atel-Vtaroandi

noisepropmityrmanrksf noperith

steps followed to detect noise are: (i) establish a criterionbased on a certain threshold, which most often is anempirical value and (ii) create a noise map. Subsequently,nt

in

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IEdoise is replaced by pixels obtained from a specicransformation.Currently, the requirements fullled to suppress noise in an

mage are not clearly dened. In [19], the conditionsecessary to eliminate noise are: (i) perceptually at regions

ToTnaT Image Process., 2014, Vol. 8, Iss. 3, pp. 131141oi: 10.1049/iet-ipr.2012.0615 This is an open access artfor grey-level images and proposals are presented in Section 3.In Section 3.1, the contrast measure provided in [24] isreplaced by a new one in terms of a top-hat byreconstruction transform, also called numerical residue [25].he new contrast measure involves the image backgroundbtained from a closeopen by reconstruction transformation.his proposal gives rise to a toggle mapping to attenuateoise. The toggle mapping uses as primitives the originalnd the morphological openclose by reconstructionfrom complete, given the intensive research on this topic.In several of the above mentioned papers, the common

measure. The contributions introduced in this paper will be

fuzzy logic is proposed in [18]. The list of references is far1 Introduction

A common problem in image processing is the detection andreduction of noise in digital images. As a result, severalmethods have been proposed. In [1], grey-level images arecleaned using top-hats [2] and smoothed images. Smoothedimages are obtained from alternated lters [3]. In [4, 5], adistance criterion to detect noise given by the top-hattransformation is applied, followed by an openclosesequence to remove noise. In [6], a distance criterion interms of the openclose and closeopen lters is presented,and noisy pixels are replaced by those obtained from anadaptive switching median lter. In [7], the Laplacian andgradient masks are used to estimate noise. In [8], theauthors present a method to extract the relationship betweenimage intensity and noise variance. In [911], differenttechniques to detect noise are provided. In [12, 13], noisypixels are replaced by the mean value of the neighbouringpixels. Methods based on wavelet transformations can befound in [1416]. A lter working similar to a median lterto suppress noise is presented in [17]. A technique based on

must be as smooth as possible, (ii) contours must bepreserved, (iii) texture detail must be maintained and (iv)global contrast must remain unchanged.These points are in conict, because when noise is

removed, global contrast is affected given that noise isreplaced by pixels obtained from a smoothed image.Furthermore, edges are also modied notwithstandingwhether the smoothed image was ltered by an adaptivetransformation. Owing to this situation, in this paper we aregoing to consider only two aspects that must be fullledwhen denoising an image: (a) modifying image contours aslittle as possible and (b) the resulting cleaned images musthave an improvement in the contrast. The mathematicalexpressions introduced throughout the paper make use ofthe following transformations: top-hat by reconstruction[20], internal gradient [21], toggle mappings [22] and theopening by reconstruction [23]. These will be presented inSection 2. The ideas to detect noise were inspired by theproposal in [24], which is briey presented in Section 2.5.In [24], two interesting things are proposed: (i) aformalisation to detect a threshold PC to identify noise and(ii) noise detection is carried out by means of a contrastPublished in IET Image ProcessingReceived on 16th June 2012Revised on 22nd March 2013Accepted on 21st April 2013doi: 10.1049/iet-ipr.2012.0615

Filtering of mixed Gaussianmorphological contrast deJorge D. Mendiola-Santibaez1, Ivn R. Tero1Doctorado en Ingeniera, Universidad Autnoma de Quer2CIDETEQ, S.C., Parque Tecnolgico Quertaro S/N, San FE-mail: mendijor@uaq.mx

Abstract: Several morphological transformations to detectprocedure presented previously in the current literature. The(i) using a contrast measure and (ii) applying different proxitwo of the proposals given in this study yield a better perfoHowever, although the methodology to identify noise wostructuring element. In Section 4, an image with two types oGaussian noise with 0.01 variance and 5% of salt and pepperformance is selected; subsequently, this is compared wconnected rank max opening and amoebas.ISSN 1751-9659

nd impulsive noise usingctorsillalobos2

, CP 76010, Quertaro, Mxicola-Pedro Escobedo, Quertaro 76703, Mxico

are introduced. The initial method is a modication of aosals given in the study allow to detect noise in two ways:criteria into several proposed toggle mappings. In the end,ce with respect to methods in which this research is based.adequately, the results are limited due to the use of theise is cleaned. Such image is contaminated with zero meannoise. From this experiment, the proposal giving the bestother recent operators as PDEs, wavelets, morphological131icle published by the IET under the Creative Commons Attribution

