Signal Processing: Image Communication 17 (2002) 509–524

Adaptive and global optimization methodsfor weighted vector median filters

Laurent Lucata,1, Pierre Siohanb,*,1, Dominique Barbac

aConexant, 2 avenue P. Piffault, 72100 Le Mans, Franceb IRISA-INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France

c IRCCyN, 63, Rue Christian Pauc, La Chantrerie-BP 60601, 44306 Nantes cedex, France

Received 4 December 2001; received in revised form 1 March 2002; accepted 8 March 2002

Abstract

Weighted vector median filters (WVMF) are a powerful tool for the non-linear processing of multi-components

signals. These filters are parametrized by a set of N weights and, in this paper, we propose two optimization techniques

of these weights for colour image processing. The first one is an adaptive optimization of the N � 1 peripheral weights

of the filter mask. The major and more difficult task is to get a mathematical expression for the derivative of the WVMF

output with respect to its weights; two approximations are proposed to measure this filter output sensitivity. The second

optimization technique corresponds to a global optimization of the central weight alone, the value of which is

determined, in a noise reduction context, by an analytical expression depending upon the mask size and the probability

of occurrence of an impulsive noise. Both approaches are evaluated by simulations related to the denoising of textured,

or natural, colour images, in the presence of impulsive noise. Furthermore, as they are complementary, they are also

tested when used together. r 2002 Elsevier Science B.V. All rights reserved.

Keywords: Colour images; Impulsive noise removing; Non-linear filters optimization; Vector filters; Weighted vector median filters

1. Introduction

Non-linear filters using order statistics are usefultools in digital image and video processing.Among them, the median filter is well known forits impulsive noise removal and ability to preserveedges. For a median filter the only tunableparameter being its sliding window, shape or size,

it may be impossible to profile it properly for agiven application. Hence, more flexible filters, suchas the weighted median filters (WMFs) have beenproposed: WMFs are parametrized by a set of N

weights, where N is the number of samplesincluded in the window; a full analysis of WMFscan be found in [21]. A judicious adjustment of theweights allows one, for instance, to reach a betterpreservation of desired patterns. This emphasizesthe need to develop techniques for adequatelychoosing these parameters. A structural constraintapproach [20] or an error gradient back-propaga-tion algorithm [14,12] are efficient ways to adaptthe WMF parameters. As the adaptive WVMFs

*Corresponding author.

E-mail address: pierre.siohan@irisa.fr (P. Siohan).1Dr. Laurent Lucat and Dr. Pierre Siohan performed the

initial part of this work while being at France Telecom R&D, 4,

rue du Clos Courtel, 35512 Cesson-S!evign!e cedex, France.

0923-5965/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 3 - 5 9 6 5 ( 0 2 ) 0 0 0 2 3 - 1

allow to maintain a good compromise betweenimpulse-like noise reduction and details preserva-tion, they are often used for still and moving imageprocessing [7].Another class of non-linear order statistics

filters is the L-filter [4], whose output is definedas a linear combination of the order statistics (i.e.the sorted samples). L-filters are also parametrizedby a set of N weights, which allows the filter toreach various profiles, including the mean or themedian behaviour. An optimization scheme, basedon an error gradient back propagation has alsobeen proposed by Pitas [13].In order to efficiently process multi-component

signals, such as colour images, satellite images, ormotion vector fields in video coders, scalar non-linear filters, such as median, WMF, L-filter, havebeen extended to the multi-variate case, leading tothe vector median filter (VMF), the WVMF [3]and the multi-channel L-filter, respectively. Theseextensions involve specific definitions related to theordering of multi-valued data [5]. Thus, the VMFand WVMF involve a reduced ordering, whilemulti-channel L-filters have been proposed usingeither a reduced [6,11] or the marginal [8] ordering.This reduced ordering can also be based on colourvector angles as fully described in [18].Unlike for the scalar case, there is a lack of

optimization procedures for vector non-linearfilters. Indeed, the underlying mathematical modelis less tractable than in the scalar case and leads toa difficult optimization problem. To the best ofour knowledge, the only proposed optimizationscheme is devoted to multi-channel L-filters, basedon the reduced [11] and marginal [8] ordering,where the numerous NP2 parameters (P denotesthe number of channels) are optimized accordingto a square error minimization, and under straightassumptions.Here we consider the WVMFs, which are

efficient tools in colour image processing. Similarto the scalar WMF, they are parametrized by a setof N weights which have to be appropriatelydetermined in order to reach high filteringperformances. In [19], typical sets of weights areproposed, which allow a preservation of allstationary regions in image sequences. An adap-tive WVMF is proposed in [2] to process motion

vector fields, where the weights are ‘‘judiciously’’adjusted according to measures of the motionassociated to each pixel included in the slidingwindow. An adaptive WVMF is also used in [1] toregularize optic flow estimates. However, no‘‘optimal’’ WVMF has yet been derived.In this paper, we propose two new approaches

