, n�,(17)

where

bK�1�m, n� � �k�1

K

wkbk�m, n�, (18)

vK�1�m, n� � �k�1

K

wkvk�m, n�. (19)

In the z domain, this leads to

BK�1�z1, z2� � �k�1

K

wkBk�z1, z2�. (20)

It can be proved that BK�1�z1, z2� is coprime with allBk�z1, z2� for k � K. Dividing BK�1�z1, z2� by Bk�z1, z2�where k � K gives

BK�1�z1, z2�Bk�z1, z2�

�w1B1�z1, z2�

Bk�z1, z2��

w2B2�z1, z2�Bk�z1, z2�

� · · ·

� wk � · · · �wKBK�z1, z2�

Bk�z1, z2�, (21)

Since Bi�z1, z2� and Bk�z1, z2� are coprime functionsfor i � k and i, k � K; thus

Bi�z1, z2�Bk�z1, z2�

� Qi�z1, z2� �Ri�z1, z2�Bk�z1, z2�

, (22)

where Qi�z1, z2� is the quotient, Ri�z1, z2� is the re-mainder, and Ri�z1, z2� � 0.

Substituting Eq. (22) into Eq. (21) yields


BK�1�z1, z2�Bk�z1, z2�

� wk � �i�1i�k

K

wiQi�z1, z2� �Ri�z1, z2�Bk�z1, z2� .

(23)

Thus

BK�1�z1, z2�Bk�z1, z2�

� Qt�z1, z2� �Rt�z1, z2�Bk�z1, z2�

, (24)

where

Qt�z1, z2� � wk � �i�1i�k

K

wi�Qi�z1, z2��, (25)

Rt�z1, z2� � �i�1i�k

K

wiRi�z1, z2� � 0. (26)

This leads to the conclusion that BK�1�z1, z2� iscoprime with all Bk�z1, z2� for k � K.

Thus the 2-D GCD algorithm can be carried outbetween yK�1�m, n� and any yk�m, n� where k � Kto give good estimates of X �exp��j�2�m�P��,exp��j�2�n�P��.

The relation

vK�1�m,n� � �

k�1

K

wkvk�m, n�

leads to an image with noise variance �K�12 given by

�K�12 � �

k�1

K

wk2�k

2. (27)

For equal weight averaging we have w1 � w2 � · · ·� wK � 1�K:

�K�12 � �

k�1

K �k2

K2 . (28)

Assuming that all observations are taken in the samenoisy environment leads to

�K�12 �

�k2

K . (29)

The above equation leads to an improvement of SNRin the image yK�1�m, n� by a factor of K. This increasein SNR enables a robust application of the 2-D GCDalgorithm between yK�1�m, n� and any other obser-vation yk�m, n� since this 2-D GCD algorithm is sen-sitive to the presence of noise.

We summarize the steps of our suggested algo-rithm as follows:

(1) Begin with multiple observations blurred byrelatively coprime blurring operators in the presenceof noise.

(2) Generate a new image by a weighted averag-ing process of the available observations.

(3) Carry out the 2D GCD algorithm between thegenerated image and each observation.

(4) Carry out a wavelet-based image fusion pro-cess on the results obtained in step (3).

(5) Carry out a maximum entropy image interpo-lation process on the obtained result of step (4).

5. Wavelet-Based Image Fusion

Image fusion is a process by which information fromdifferent observation images is incorporated into asingle image. The importance of image fusion lies inthe fact that each observation image contains com-plementary information. When this complementaryinformation is integrated with that of another obser-vation, an image with the maximum amount of in-formation is obtained. Different approaches havebeen adopted for multisensor or multiple observationimage fusion, from a simple image averaging ap-proach to the wavelet transform image fusionapproach.29,30

We adopt the wavelet transform image fusion ap-proach to integrate the data from the multiple out-

Fig. 1. Schematic diagram of the wavelet image fusion process;DWT, discrete wavelet transform.

