EdgeDetectioninColorImagesBasedonDSmTJeanDezertZhun-gaLiuGrgoireMercierAbstractIn this paper, we present a non-supervised method-ology for edge detection in color images based on belief functionsand their combination. Our algorithm is based on the fusion oflocal edge detectors results expressed into basic belief assignmentsthanks to a exible modeling, and the proportional conict redis-tributionruledevelopedinDSmTframework. Theapplicationof this new belief-based edge detector is tested both on original(noise-free) Lenaspictureandonamodiedimageincludingarticial pixel noises toshowthe abilityof our algorithmtowork on noisy images too.Keywords: Edge detection, image processing, DSmT, DST,fusion, belief functions.I. INTRODUCTIONEdgedetectionis oneof most important tasks inimageprocessinganditsapplicationtocolorimagesisstillsubjecttoaverystronginterest [8], [10][12], [14]forexampleinteledetection, inremotesensing, target recognition, medicaldiagnosis, computer visionandrobotics, etc. Most of basicimageprocessingalgorithmsdevelopedinthepast forgray-scale images have been extended to multichannel images. Edgedetection algorithms for color images have been classied intothreemainfamilies[15]: 1)fusionmethods, 2)multidimen-sional gradient methods and 3) vector methods depending onthepositionofwheretherecombinationstepapplies[7]. Inthis paper, the method we propose uses a fusion method withamultidimensional gradient method. Our newunsupervisededge detector combines the results obtained by gray-scaleedge detectors for individual color channels [3] to denebbas fromthe gradient values which are combined usingDezert-Smarandache Theory[17] (DSmT) of plausible andparadoxical reasoning for information fusion. DSmT has beenprovedtobeaseriousalternativetowell-knownDempster-Shafer Theoryof mathematical evidence [16] speciallyfordealingwithhighlyconictingsources of evidences. Somesupervised edge detectors based on belief functions computedfromgaussianpdfassumptionsandDempster-ShaferTheorycan be found in [1], [21]. In this work, we show through verysimple examples how edge detection can be performed basedon DSmTfusion techniques with belief functions withoutlearning (supervision). The interest for using belief functionsfor edge detection comes from their ability to model more ade-quately uncertainties with respect to the classical probabilisticmodelingapproach, andtodeal withconictinginformationduetospatial changesintheimageornoises. Thispaperisorganized as follows: In section 2 we briey recall the basicsof DSmT and the fusion rule we use. In section 3, we presentin details our new edge detector based on belief functions andtheir fusion. Results of our new algorithm tested on the originalLenaspictureanditsnoisyversionarepresentedinsection4withacomparisontotheclassical Cannysedgedetector.Conclusions and perspectives are given in section 5.II. BASICS OF DSMTThepurposeofDSmT[17]istoovercomethelimitationsof DST[16] mainlybyproposingnewunderlyingmodelsfor the frames of discernment in order to t better withthe nature of real problems, and proposing newefcientcombination and conditioning rules. In DSmT framework, theelements i, i =1, 2, . . . , nof a given frame are notnecessarily exclusive, and there is no restriction on i but theirexhaustivity. Thehyper-power set DinDSmT, thehyper-powersetisdenedasthesetofallcompositepropositionsbuilt from elements of with operators and . For instance,if = {1, 2}, thenD= {, 1, 2, 1 2, 1 2}. A(generalized) basic belief assignment (bba for short) is denedas the mappingm:D[0, 1]. The generalized belief andplausibilityfunctionsaredenedinalmostthesamemanneras in DST. More precisely, from a general frame , we deneamapm(.) : D[0, 1] associatedtoagivenbodyofevidence Basm() = 0 and

ADm(A) = 1 (1)The quantity m(A) is called the generalized basic beliefassignment/mass (or just bba for short) ofA.Thegeneralizedcredibilityandplausibilityfunctionsarede-ned in almost the same manner as within DST, i.e.Bel(A) =