to optimize the WVMF weights. The first oneconsists of an adaptive scheme, where the weightsare regularly updated in order to adapt to the localsignal content. The second approach can be seenas a global optimization, where the central weightof the filter mask is chosen so as to satisfy astatistical criterion; the underlying context con-sidered here is the channel-independent impulsivenoise removing in colour images. Finally, bothmethods are combined, providing a powerfulprocessing tool, which is evaluated in the filteringof noise into colour textured images.Our paper is organized as follows. In Section 2,

the adaptive optimization scheme is described; twoversions are proposed, involving different expres-sions of the filter output sensitivity. Performanceevaluation are based on a weight convergence testand on results of colour image denoising. Theglobal optimization is described in Section 3;results of the proposed approach are comparedto those of an empirical minimization of the meanabsolute or quadratic error, conducted on a colourimage, and the enhancement due to the WVMFoptimization is illustrated through a comparisonwith the unoptimized filter results. Both ap-proaches are then combined in Section 4 andconclusions are given in the last section.

1.1. Nomenclature

X ¼ fxigi¼1;y;N : set of input vector samplesincluded in the filter’s window

xpi : pth component of the input

vector xi ð1pppPÞxc: central vector of the filter’s

windowfwigi¼1;y;N : set of scalar weights of the

filter’s windowwk

i : value of the scalar weight wi atiteration k

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524510

sgnvðaÞ ¼ðsgnða1Þ;y;sgnðaPÞÞT:

sign function for a vector a

LMS: least mean squaresmL1E: mean L1 errormL2E: mean L2 errorRGB: red, green, blueSSP: sample selection probabilityVML1: vector median with the L1

normVML2: vector median with the L2

normWVML1: weighted vector median with

the L1 normWVML2: weighted vector median with

the L2 norm

2. Adaptive optimization of WVMF

2.1. WVMF optimization formulation

The WVMF is defined [3] by

y ¼ arg minxjAX

XN

i¼1

wi jjxj � xijjLp; ð1Þ

where X ¼ fxigi¼1;y;N is the set of input vectorsamples included in the window, fwigi¼1;y;N arethe associated weights, Lp is a norm (usually L1 orL2), and y is the filter output. We denote by P thenumber of components of these input and outputvectors, with P being equal to 3 for RGB colourimage processing. When the min in Eq. (1) is notunique, an additional rule is required to choose theWVMF output sample. In this paper, the filtermask is supposed being specified with respect to itsshape and size, these parameters are not optimizedas it may be the case for some scalar filters, as forinstance for the rank order-based filter (ROBF)in [16].The weights of the optimal WVMF are derived

from the minimization of a cost function, whichcorresponds to the ML1E or ML2E criteria. Thelatter [19] are respective extensions to vector dataof the well-known minimum absolute error (MAE)and minimum square error (MSE). That means,denoting by d the desired output, which may

be represented by a set of M noise-free vectorsd ¼ fdmgm¼1;y;M ; and by fymgm¼1;y;M the set ofcorresponding estimates at the filter output, themean L1 error and mean L2 error are given byML1E ¼ ð1=MÞ

PMm¼1 jjdm � ymjjL1

and ML2E ¼ð1=MÞ

PMm¼1 jjdm � ymjj

2L2; respectively. The opti-

mization is conducted independently for eachweight wi and is based on a gradient back-propagation algorithm [10]. Considering theML2E criterion and using the stochastic approx-imation which leads to an LMS-like algorithm(least mean squares), it can be summarized by

wkþ1i ¼ wk

i þ 2mðd � yÞ@y

@wki

; 1pipN ; ð2Þ

where m is the adaptation step, and wki denotes the

value of the weight wi at the kth iteration. Whenthe ML1E criterion is used, this leads to

wkþ1i ¼ wk

i þ m sgnvðd � yÞ@y

@wki

; 1pipN ; ð3Þ

where sgnvðaÞ ¼ ðsgnða1Þ;y; sgnðaPÞÞT; sgn denot-ing the sign function and a ¼ ða1;y; aPÞT:Thus, the central problem in both cases is

finding, at each iteration k; a mathematicalexpression for the derivatives @y=@wk

i ; 1pipN:When the WVMF output is uniquely defined (i.e.

there is a single sample corresponding to the min inEq. (1)), it can be proved that the filter output isinvariant to an infinitesimal variation of its weights.Hence, the mathematical derivative @y=@wk

i is nullfor each i; therefore it does not provide relevantinformation on the filter output sensitivity.It is worth mentioning that the approach based

on the implicit function theorem, which has beenused in the scalar case [16,17], is no longer valid inthis vector case. In order to get suitable expressionsfor the derivatives, two solutions are proposed,whose validity is illustrated afterwards by aconvergence test and filtering results on colourimages. Note also that, as the optimization iscarried out independently for each weight, and asthe computation principle is the same at eachiteration, in order to alleviate the notation the lowerand upper indexes i and k; respectively, are omittedin the description of the two solutions we propose.