License(http://creativecommons.org/licenses/by/3.0/)

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images, which do not create new contours. However, giventhat the critical point PC represents the mean value denedin [24], this idea is extended to a global parameter fordetermining when a pixel is noise. The proximity criterionin the toggle mapping is modied and a new one isintroduced in Section 3.2 in terms of the top-hat byreconstruction transformation. In Section 3.3, noisy pixelstake values in accordance with a rank lter [26] to produceenhanced images with different contrast. In Section 3.4, themean lter is approximated by morphologicaltransformations, and the toggle mappings proposed so farcan be rewritten as purely morphological transformations.In Section 3.5, the top-hat criterion is replaced by theinternal gradient operator, and the primitive denoted as isutilised. The lter is given in terms of the openclose andcloseopen by reconstruction transformations. This ltershows a regular performance when compared to the medianlter. According to the experiments, two transformationsintroduced in this paper yield a better performance withrespect to the method proposed in [24] to detect noise. Toselect the transformations producing the best performance,the results obtained from the following indexes areanalysed: (i) mean square error (MSE), (ii) peaksignal-to-noise ratio (PSNR) and (iii) morphologicalcontour preservation index (MCPI).Once the two transformations producing better results are

detected, one of them is selected to be compared with thefollowing operators reported in the current literature: PDEs[27], wavelets [28], morphological connected rank maxopening [29] and connected amoebas [30]. In thecomparison process, a set of natural images [31]contaminated with 10% of Gaussian noise and 5% of saltand pepper noise were used. Conclusions are presented inSection 5.

2 Morphological contrast detectors andopening by reconstruction

2.1 Denitions of some morphologicaltransformations

In mathematical morphology (MM), increasing andidempotent transformations are frequently used.Morphological transformations complying with theseproperties are known as morphological lters [32]. Thebasic morphological lters are the morphological openingB( f )(x) and closing jB( f )(x) using a given structuringelement B. In this paper, a square structuring element isemployed, where B represents the basic structuring elementof size 3 3 pixels, which contains its origin. Whereas B

^

isthe transposed set with respect to its origin,B^ = {x:x [ B}, and is a size parameter. The number ofelements within a structuring element of size is (2 + 1)2.Formally, the morphological opening B( f )(x) is expressedas follows

gmB(f )(x) = dmB^ 1mB(f )( )

(x)

By duality, the closing jB( f )(x) is dened as

wmB(f )(x) = gmB f c( )

(x)[ ]c

where f c(x) = 255 f(x). The morphological erosion isexpressed as 1mB(f )(x) = ^{ f (y):y [ mB

^

x}. Here, is the132This is an open access article published by the IET under the Creative CLicense(http://creativecommons.org/licenses/by/3.0/)inf operator. By duality, the morphological dilation iswritten as B( f )(x) = [B( f

c)(x)]c.

2.2 Opening and closing by reconstruction

Transformations by reconstruction are useful operatorsintroduced in MM. These transformations allow theelimination of undesirable regions without substantiallyaffecting the remaining structures of the image, given thatthese transformations are built by means of geodesictransformations [23]. The geodesic dilation d1f (g)(x) of sizeone is given by d1f (g)(x) = f (x) ^ dB(g)(x) with g(x) f (x),g represents the marker. The reconstruction transformationfor grey-level images is dened as follows

R(f , g)(x) = d1f d1f . . . d1f (g)(x)NameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeNameMeuntil stability

= limn1 d

nf (g)(x)

When the marker g(x) is equal to the erosion of the originalfunction B( f )(x), and the geodesic dilation is iterated, theopening by reconstruction gmB(f ) is obtained. This can bewritten as

gmB(f )(x) = R(f , 1mB(f ))

By duality, the closing by reconstruction wmB(f ) is expressedas

wmB(f )(x) = gmB f c( )

(x)[ ]c

2.3 Top-hat by reconstruction

The white top-hat transformation is used for detecting peaksof certain height and thickness and it is dened as thearithmetical difference between the original and themorphological opening images, that is, TWB( f )(x) = f (x)B( f )(x). The white top-hat by reconstruction TWmB(f ) isbuilt in a similar way, but using the opening byreconstruction transformation. This connectedtransformation is dened as follows

TWmB(f )(x) = f (x) gmB(f )(x)

Connected transformations have the property of preservingthe contours of the processed image.