Fig. 2. Original undegraded building image and its maximumentropy interpolation: (a) original image and (b) maximum entropyinterpolation of the original image.


puts of the multichannel GCD image restoration step.In the application of the simple wavelet image fusionscheme, the wavelet packet decomposition is calcu-lated for each observation to obtain the multiresolu-tion levels of the images to be fused. In the transformdomain, the coefficients belonging to all resolutionlevels whose absolute values are largest are chosenfrom among the available observations. This rule isknown as the maximum frequency rule.29,30 Usingthis method, fusion takes place on all resolution lev-els and the dominant features at each scale are pre-served.

Another alternative to the maximum frequencyrule is the area-based selection rule, also called thelocal variance or the standard deviation rule. Thelocal variance of the wavelet coefficients is calculatedas a measure of local activity levels associated witheach wavelet coefficient. If the measures of activity ofthe wavelet coefficients in each of the two images tobe fused are close to each other, the average of the twowavelet coefficients is taken; otherwise, the coeffi-cient with the maximum absolute is chosen.29,30

Generally, the process of wavelet packet imagefusion, shown in Fig. 1, can be summarized in thefollowing steps:29,30

(i) The available images are first registered (thisis achieved for the outputs of the multichannel GCDrestoration step).

(ii) The wavelet packet decomposition of the ob-servations is calculated using a suitable basis func-tion and decomposition level.

(iii) A suitable fusion rule is used to select thewavelet coefficients from the source observations.

(iv) A decision map is created according to thefusion rule.

(v) A wavelet packet reconstruction is performedon the combinational created coefficients.

6. Maximum Entropy Image Interpolation

This step seeks to interpolate the resulting imagefrom the fusion step. The relationship between thelow-resolution (LR) fused image and the requiredhigh-resolution image can be represented by31,35

x � Df � n, (30)

where f, x, and n are lexicographically ordered vec-tors of the unknown high-resolution image, themeasured fused image, and additive noise values,respectively. The vector f is of size �RP�2 � 1 and thevectors x and v are of size P2 � 1. The matrix Drepresents a filtering and downsampling process ofdimensions P2 � �PR�2, where R is the resolutionenhancement factor in both directions.

Under a separability assumption, the matrix Dthat transforms the RP � RP high-resolution image

Fig. 3. Available observations 5 � 5 blur operator SNR of 60 dB: (a) observation (1), (b) observation (2), (c) observation (3).

Fig. 4. Results of the proposed algorithm: (a) fused image and (b)obtained high-resolution image peak SNR of 25.26 dB, CPU of 55 son a 1 GHz processor.

Fig. 5. Original undegraded MRI image and its maximum en-tropy interpolation: (a) original image and (b) maximum entropyinterpolation of the original image.


to the P � P low-resolution image is given by31–35

D � D1 � D1, (31)

where � represents the Kronecker product, and thematrix D1 represents the 1-D low-pass filtering anddownsampling. When R � 2, the matrix D1 will begiven by31–35

D1 �12�

1 1 0 0 . . . 0 00 0 1 1 . . . 0 0É É É É Ì É É

0 0 0 0 . . . 1 1�. (32)

From the above model, it is clear that the process ofobtaining a high-resolution image from a LR image isan inverse problem that requires inverting the oper-ator D. This inversion process is impossible becauseD is not a square matrix. The objective of imageinterpolation is to estimate the vector f given thesamples of the recorded image x.

In this section a mathematical model for imageinterpolation is derived based on the maximization ofthe entropy of the high-resolution image a priori. Ifthe samples of the required high-resolution imageare assumed to have unit energy, they can betreated as if they are probabilities, possibly of somany photons that are present at the ith sample ofthe required high-resolution image.27,33 The re-quired high-resolution image is assumed to betreated as light quanta associated with each pixelvalue. Thus the entropy of the required vector f isthen defined as follows:

E � ��i�1

p2

fi log2 �fi�, (33)

where E is the entropy and fi is the ith pixel value ofthe required vector f. This equation can be written invector form as follows:

E � �f t log2�f�. (34)

For image restoration, to maximize the entropy sub-ject to the constraint that �x � Df�2 � �n�2, the fol-lowing cost function must be minimized:

W�f� � f t log2�f� � ��x � Df�2 � �n�2�, (35)

where is a Lagrangian multiplier. Differentiatingboth sides of Eq. (35) with respect to f and equatingthe result to zero:

W�f�f � 0 �

1ln�2� �1 � ln�f̂�� 2Dt�x � Df̂��.