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524 511

2.2. First solution: local approximation for @y=@w

In order to get a suitable expression, weconsider, at first, the derivative as a local

approximation of the output increasing rate,leading to the following formulation:

@y

@w

� �s1

¼ limjdwj-0dya0

dy

dw; ð4Þ

where the subscript s1 corresponds to solution 1.Let us look for the smallest increment dw of theweight w such that dy does not become null.Hence, the output of the WVMF switches from thesample xj0 to a sample xj1 due to a weightincrement equal to dw j1 : Using these notations,the derivative approximation can be written as

@y

@w

� �s1

¼xj1 � xj0

dw j1: ð5Þ

The problem is now to find the index j1 and theweight increment dw j1 : Let us define f ðxjÞ ¼PN

i¼1 wijjxj � xijj: The WVMF expression inEq. (1) can be rewritten as y ¼ arg minxjAX f ðxjÞ:The partial derivative of f ðxjÞ with respect to w is

@f ðxjÞ@w

¼ jjxj � xjj: ð6Þ

Denoting by f 0ðxjÞ the aggregate weighted distancef ðxjÞ after a weight incrementation dw; we get

f 0ðxjÞ ¼ f ðxjÞ þ dwjjxj � xjj: ð7Þ

Hence, it can be seen that ff 0ðxjÞgj¼1;y;N corre-sponds to a set of affine functions of the weightincrement dw: In Fig. 1, these affine functions arerepresented through the set of N straight linesfðDjÞgj¼1;y;N :The intersection points of each ðDjÞ with ðDj0Þ

are the key issue of the problem; they arecomputed for 1pjpN (and for each weight wi)through

f 0ðxjÞ ¼ f 0ðxj0 Þ

3 dw j ¼f ðxjÞ � f ðxj0Þ

jjxj0 � xjj � jjxj � xjj: ð8Þ

Thus, we know each minimal weight incrementdw j of the weight w implying f ðxjÞ becomes lowerthan f ðxj0Þ: at this point, the straight line ðDjÞbecomes under ðDj0 Þ: Hence, dw j1 ¼ minj jdw j j

corresponds to the minimal weight increment(or decrement) of w which allows the WVMFoutput to switch from the sample xj0 to anothersample, namely xj1 ; corresponding to the local

sensitivity. In the example presented in Fig. 1,j0 ¼ 4 and j1 ¼ 2:

2.3. Second solution: formulation of @y=@w based

on a mean sensitivity

The objective of this alternative is to estimate amore global behaviour of the filter when a weight ismodified. This global analysis is motivated by thefact that a gradient approach, such as the onestudied in Section 2.2, is confined to the search of alocal minimum of the cost function; however,a local variation of the WVMF output due to asmall weight increment, or decrement, may not berepresentative of the filter behaviour for largerfluctuations. The alternative solution principle isto take into account the increments dw j ; jaj0; ofthe weight w which are necessary to make theWVMF output switch from the sample xj0 to each

sample xjaj0 ; when these switches are possible.These required weight increments can be seenas critical values in the filter output evolution,in the way that each one indicates a new direction(in the vector component space) of the WVMF

(D )5

(D )2

(D )4 j o(D )=

(D )1

(D )3 = (D )i

δwi

δw1i

δw3i

δw2i

δw5i

jf ’(x )

0

Fig. 1. Evolution of the aggregate weighted distances according

to the weight increment wi ð1pipNÞ:

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524512

output variation. Then, the global behaviour maybe illustrated by the mean WVMF output increas-

ing rate, computed from all critical points. Thisleads to the following expression of the filteroutput derivative:

dy

dw

� �s2

¼1

cardðSÞ

XxjAS

xj � xj0

dw j; ð9Þ

where S denotes the whole set of potential outputvectors involved when increasing or decreasing theweight w; and the subscript s2 corresponds tosolution 2. This second solution is expected to beless reactive than the first one, but the weightsevolution may be more regular because of themean behaviour.The problem is now to find the location of the

critical points. These points correspond to eachweight increment implying that a given f ðxjÞbecomes lower than all other f ðxiÞ (the straightline (Dj) is then under the other lines (Di)).This leads to the search of a convex curve, such

as the one illustrated in Fig. 2.The intersection points are iteratively computed;

this search starts with the straight line ðDj0Þ and isindependently conducted for negative versus posi-tive weight increments. At each step, the computa-tion of the intersections between the current

reference line (Djn) and the others ðDjÞ is obtainedin the same way as in Section 2.2, using Eq. (8)where the index j0 is then replaced with the currentindex jn: In the example of Fig. 2, the meanWVMF output increasing rate is then given by

dy

dw

� �s2

¼1

4

X4X5���!dw5

þX4X2���!dw2

þX4X6���!dw6

þX4X3���!dw3

!;

where XiXj��!

¼ OXj��!

� OXi��!