2.4 Morphological gradients

The morphological gradient is an edge detector. Thistransformation is expressed as

gmmB(f )(x) = dmB(f )(x) 1mB(f )(x)

where f represents the original image, whereas B( f ) andB( f ) represent the morphological dilation and erosion size, respectively. Other expression to obtain the edges of animage is given below. This transformation is called internalgradient

gimB(f )(x) = f (x) 1mB(f )(x) (1)

In this paper, the following expression will be considered,ommons AttributionIET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141

doi: 10.1049/iet-ipr.2012.0615

Once the contrast measure C(x) has been established alongwith PCMm , the following toggle mapping is proposed to lter

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B = , to simplify the notation. Also, when = 1, one hasgm(f )(x) = g(f )(x).

2.5 Methodology introduced in the currentliterature to detect noise

Amethodology to identify noise was presented in [24]. In thisresearch, a local analysis is developed considering a windowW of 3 3 pixels or size = 1; however, it can be extended toother sizes of . By applying a statistical analysis, the authorsdetect noise at point x of the grey-level image f if

P(x) . PC

where

P(x) = C(x)n1x=1 C(x)

is the contrast probability at point x andn1

x=1 C(x) considersthe eight neighbouring points surrounding point x. C(x) is alocal contrast measure

C(x) = f (x) f (x)

f (x)(2)

where f (x) is the background and represents the mean value ofthe pixels surrounding x. Also, it was demonstrated that PCtakes the value

PC =1

8for m = 1

This critical probability represents a mean, and all pixelvalues markedly different from this quantity are consideredas noisy elements.

3 Noise detection

3.1 Proposed method to detect noise

An operator that computes the background in terms of theopening and closing by reconstruction is proposed as follows

bm(f )(x) = wmgm(f )(x) (3)

where b is the background of size , and gm and wm are theopening and closing by reconstruction size , respectively.The background operator in (3) indicates that if the size ofthe structuring element is increased, more maxima andminima will merge. Other expressions for the imagebackground in terms of morphological operators can befound in [33, 34]. By replacing (3) in (2), the new contrastmeasure is rewritten as

Cm(x) =f (x) bm(f )(x)

bm(f )(x)

=Rm(f )(x) bm(f )(x)

(4)

where Rm(f )(x) = f (x) bm(f )(x) is a residue transformation.The parameter PC introduced in [24] can be rewritten asIET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141doi: 10.1049/iet-ipr.2012.0615 This is an open access artnoise

hwgPCMm(f )(x) = f (x), P(x) , PCMm

wg(f )(x), otherwise

{(5)

where w and g are the closing and opening by reconstructionsize 1, and P(x) , PCMm is a noise criterion. The proposed

toggle mapping hwgPCMmnot only lters noise but also

improves the contrast in the output image without

introducing new contours. However, hwgPCMmhas the

disadvantage of eliminating unconnected narrow regions inwhich the structuring element does not t. Notwithstandingthis drawback, there are many images in which (5) can beapplied with good results. Figs. 1bd illustrate theperformance of (3), (4) and (5) using = 5. Image inFig. 1e presents the output image following the procedureexplained in [24], where noisy points are replaced by thoseprovided by the median lter of size = 5, PC =PCMm=5 = (1/120), and the mean lter f also with a sizewindow = 5. Image in Fig. 1e exhibits more smoothnessthan the image in Fig. 1d together with the elimination ofnarrow regions.The output image associated with the operator hwgPCMm

has a

well-dened contrast, which may be explained as follows.

The use of the noise criterion to build (5) allows theclassication of the points in the domain of denition of fin two sets: (i) a set SP(x),PCMm

(f ) composed by the regions

of enhanced contrast, where x [ SP(x),PCMm (f ), P(x) ,PCMm for h

wgPCMm

and (ii) the set ScP(x),PCMm(f ) which is the

complement of SP(x),PCMm(f ), where for all points x

ScP(x),PCMm(f ), P(x) PCMm for h

wgPCMm

. The denition listed

below is a consequence of the noise criterion, P(x) , PCMm ,

used in (5).