(36)

Solving for the estimated vector f,

ln�f̂� � �1 � ln�2�<2Dt�x � Df̂�=. (37)

Thus

f̂ � exp<�1 � ln�2�<2Dt�x � Df̂�= = . (38)

Expanding the above equation using the Taylor ex-pansion and neglecting all but the first two terms(since x � Df̂ must be a small quantity) leads to thefollowing form:

f̂ � ln�2�<2Dt�x � Df̂�=. (39)

Solving for f̂ leads to

f̂ �DtD � �I��1Dtg, (40)

where � � �1��2 ln�2��.This solution is based on the direct inversion of the

term DtD � �I. This matrix inversion can be per-formed easily depending on the special nature of thismatrix, which is a sparse diagonal matrix. The ma-trix DtD can be proved to be diagonal. This is easilyverified by noting that the operation DtD stands fordecimation followed by interpolation. Thus, if D dec-imates by a factor of R, applying DtD causes all po-sitions �1 � mR, 1 � nR� for integers (m, n) to remainunchanged, whereas the remaining pixels are re-placed by zeros. Thus DtD represents a maskingoperation, which is represented by a diagonal ma-trix.31,33 The effect of the term �I is to remove theill-posedness nature of the inverse problem of the

Fig. 6. Available observations 5 � 5 blur operator SNR of 60 dB: (a) observation (1), (b) observation (2), (c) observation (3).


term DtD by redistributing its eigenvalues to avoidsingularity. It is clear that the term DtD � �I rep-resents a sparse diagonal matrix, which can be easilyinverted.

7. Experimental Results

In this section the suggested blind high-resolutionimage reconstruction approach is tested. Two exper-iments have been conducted to test the suggestedimplementation. In the first experiment, three de-graded observations of the building image blurredwith different coprime blurring operators are used totest the suggested algorithm. Each observation is ofsize �128 � 128� pixels and the SNR in each obser-vation is 60 dB. The original image and its maximumentropy interpolated version are given in Fig. 2. Thedegraded observations are given in Fig. 3. A combi-national image is constructed from the available de-graded observation by equal weight averaging. TheGCD is estimated by the obtained combinational im-age and each of the available observations. The mul-tiple outputs obtained from this step are fused on awavelet basis. The image obtained from the wavelet-based fusion step is given in Fig. 4(a). The rule used inimage fusion is the maximum frequency rule and thefusion process is performed in a one-decompositionlevel. Applying the maximum entropy image interpo-lation algorithm on that image in Fig. 4(a) gives thehigh-resolution image in Fig. 4(b).

Another experiment on the magnetic resonance im-aging (MRI) image has been carried out to test thesuggested algorithm. The results of this experimentare given in Figs. 5–7. The computation time of eachof the two experiments is 55 s on a 1 GHz processor.It is clear that the suggested algorithm has succeededin obtaining a high-resolution image with good visualquality as compared to the available observationsand a high peak SNR in a short time.

8. Conclusion

We have developed a blind reconstruction algorithmfor high-resolution images. This algorithm is based on

estimating the 2-D GCD between each available ob-servation and a new generated combinational image.The results of each 2-D GCD operation are fused to-gether using the wavelet fusion scheme. The fusedimage is then interpolated using the maximum en-tropy algorithm derived in this paper. The robustnessof the suggested algorithm is guaranteed, since thecoprimeness of the blurring operator in the generatedimage with that of each observation is guaranteed. Thesensitivity of the 2-D GCD to noise is reduced by theaveraging process used to generate the combinationalimage. The suggested blind image reconstruction algo-rithm has succeeded in obtaining a high-resolution im-age with good visual quality and a high peak SNR in amoderate time.

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blind multichannel reconstruction of high-resolution images using wavelet fusion

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