¼ xj � xi: As a matter ofcomparison, the solution of Section 2.2 wouldresult in ðdy=dwÞs1 ¼ X4X2

���!=dw2; dw2 being the

smallest increment of the weight w implying aWVMF output switch.Note that, in order to maintain the curve

convexity, when a multi-intersection occurs, suchas the point I4;5 in Fig. 2, the new reference line isthe one having the larger slope for negative dw andthe lower one for positive dw:

2.4. Parameter convergence test

The aim of the test is to study the evolution ofthe filter’s parameters. However, as the proposedoptimization aims to regularly adapt to any givensignal content, in general, no convergence to agiven set of weights can be observed. Nevertheless,if for a given input signal such an optimal set ofweights exists, it is of great interest to assesswhether the parameters can converge to desiredvalues, using the proposed adaptation schemes.Thus, we consider the test illustrated in Fig. 3.The input signal consists of a one-dimensional

(1D) fragment (70 pixels) extracted from a texturedcolour image, and periodically duplicated. Theadaptive and the target filters are 1D WVMFdefined using the L1 norm as distance metric andsuch that N ¼ 5: The set of target weights isf1; 2; 3; 4; 5g and the optimization of the adativeWVMF is based on solution 1, with a step m; inEq. (2), set to 10�7: In Fig. 4, we give the results ofthe convergence test. It can be observed that theweights converge to values close to the normalizedtarget weights. At the convergence, which isreached in this example after 620 iterations, theoutput y is equal to the desired output d:Other simulations with different sets of target

weights, as well as using solution 2, lead to a

(D )2

(D )4 jo(D )=

δwi

(D )7

(D )1

δw2i

δw5i

δwi6 δw

i3

(D )5

(D )3 = (D )i

4,2I

4,5I

(D )6

0

II

2,6

6,3

jf ’(x )

Fig. 2. Iterative search of the intersection points for the second

solution and for the weight wi ð1pipNÞ:

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524 513

weight convergence, with again y ¼ d; for lengthpattern of up to 70 pixels. This ‘‘positive’’ testillustrates the validity of our optimization ap-proach.

2.5. Application to colour image filtering

In this section, we evaluate the performance ofthe adaptive WVMF for removing impulsive noise.The input image is a textured image embedded in aR–G–B channel-independent impulsive noise. Theobjective of the filtering is to remove the impulseswhile preserving the noise-free patterns; in thisway, the WVMF weights should adapt to thesepatterns. In Fig. 5, we present illustrations oforiginal noisy (channel-independent impulsivenoise of probability p ¼ 0:2) images and therespective results of the adaptive WVMF andVMF,2 using a 5� 5 filter mask, the L1 norm and

optimization solution number 2. This exampleemphasizes the benefit of using adaptive weightsfor a better preservation of fine details.Other simulations, using solution 1 and the L2

norm as distance metric have also shown thesignificant improvements due to our adaptiveoptimization. According to our simulations, andas shown with the ML2E measures presented inTable 1, the WVML2 filter seems to work betterwith solution 1 while the WVML1 filter performsbetter using solution 2.In the present case of impulsive noise, the

optimization can be performed without anyreference to the noise-free image; the corruptedimage is then considered as the reference. Furthersimulations have shown that the resulting degra-dation of the performances is weak, with regard tothe gain reached with the adaptive filtering. This isdue to the fact that, at least up to a probability ofthe impulse noise such that p ¼ 0:2; the ‘‘cor-rupted’’ adaptation steps where the reference datacorrespond to impulses are not correlated amongthem, and thus do not influence the globalevolution of the weights [9].Finally, we have to note that the central pixel of

the mask, denoted wc; has been ignored here ðwc ¼0Þ; because a positive wc quickly involves a highvalue of the central weight leading to an identity

filter. Therefore, an appropriate specification of wc

is expected to further improve the filteringperformances.

3. Global optimization

3.1. Global approaches

In this section, rather than adaptively optimiz-ing the parameters of the WVMF filter in order tobe locally adapted to the signal, we look for fixedparameters in adequation to some global constantfeatures of the signal. The proposed approach is astatistical one. It takes into account the fact thatthe weight values, wj ; naturally have a strongimpact on the filter output y; and, consequently,on the probabilities Pr½y ¼ xj ; usually denotedSSPðxjÞ (sample selection probability) in the scalarcase. This terminology can be straightforwardly

0 100 200 300 400 500 600 700 800 900 10000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Weight 1

Weight 2

Weight 3

Weight 4

Weight 5

iterations

norm

aliz

ed w

eigh

ts

Fig. 4. Evolution of the weights.

WVMFwith adaptive

weights outputsignal

WVMF

weightswith target

inputsignal

+ -d

yx

Fig. 3. Schematic illustration of the convergence test.

2Which is equivalent to an unoptimized WVMF whose

weights are equal to 1.

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524514

extended to the vector case. If the desired SSPvalues are known, and if a mathematical formula-tion, or a conversion algorithm, is available toderive the weights from the SSPs, then we are able

to express, or compute, the optimal weights. Suchan approach has been used in the scalar case, forinstance to determine the parameters of the WMFand weighted order statistics filter (WOSF) [15]. Itinvolves a direct account of the SSP, the desiredimage being assumed to be known.However, even if a similar approach could be

investigated with WVMF its complexity wouldprobably be very high. In the following, we preferto focus on a particular case related to thestatistical optimization of the central weight, wc;according to the SSP of the central sample xc: Thefirst and straightforward advantage is to have asingle weight to optimize and a second one is thatan estimation of the SSP is no longer required,which also means that a priori knowledge of thedesired image is not necessary.

(a) Noisy images ( p = 0.2) (b) VMF 5×5 (c) Adaptive WVML1 5×5

Fig. 5. Illustration of the benefit due to the WVML1 adaptation (solution 2) for texture 1 (at the top), texture 2 (in the middle) and

texture 3 (at the bottom).