Denition 1: A denoised image is said to have a well-denedcontrast if one can classify its points according to a noisecriterion.It is noteworthy to mention that the toggle mappings proposedin this paper full Denition 1. In the next subsection, a newproposal is introduced which is an extension of the one givenin this part.

3.2 Noise elimination based on the top-hattransformation

The white top-hat by reconstruction TWm(f ) allows thedetection of clear components. Hence, this transformation isan excellent operator to detect pixels with high intensitylevels. The criterion to detect noise using the white top-hatPCMm , which considers the size of the structuring element.Then PCMm = 1/ (2m+ 1)(2m+ 1) 1

( )( )with 0. For

example, if = 1, PCMm=1 = (1/8) and for = 2,PCMm=2 = (1/24).The contrast measure in (4) produces a family of ordered

transformations, that is, for m1 m2 mm andCm1 (x) Cm2 (x) Cmm (x).133icle published by the IET under the Creative Commons Attribution

License(http://creativecommons.org/licenses/by/3.0/)

www.ietdl.orgFig. 1 Modied method for noise detectiona Original imageb Background with = 5 by applying (3)c Detected contrast using (4)d Filtered image using (5)e Output image according to the proposal introduced in [24] with = 5by reconstruction can be established by analysing theconsiderations given in [24]. In that study, the criticalprobability to detect noise is 1/8 when = 1. In otherwords, the central pixel whose value is above the mean ofits neighbours is a noisy pixel. This criterion can beextended to one, that is, convenient for detecting noise inthe high intensity levels. Thus, f (x) is not a noisy grey level if

TWm(f )(x) TWm(f )(x)

where TWm(f ) represents the mean white top-hat byreconstruction. This noise criterion allows to detect noiseonly in white components. Alternatively, the complementimage must also be processed to lter the dark componentsusing the same criterion. The following toggle mapping isproposed to suppress noise in the foreground

hwgTm(f )(x) = f (x), TWm(f )(x) TWm(f )(x)

wg(f )(x), otherwise

{(6)

To clean noise in the complement image, (7) should beapplied

kwgTm(f )(x) = hwg

Tm(f c)(x)

( )c(7)

The differences between (5) and (6) are: (a) Equation (6) doesnot use a contrast measure nor a probability criterion; and (b)according to (7), (6) must be applied twice, once in theforeground and once in the complement. On the other hand,(5) is applied only once because the contrast measure

134This is an open access article published by the IET under the Creative CLicense(http://creativecommons.org/licenses/by/3.0/)detects important changes of intensity between the centralpixel and the mean of its neighbours. The noisy centralpixel is corrected by the lter wg(f ), that is, if the centralpixel is a white pixel, the opening by reconstruction comesinto action, otherwise the closing by reconstruction isemployed. The lter wg(f ) modies white and darkcomponents when is applied.

3.3 Other proposals for selecting the central pixel

When noisy pixels are detected, the contrast of the image ismodied depending on the selection of the central pixel thatsubstitutes noise. For example, if the intention is to obtainhigh contrast, a suggested central pixel can be obtainedfrom the morphological erosion. Then (6) is rewritten as

h1Tm (f )(x) =f (x), TWm(f )(x) TWm(f )(x)

1(f )(x), otherwise

{(8)

In Fig. 2, the performance of (8) is illustrated. Images inFigs. 2a, b are smoother than the picture in Fig. 2c. Thesubstitution of the central pixel by the morphologicalerosion produces more intense contours. To corroborate thissituation, compare the images in Figs. 2df. Contours wereobtained by (1). The erosion used as primitive in (8) is veryinteresting, because it suggests that the central pixel cantake any value inside the window W given by thestructuring element B. If the set of pixels in the structuringelement is sorted in ascending way, a rank lter B,k( f ) isobtained [26], in which ,k = 1( f ) = ( f ), where is thesize of the structuring element, and k represents the numberof applied lter. For this research, = 1, hence 1 k 9.

ommons AttributionIET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141

doi: 10.1049/iet-ipr.2012.0615

www.ietdl.orgTo simplify the notation, let us consider that , k( f ) = k( f ).Given that the rank lters are ordered, then the followingproposed contrast mappings