Table 1

ML2E measures for the VMF and the optimized adaptive

WVMF defined with L1 or L2

Norm for VMF WVMF WVMF

the filters solution 1 solution 2

Texture 1 L1 1123 995 865

L2 1162 880 923

Texture 2 L1 2229 2153 1906

L2 2268 1866 1956

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524 515

3.2. Statistical optimization of the central weight

3.2.1. General formulation

In the context of impulsive noise reduction forcolour images, our goal is to find an optimal valuefor the central weight of the WVMF, wc; or for itsnormalized version, wn

c ¼ wc=PN

i¼1 wi: This com-putation naturally takes into account all the otherweights included in the filter mask.Indeed, in the case of an impulsive noise, the

central sample xc plays a particularly importantrole: if it is not affected by an impulse, it isdesirable that the filter output takes its value,therefore it is then required that the central weightwc be high. On the contrary, when the central pixelis affected by an impulse, it is better if the centralweight is small.Assuming no impulse detection is carried

out, which is the case in our study, we have noa priori knowledge to tell us if the target pixelis noisy or not. In fact the only information,which is assumed to be known, is statistical:it is the probability of occurrence of noise, denotedby p: The ‘‘optimal’’ choice for wc has to bebased on the input data and, especially, on theparameter p:Thus, this problem is directly related to

the computation of SSPðxcÞ: We have to findthe adequate relation between SSPðxcÞ and p;such as

SSPðxcÞ ¼ probability that xc be not noisy: ð10Þ

Otherwise said, the optimization has to find thevalue of wc providing an SSPðxcÞ such that Eq. (10)is satisfied. It is assumed that the impulsive noise isscalar and independent for each one of the threecomponents of the colour images. The probabilitythat the kth component is not noisy is ð1� pÞ:Therefore, the vector xc will not be noisy if eachone of the component is not; the probability of thisevent is ð1� pÞ3: The second and harder remainingproblem consists of finding the left member inEq. (10), i.e. the probability P ¼ SSPðxcÞ; as afunction of the weights.

3.2.2. Computation of the SSPðxcÞIn a probabilistic framework the input samples

fxigi¼1;y;N are realizations of a random variable

X : The weights associated to them are determinis-tic parameters. The SSPðxcÞ can then be formu-lated as

P ¼ Pr½xc ¼ WVMFðfxi;wigi¼1;y;NÞ :

Taking into account the definition of theWVMF, see Eq. (1), it can be rewritten as

P ¼Pr xc ¼ arg minxj

XN

i¼1

wijjxj � xijj

" #

¼PrXN

i¼1

wijjxc � xi jjoXN

i¼1

wi jjxj � xi jj; 8jac

" #:

If we want to be able to go beyond this first step,simplifying assumptions cannot be avoided.The samples fxjgj¼1;y;N are supposed to be

independent, although some strong relations mayexist between the samples included in the mask.Furthermore, we also assume that the measuresfPN

i¼1 wijjxj � xijjgj¼1;y;N are independent and wewill see later on that, even if such a strongassumption is required, it nevertheless leads touseful results. According to these assumptions, wecan express P as follows:

P ¼YN

j¼1;jac

PrXN

i¼1

wi jjxc � xi jjoXN

i¼1

wi jjxj � xi jj

" #:

For xjaxc; P can be written as

P ¼YN

j¼1ðjacÞ

Pr

PNi¼1ðiaj;cÞ wiðjjxc � xijj � jjxj � xi jjÞ

jjxc � xj jj

"

owc � wj

#: ð11Þ

The problem is then to find the probability

density function (pdf) of the random variable Z

given by

Z ¼

PNi¼1ðiaj;cÞ wiðjjxc � xi jj � jjxj � xi jjÞ

jjxc � xj jj:

The derivation of this density function can bedecomposed in four steps:

1. Find the pdf of jjxj � xi jj: This pdf naturallydepends on the pdf of the samples but is alsorelated to the norm under consideration. Thesubstraction ðxj � xiÞ and the norm computa-tion imply convolutions of the pdf of the

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524516

random variable X : Consequently, the pdffor jjxj � xijj is close to a Gaussian distribution.Later on we will assume that jjxj � xi jj hasa Gaussian distribution Nðmn;snÞ; wherethe subscript n is related to the selected norm.The validity of this assumption, justifiedin Appendix A, can be visually checked inFig. 6.

2. Deduce the pdf of jjxc � xijj � jjxj � xi jj: Thesamples being assumed independent we haveto consider the substraction of two randomvariables (X1 � X2), where X1 and X2

admit pdfs given by N1ðmn;snÞ andN2ðmn;snÞ; respectively. ðX1 � X2Þ is thereforeanother random variable with a density func-tion given by the convolution of N1 and N2;which leads again to a normal distribution:N3ð0;

ffiffiffi2

psnÞ:

3. Calculate the pdf ofP

wiðjjxc � xi jj � jjxj �xi jjÞ: Each element in the summation corre-sponds to a random variable with distributionNið0;

ffiffiffi2

pwisnÞ: To obtain the pdf of the sum we

have to convolve all Ni distributions. This‘‘multiconvolution’’ again leads to a normal

distribution: Nð0;ffiffiffi2

psn

ffiffiffiffiffiffiffiffiffiffiffiffiffiPi w2

i

qÞ:

4. Find the pdf of the quotient.

Denoting W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiP

i w2i

qand K 0

n ¼ mn=ffiffiffi2

psn; it

can be shown [9] that the pdf is

qðzÞ ¼exp� K

02n 1�

1

1þ z2=2W 2

� �ffiffiffi2

ppW

K 0n

ffiffiffip

p1þ

z2

2W 2

� ��3=2"

� erf K 0n 1þ

z2

2W 2

� ��1=2 !

þ 1þz2

2W 2

� ��1

� exp �K02n 1þ

z2

2W 2

� ��1 !#

: ð12Þ

The general form of the probability expressed inEq. (11) is such as Pr½Zowc � wj :Denoting by QðzÞ the cumulative distribution

function (cdf) of the random variable Z; wefind that the corresponding probability isQðwc � wjÞ:To find Q we have to integrate qðzÞ: But beforeproceeding to this numerical integration wehave to determine K 0

n and, therefore,the parameters mn and sn corresponding tothe Gaussian-like distribution of jjxj � xijj;these equivalent parameters are also given inAppendix A.

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

3

3.5

4x 10 _3

0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7x 10 _3

(a) L1 Norm (b) L2 Norm

Fig. 6. Theoretical distribution (solid line) and ‘‘equivalent Gaussian distribution’’ (dotted line) for jjxj � xi jj:

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524 517

3.2.3. Central weight of the WVMF and probability

of occurrence of the impulsive noise

As indicated in Section 3.2.1 we are looking fora single parameter, the central weight, beingoptimal and determined once for all with respectto global characteristics of the signal, moreprecisely according to the occurrence probabilityof the impulses in the input image to be filtered.On the other hand, taking into account theexplanations given in Section 3.2.2, Eq. (11) canbe rewritten as

P ¼ ð1� pÞ3 ¼YN

j¼1ðjacÞ

Qðwc � wjÞ: ð13Þ

The computation of P is therefore dependent onthe values of the peripheral weights wj ðjacÞ: Inorder to simplify the use of the WVMF filter, wepropose computing the function Q for a set ofstandard weights fwjacg: For example, we imposethat all the weights be equal to 1. Nevertheless, as

qðzÞ depends on W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1ðiaj;cÞ w2i

q; and there-

fore on N; a computation of QðzÞ has to be donefor each desired value of N: With this standard setof weights, qðzÞ and QðzÞ are independent from j;so we can rewrite Eq. (13) as

P ¼ ð1� pÞ3 ¼ ðQðwc � 1ÞÞN�1:

This formulation allows us to express the centralweight value, wc; or the normalized central weightwn

c (the unitary normalization consists ofPNj¼1 wn

j ¼ 1), as a function of the probabilityoccurrence of the impulses

wc ¼ 1þ Q�1ðð1� pÞ3=ðN�1ÞÞ

or wn

c ¼1þ Q�1ðð1� pÞ3=ðN�1ÞÞ

N þ Q�1ðð1� pÞ3=ðN�1ÞÞ; ð14Þ

where the function Q simply depends on K 0n

(according to the WVMF norm) and N:In order to evaluate the relevance of the

proposed method for the determination of theWVMF central weight, simulations have beencarried out. For that we used a textured imageand distorted it with an impulsive independentnoise on each component R, G and B; theprobability of occurrence of the impulses were p ¼0:05; 0:1; 0:2 and 0:5; respectively; for each value of

p different values of the normalized central weighthave been tested and the ML1E and ML2E errorshave been computed. These simulations allowed usto find the optimal empirical weights with anaccuracy of 10�2: For each value of p underconsideration we have compared these optimalempirical weights with the ones given by ourmethod (cf. Eq. (14)) for masks with sizes 3� 3;5� 5; 7� 7 and for the two norms L1 and L2 ofthe WVMF filter.The results are synthesized in Fig. 7.As in the previous section the filter size and

shape are assumed to be known. Naturally, if thissize is increased too much, we get a loss in fineimage details as well as higher computation time.Based on our experiments [9], it seems that a sizeof 5� 5 leads to a good tradeoff.These simulations show that the proposed

technique allows a good estimation of the centralweight, taking into account the probability ofoccurrence of the impulses. Nevertheless, it is alsoclear that our method is less efficient whenp-0 ðpo0:05 or 0:1). Indeed our model doesnot take into account a typical feature of theWVMF, which is that a normalized weight wn

i ; sothat wn

i X0:5; implies that the output is necessarilyxi: Therefore, the optimal empirical weightsconverge towards 0.5 while the curve becomesclose to 1. It is worth noting that in such asituation one has rather to proceed in two steps inorder to get rid of the impulses: detecting them in afirst step and filtering them in the second one.