Fig. 2 Erosion used as primitive to produce high contrasta Original imageb Image processed with (6) taking = 1c Image processed with (8) considering = 1df Contours of the images in Figs. 2achrkTm(f )(x) = f (x), TWm(f )(x) TWm(f )(x)

rk(f )(x), otherwise

{(9)

are ordered, that is

hr1Tm(f ) hr2

Tm(f ) hr9

Tm(f )

In this case, the inverted image must also be cleaned toobtain a ltered image. This is expressed as follows

krkTm(f )(x) = hrk

Tmf c( )

(x)

( )c

The lter krkTm(f ) maintains the following order

kr1Tm(f ) kr2

Tm(f ) kr9

Tm(f )

However, it is not convenient to change the central pixel for avalue higher than the one obtained with the median lter,since the proposals to suppress noise in this study considerthat the values of the pixels in a small region given by thesize of the structuring element B are not superior to theirmean value. Therefore, the following transformations fullsuch requirement

kr1Tm(f ), kr2

Tm(f ), kr3

Tm(f ), kr4

Tm(f ) and hr5Tm (f )

IET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141doi: 10.1049/iet-ipr.2012.0615 This is an open access art3.4 Alternative approximation to the mean lter

On the other hand, to eliminate TW (f ) from (9), the meanmtop-hat by reconstruction can be approximated by othermorphological transformation. Consider the followingrelation [26]

wg(f ) f gw(f ) (10)

The order relation in (10) is true for the opening and closingby reconstruction, that is

wg(f ) f gw(f ) (11)

where f is the mean function and wg(f ) and gw are thealternated lters by reconstruction. The next approximationis carried out from (11)

f gw(f )

so that, TWm can be replaced by jTHm (f ) = (gw(TWm(f )) in

(6), (8) and (9). For example, (6) can be rewritten as

hwg

jTHm

(f )(x) = f (x), TWm(f )(x) jTHm (f )(x)wg(f )(x), otherwise

{

To lter noise in the dark components, the next expression

135icle published by the IET under the Creative Commons Attribution

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must be applied

kwg

jTHm

(f )(x) = hwgjTHm

f c( )

(x)

{ }c

3.5 Noise elimination based on the gradienttransformation

The internal gradient dened in (1) can be used asmorphological contrast criterion Cgim to detect noise.Consider that

Cgim (x) = gim(f )(x)

Here, a variation in the method to detect noise similar to thatdened in Section 3.1 is applied, but now the contrastprobability is computed directly on the internal gradient.Let a(x) = (1/2)wg(f )(x)+ (1/2)gw(f )(x) be a primitive. Asimilar primitive without reconstruction transformations wasused in [1]. A new contrast mapping is dened as follows

ha = f (x), PCgim (x) PCMma(x), otherwise

{(12)

where PCgim(x) is the probability computed on Cgim (x).

Equation (12) has the following advantages: (i) it detectsnoise efciently by employing the PCgim

(x) PCMmcriterion; (ii) it is applied only once to lter the image and

(iii) the primitive (x) allows less smoothing than when theopenclose by reconstruction is used as primitive.Nevertheless, the elimination of narrow regions remains anissue. In the case that noise is ltered rst for the whitecomponents and then for the dark ones, the followingtoggle mapping in terms of the internal gradient is useful.This toggle does not use a contrast measure. Let

j gim (f ) = gw(gim(f )) and

hwg(f )(x) = f (x), gim(f )(x) jgim (f )(x)

wg(f )(x), otherwise

{(13)

To lter noise for the dark components, the next expression

must be applied

kwg(f )(x) = hwg f c( )(x)( )cThe performance of (13) is illustrated in Fig. 3.

4 Experimental results

4.1 Detection of the best operators to removenoise

Fig. 4a presents the original image. The original image iscontaminated with two classes of noise: zero meanGaussian with 0.01 variance and 5% of salt and peppernoise. The resulting image is displayed in Fig. 4b. Theimage in Fig. 4b will be used as input image to testdifferent transformations to remove noise. The resultingimages are presented in Figs. 4cl. The indexes, MSE,

www.ietdl.orgFig. 3 Noise elimination based on the gradient transformationa Original imageb Application of (13) with = 1c Inversion of the image in Fig. 3bd Application of (13) to the image in Fig. 3c with = 1e Inversion of the image in Fig. 3d136This is an open access article published by the IET under the Creative CLicense(http://creativecommons.org/licenses/by/3.0/)ommons AttributionIET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141