3.3. Performances of the WVMF filter with

optimized central weight

We now want to evaluate the improvement ofthe quality of the filtered image using a WVMFwith a central weight optimized according to thetechnique proposed in Section 3.2. All the periph-eral weights are supposed to be fixed and all equal.Our test images are distorted by an impulsive-independent noise with probability p ¼ 0:2: Opti-mizing the central weight, as explained above,provides a very significant gain with regard to thefiltering performances, for both norms L1 and L2:In Fig. 8 we can see that, for a texture, incomparison with a VML1; the WVML1 with

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524518

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of the impulsive noise (for each component)

Nor

mal

ized

cen

tral

wei

ght

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of the impulsive noise (for each component)

Nor

mal

ized

cen

tral

wei

ght

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of the impulsive noise (on each component)

Nor

mal

ized

cen

tral

wei

ght

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of the impulsive noise (on each component)

Nor

mal

ized

cen

tral

wei

ght

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of the impulsive noise (on each component)

Nor

mal

ized

cen

tral

wei

ght

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of the impulsive noise (on each component)

Nor

mal

ized

cen

tral

wei

ght

(a) WVML1 3×3. (d) WVML2 3×3.

(b) WVML1 5×5. (e) WVML2 5×5.

(f) WVML2 7×7.(c) WVML1 7×7.

Fig. 7. Normalized central weight for our method (solid line: -), or empirically optimized in order to minimize the ML1EðþÞ andML2E (o) criteria, for the WVML1 [a,b,c] and WVML2 [d,e,f], respectively.

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524 519

identical peripheral weights and optimized centralweight leads to a clearly better compromise interms of noise reduction/detail preservation. Thecorresponding ML2E measures are reported inTable 2. In Fig. 9 another illustration is given,with the image ‘‘Boat’’, showing that a significant

improvement can also be obtained for naturalimages.

4. Adaptive and global optimization of the WVMF

The two optimization schemes presented in thetwo previous sections may be complementary. Theadaptive technique presented in Section 2 consistsof a local optimization of the peripheral weightswhile, on the contrary, the global statisticaloptimization in Section 3 only deals with thecentral weight. In the latter section, we have seenthat the computation of the function QðzÞ was onlycarried out for a standard set of identical andconstant peripheral weights. However, if we have

(a) VML1 5×5. (b) WVML1 5×5.

Fig. 8. Comparison of the VML1 and WVML1 with optimized central weight and equal peripheral weights.

Table 2

ML2E errors for the VML1 and WVML1 with optimized

central weight and identical peripheral weights (mask sizes

5� 5); input noisy images with p ¼ 0:2

Filtered

image

VML1 WVML1

Texture 1 1123 595

Texture 2 2229 1167

Texture 3 1639 864

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524520

no a priori knowledge of these weights, it is, intheory, necessary to achieve the numerical integra-tion of qðzÞ for each value of wjac; and, therefore,for each new filtering operation if these weightsare not constant. Even if it is possible, that wouldbe very time consuming from a computationalpoint of view. To combine adaptive and globaloptimizations it is in fact simpler, and just asefficient as the results will show, to use a uniquefunction qðzÞ for given N and p: Therefore, themain difference with the adaptive method pre-sented in Section 2 is that instead of being equal to0 the central weight is fixed at a value which is theresult of a global optimization. Then, at eachiteration of the adaptive algorithm, the peripheralweights are locally optimized, and normalizedafterwards to take account of the central weightvalue.The simulation results show a very significant

improvement concerning the quality of the filteredimages for the two adaptive methods described inSection 2 and for the two types of vector filtersWVML1 and WVML2: It also appears that forthe two optimization methods of the peripheralweights, the WVML1 provides better results, interms of visual quality and ML2E measures,than the WVML2; even if the noisy image is usedas a reference to compute the error at the filteroutput.For the overall set of possible situations, in

which the central weight is optimally computed,the WVML1 filter provides the better results, withsolution 1 if the reference image is noisy, andotherwise with solution 2. In Fig. 10 we illustrate,with the WVML1 and using solution 2, the interestof optimally specifying the central weight. Indeed,the optimal central weight leads to a picturehaving a clearly better visual quality. In Table 3we have reported the whole set of results for thecase of non-noisy reference. These numericalresults, firstly, show the significant advantageof optimized WVMF over VMF, the signal tonoise improvement being of, at least, 2:4 dB: Onthe other hand, it also appears that there aresignificant differences, up to nearly 1 dB; accord-ing to the filter, WVML1 performing better thanWVML2; and to the adaptive optimization solu-tion, 1 being better than 2, used.

Original noisy "Boat" image ( p = 0.2).

Image boat filtered by a 5×5 VMF.

Image boat filtered by a 5×5 WVML1 withoptimal central weight.

Fig. 9. Comparison of the VMF and of the WVMF with

optimal central weight for impulsive noise filtering.

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524 521

5. Conclusion

If various optimization techniques are availablefor non-linear scalar filters, it is not the case fortheir vector counterpart. In this paper, wehave proposed two optimization techniques tocompute the weights of the WVMF. The firsttechnique provides an optimization of the WVMFperipheral weights, which takes into accountthe non-stationary behaviour of the signal. Two

variants of an adaptive algorithm minimizing theML2E criterion have been tested for impulse noisereduction. The results obtained with texturedimages have shown that the noise could bestrongly reduced while keeping a good preserva-tion of fine details, even when using a noisyreference. According to our simulations WVML2filters gave better results for the first variant of ouradaptive algorithm (solution 1) and WVML1filters performed better when using solution 2.