doi: 10.1049/iet-ipr.2012.0615

Fig. 4 Detection of the best operators to remove noisea Original imageb Noisy imagec Mean lter with = 1d Median lter with = 1e Method proposed in [24] using the median value to substitute noise taken from a window size = 1f Method proposed in [24] using the (x) lter to substitute noise taken from a window size = 1g Application of (13) taking as primitive the median lter size = 1h Application of (13) taking as primitive the (x) lter size = 1i Application of (5) taking as primitive the median lter size = 1j Application of (5) taking as primitive the (x) lter size = 1k Application of (7) taking as primitive the (x) lter size = 1l Application of (7) taking as primitive the median lter size = 1

www.ietdl.org

IET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141doi: 10.1049/iet-ipr.2012.0615

137This is an open access article published by the IET under the Creative Commons Attribution

License(http://creativecommons.org/licenses/by/3.0/)

www.ietdl.orgPSNR and morphological contour preservation will be used todetect the best transformation to suppress noise.

4.1.1 Mean-square error (MSE): This index iscomputed by analysing the global contrast of the outputimages with respect to the original image. The betterdenoised images must full two conditions: high contrast

Fig. 5 Image in Fig. 4b is used as the input image to test different tranThis image has 10% of zero mean Gaussian noise together with 5% of salt and pea Table with the computed contour preservation index, MSE and PSNR valuesb Graph corresponding to MSEc Graph corresponding to PSNRd Graph corresponding to the MCPI

138This is an open access article published by the IET under the Creative CLicense(http://creativecommons.org/licenses/by/3.0/)and highest resemblance to the original image. Equation (4)is extended to measure the contrast as a global measure,which is expressed as follows

xm(f )(x) =1

vol(f )

x[Df

Cm(x), if vol(f )= 0

sformations

pper noise

ommons AttributionIET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141

doi: 10.1049/iet-ipr.2012.0615

literature. This transformation is expressed as follows

hMm (f )(x) = f (x), P(x) , PCMmMm(f )(x), otherwise

{(14)

where M( f )(x) represents the median lter.

Fig. 6 Comparison of transformation to suppress noise

www.ietdl.org

where vol(f ) = x[Df f (x) represents the sum of all greylevels of the original image and is used to normalise thecontrast values, whereas Df represents the denition domainof the original image. The MSE index is obtained asfollows [34]

MSE = xm xmRef( )2

where represents the contrast measure of each processedimage and xmRef is the contrast measure of the original image.

Fig. 5a shows the computed values of MSE for each imagepresented in Fig. 4. The graph of the MSE values is given inFig. 5b. The best values of MSE are those nearest to 0.

4.1.2 Peak signal-to-noise ratio (PSNR): The PSNRvalues can be observed in Fig. 5a, and its correspondinggraph in Fig. 5c. The best PSNR measures are associatedwith the greatest values.

4.1.3 Morphological contour preservation index(MCPI): This index is used as a measure to survey thecontours of the output images with respect to the edges ofthe original image. To obtain the MCPI index, it isnecessary to compute the internal gradient, which wasdened in (1), and the edge preservation parameter (EPP).The EPP is obtained from the convolution (denoted as *) oftwo kernels of size 3 3 with the morphological internalgradient. These kernels allow to detect horizontal, Gh, andvertical, Gv, changes for each point in the internal gradient.EPP is expressed as follows

EPP =

x[Dgradi(f )

Gh(x)+

x[Dgradi(f )

Gv(x)

With

Gh =1 1 1

0 0 0

1 1 1

gradimB(f )

and Gv =1 0 11 0 11 0 1

gradimB(f )

where Dgradi( f ) represents the denition domain of theinternal gradient. The EPP enables us to compute the MCPIas follows

MCPI = EPPnoisyEPPoriginal

where EPPnoisy is obtained from the noisy image andEPPoriginal from the original image. The computed MCPIvalues for images in Fig. 4 are presented in Fig. 5a and thecorresponding graph in Fig. 5d. The best MCPI values arethose nearest to 1.According to the graphs in Figs. 5b, c and d, the images

presenting the best characteristics MCPI nearest to 1, highPSNR values and MSE measure close to 0 are the imagesin Figs. 4i, j and k. In particular, operator in (5) with themedian lter size = 1 as primitive and associated with theimage in Fig. 4i will be used to verify its performance withrespect to other transformations reported in the currentIET Image Process., 2014, Vol. 8, Iss. 3, pp. 131141doi: 10.1049/iet-ipr.2012.0615 This is an open access art(O-a,b,c) Original images(N-a,b,c) Images in Figs. 6(O-a,b,c) contaminated with zero mean Gaussianwith 0.01 variance and 5% of salt and pepper noise(W-a,b,c) Noise elimination using wavelets(P-a,b,c) Noise elimination using PDEs(R-a,b,c) Noise elimination using Rank max connected opening(OP-a,b,c) Noise elimination using our proposal(A-a,b,c) Noise elimination using morphological amoebas139icle published by the IET under the Creative Commons Attribution

License(http://creativecommons.org/licenses/by/3.0/)

4.2 Comparison of different transformations toremove noise

Three images with the codes 66075, 35008 and 65010 wereselected from [31] to compare the performance of (14) withother transformations reported in the literature.The three selected images are presented in Figs. 6O-a, b

and c. The original images are contaminated with twoclasses of noise, zero mean Gaussian with 0.01 varianceand 5% of salt and pepper noise. These images aredisplayed in Figs. 6N-a, b and c and will be used to testour operator. Two transformations, the curvature preservingPDE [27] and the wavelet GSM [28] are applied and theresults are presented in Figs. 6P-a, b, c, W-a, b and c. Forcurvature preserving PDE, the smooth anisotropic functionis applied with the following parameters: amplitude 60,sharpness 0.70, anisotropy 0.26, gradient smoothness 0.63,tensor smoothness 0.29, spatial precision 0.80, angularprecision 30, value precision 2, iterations 1 and tiles1. Whereas the wavelet function is used with defaultparameters dened in the MATLAB code provided freelyby the authors. In Figs. 6R-a, b and c, the morphologicalconnected rank max opening [29] with parameters = 5 andk = 12 is utilised; this operator is an adaptivetransformation. The morphological connected rank max

produce an erroneous selection.

Diego Rodrigo and Daro T.G. for their great

www.ietdl.orgopening has the capacity of eliminating white connectedregions of certain size. In this case structures smaller than12 pixels are suppressed. Images in Figs. 6OP-a, b and care obtained by applying (14) with = 1, whereas those inFigs. 6A-a, b and c employ morphological amoebas [30] tosuppress noise. In this paper, amoebas use a window of size = 5 and = 0.05. Morphological amoebas are alsoadaptive transformations. In order to elucidate which outputimage presents a better behaviour, PSNR and structuralsimilarity (SSIM) indexes [35] were computed. Theseindexes are shown in Tables 1 and 2. In terms of PSNR,our technique outperforms the other algorithms. On theother hand, the SSIM index corresponding to the image inFig. 6P-a indicates that PDEs transformations have a betterperformance; however, notice that salt and pepper noise hasnot been completely eliminated.

Table 1 PSNR for the three analysed images

PSNR

(a) (b) (a)

noisy images (N) 23.49 21.63 23.98wavelets (W) 24.19 21.63 24.31PDEs (P) 25.22 22.33 24.87rank max opening (R) 24.27 21.42 23.57our proposal (OP) 25.84 22.76 25.71amoebas (A) 22.24 22.22 21.47

Table 2 SSIM for the three analysed images

SSIM

(a) (b) (a)

noisy images (N) 0.31 0.23 0.36wavelets (W) 0.35 0.23 0.39PDEs (P) 0.46 0.29 0.44rank max opening (R) 0.40 0.26 0.39our proposal (OP) 0.45 0.30 0.47amoebas (A) 0.32 0.25 0.27140This is an open access article published by the IET under the Creative CLicense(http://creativecommons.org/licenses/by/3.0/)encouragement. This work was funded by the governmentagency CONACyT-Mxico under the grant 133697.

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6 Acknowledgments

The author Ivn R. Terol-Villalobos would like to thank5 Conclusions

At the beginning of this paper the methodology studied in[24] was presented. From this technique, two concepts wereexploited: (i) PC, the critical threshold to detect noise and(ii) the contrast measure. When the parameter PCMm is usedas proximity criterion in the toggle mappings, noise isdetected and ltered in one pass; however, it is necessary tocompute a contrast probability. If toggle mappings use theextended PCMm criteria, then to supress noise it is necessaryto lter the foreground and the inverted image, as typicallyoccurs in MM.When the primitives applied to lter the images are the

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