(a) Central weight equal to 0 (b) Optimized central weight

Fig. 10. Comparison between WVML1 (solution 2, mask size 5� 5; noise free reference) with and without central weight.

Table 3

Comparison of the WVML1 and WVML2 with globally optimized central weight and peripheral weight optimized according to

solution 1 or solution 2 (mask size 5� 5; input noisy images with p ¼ 0:2; noise-free references)

Filtered image WVML1 WVML2

Solution 1 Texture 1 ML2E ¼ 570 ML2E ¼ 669

SNRi=VML1¼ 2:94 dB SNRi=VML2

¼ 2:40 dBTexture 2 ML2E ¼ 1127 ML2E ¼ 1248

SNRi=VML1¼ 2:96 dB SNRi=VML2

¼ 2:59 dBSolution 2 Texture 1 ML2E ¼ 517 ML2E ¼ 616

SNRi=VML1¼ 3:37 dB SNRi=VML2

¼ 2:76 dBTexture 2 ML2E ¼ 1042 ML2E ¼ 1166

SNRi=VML1¼ 3:30 dB SNRi=VML2

¼ 2:89 dB

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524522

For this first set of optimization techniques thefilter central weight was assumed to be zero. In thesecond approach our purpose was to achieve aglobal optimization of the central weight, all theperipheral weights being fixed and equal; theoptimization issue being to remove as much aspossible an additive impulsive noise affecting thecolour images. The principle of our statisticaloptimization consisted of constraining the prob-ability that the filter output corresponding to thecentral sample of the filter mask be equal to theprobability that this sample be not distorted by animpulse. Using some simplifying assumptions, wewere able to provide an analytical expression ofthe optimal central weight as a function of theprobability of occurrence of the noise. Ouranalytical approach has been validated by manysimulation results, showing also the efficiency ofthe method for denoising colour images, texturesand natural images as well.These two techniques, the local and the global

ones, can be combined together and we finallyshowed that their respective benefits can be addedto each other.

Appendix A. Parameters of the equivalent Gaussian

distributions

First of all, let us examine the case of the L1

norm. It can be expressed as

jjxj � xi jj ¼XP

k¼1

jxkj � xk

i j;

where xki denotes the kth component of the vector

xi and P the number of components. In thefollowing, we focus on colour image processingand therefore we set P ¼ 3:We assume that xk

i is uniformly distributed inthe range ½0; 255 for images coded using 8 bits percomponent and, therefore, 24 bits for colourimages. As the original image is distorted by auniformly distributed impulse noise, this assump-tion becomes still more valid.Afterwards, to determine the parameters of a

Gaussian distribution being as close as possibleto the density function, jjxj � xi jjL1

; we use the

logarithmic version of the maximum likelihoodcriterion.Let us consider a collection of samples denoted

fuigi¼1;y;N ; and let us assume that this collection ischaracterized by a Gaussian distribution. Theprobability that its parametrisation be given byðm;sÞ is characterized by the ‘‘log-likelihood’’

log Lðm; sÞ ¼ � N logðffiffiffiffiffiffi2p

psÞ

�1

2s2XN

i¼1

ðui � mÞ2:

After a discretization of the continuous dis-tribution jjxj � xi jjL1

; we get a new collectionfuigi¼1;y;N containing nðkÞ samples equal to k;the nðkÞ being proportional to the density function.The ‘‘log-likelihood’’ can therefore be rewritten as

log Lðm; sÞ

¼ � N logðffiffiffiffiffiffi2p

psÞ �

1

2s2X

k

nðkÞðk � mÞ2;

withX

k

nðkÞ ¼ N:

Then it can be shown that the maximumlikelihood is reached for the following set ofparameters:

m ¼P

k knðkÞN

and s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPk k2nðkÞ

N� m2

s:

Let m1 and s1 denote their correspondingnumerical values, respectively, we find m1 ¼255:3 and s1 ¼ 103:8; which leads to K 0

1 ¼ 1:7:Let us now consider the case of the L2 norm

jjxj � xijj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXP

k¼1ðjxk

j � xki jÞ

2

r:

Again we note that the resulting distribution forjjxj � xijjL2

is close to a Gaussian distribution, eventhough our initial assumption is one of a scalaruniform distribution for fxk

i g: jjxj � xi jjL2will

therefore be supposed to follow a Gaussiandistribution and its parameters will be computedaccording to the maximum likelihood criterion.The parameters found for the L2 norm are m2 ¼

169:3 and s2 ¼ 63:4; leading to K 02 ¼ 1:9:

In Fig. 6, we show the ‘‘equivalent Gaussiandistributions’’ as defined previously in the case of

L. Lucat et al. / Signal Processing: Image Communication 17 (2002) 509–524 523

the L1 and L2 norms. It is also worth noting thatif, instead of a uniform distribution for xk

i ; we hadchosen any ‘‘bell-shaped’’ distribution the finalresult would be still closer to the Gaussian.

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