BABDm(B) and Pl(A) =

BA=BDm(B) (2)Two models1(the free model and hybrid model) in DSmTcan be used to dene the bbas to combine. In the freeDSmmodel, thesourcesof evidencearecombinedwithouttaking into account integrity constraints. When the free DSmmodel does not holdbecausethetruenatureof thefusionproblem under consideration, we can take into account someknown integrity constraints and dene bbas to combine usingthe proper hybridDSmmodel. All details of DSmTwith1Actually, Shafersmodel, consideringall elementsoftheframeastrulyexclusive, can be viewed as a special case of hybrid model.Originally published as Dezert J., Liu Z., Mercier G., Edge Detection in Color Images Based on DSmT, in Proc. Of Fusion 2011 Conf., Chicago, July, 2011, and reprinted with permission.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4343manyexamplescanbeeasilyfoundin[17]availablefreelyontheweb. Inthispaper, wewill workonlywithShafersmodeloftheframewhereallelementsiofareassumedtrulyexhaustive andexclusive (disjoint) andtherefore Dreduces the the classical power set2and generalized belieffunctions reduces to classical ones as within DST framework.Aside offering the possibility to work with different underlyingmodels (not only Shafers model as within DST), DSmT offersalsonewefcient combinationrules basedonproportionalconict redistribution (PCR rules no 5 and no 6) for combininghighlyconictingsourcesofevidence.InDSmTframework,theclassicalpignistictransformationBetP(.)isreplacedbythe by the more effective DSmP(.) transformation to estimatethe subjective probabilities of hypotheses for decision-makingsupport once the combinationof bbas has beenobtained.Before presenting our new edge detector, we just recall brieywhat are the PCR5 fusion rule and the DSmP transformation.All details, justications with examples on PCR5 and DSmPcanbefoundfreelyfromthewebin[17], Vols. 2&3andwill not be reported here.A. PCR5 fusion ruleThe Proportional Conict Redistribution Rule no. 5 (PCR5)is used generally to combine bbas in DSmT framework. PCR5transfers the conicting mass only to the elements involved intheconictandproportionallytotheirindividualmasses, sothat the specicity of the information is entirely preserved inthis fusion process. Let m1(.) and m2(.) be two independent2bbas, then the PCR5 rule is dened as follows (see [17], Vol.2for full justicationandexamples): mPCR5() =0andX 2\ {}mPCR5(X) =

X1,X22X1X2=Xm1(X1)m2(X2)+

X22X2X=[m1(X)2m2(X2)m1(X) +m2(X2)+m2(X)2m1(X2)m2(X) +m1(X2)] (3)whereall denominators in(3) aredifferent fromzero. If adenominator is zero, that fraction is discarded. AdditionalpropertiesofPCR5canbefoundin[5]. ExtensionofPCR5for combining qualitative bbas can be found in [17], Vol. 2 &3. All propositions/sets are in a canonical form. A variant ofPCR5, called PCR6 has been proposed by Martin and Osswaldin[17], Vol. 2, for combinings >2sources. Thegeneralformulasfor PCR5andPCR6rulesaregivenin[17], Vol.2also. PCR6coincideswithPCR5whenonecombinestwosources.ThedifferencebetweenPCR5andPCR6liesintheway the proportional conict redistribution is done as soon asthree or more sources are involved in the fusion. For example,lets consider three sources with bbas m1(.), m2(.) and m3(.),A B= for the model of the frame, andm1(A) = 0.6,m2(B) =0.3, m3(B) =0.1. WithPCR5thepartial con-ictingmassm1(A)m2(B)m3(B)=0.6 0.3 0.1=0.0182I.e. each source provides its bba independently of the other sources.is redistributedbacktoAandBonlywithrespect tothefollowingproportions respectively: xPCR5A=0.01714andxPCR5B= 0.00086 because the proportionalization requiresxPCR5Am1(A)=xPCR5Bm2(B)m3(B)=m1(A)m2(B)m3(B)m1(A) +m2(B)m3(B)that isxPCR5A0.6=xPCR5B0.03=0.0180.6 + 0.03 0.02857thus

xPCR5A= 0.60 0.02857 0.01714xPCR5B= 0.03 0.02857 0.00086With the PCR6 fusion rule, the partial conicting massm1(A)m2(B)m3(B) = 0.6 0.3 0.1 = 0.018 is redistributedback to A and B only with respect to the following proportionsrespectively: xPCR6A= 0.0108 andxPCR6B= 0.0072 becausethe PCR6 proportionalization is done as follows:xPCR6Am1(A)=xPCR6B,2m2(B)=xPCR6B,3m3(B)=m1(A)m2(B)m3(B)m1(A) +m2(B) +m3(B)that isxPCR6A0.6=xPCR6B,20.3=xPCR6B,30.1=0.0180.6 + 0.3 + 0.1= 0.018thus xPCR6A= 0.6 0.018 = 0.0108xPCR6B,2= 0.3 0.018 = 0.0054xPCR6B,3= 0.1 0.018 = 0.0018and therefore with PCR6, one gets nally the followingredistributions toA andB:

xPCR6A= 0.0108xPCR6B= xPCR6B,2+xPCR6B,3= 0.0054 + 0.0018 = 0.0072Fromtheimplementationpoint ofview, PCR6issimplertoimplement than PCR5. Very basic Matlab codes for PCR5 andPCR6 fusion rules can be found in [17], [18].B. DSmP transformationDSmP probabilistic transformation is a serious alternative tothe classical pignistic transformation which allows to increasethe probabilistic information content (PIC), i.e. to reduceShannonentropy, oftheapproximatedsubjectiveprobabilitymeasuredrawnfromanybba. JusticationandcomparisonsofDSmP(.) w.r.t.BetP(.) and to other transformations canbe found in details in [6], [17], Vol. 3, Chap. 3. DSmPtrans-formation is dened3byDSmP

() = 0 and X 2\ {}DSmP

(X) =

Y 2

ZXY|Z|=1m(Z) + |X Y |

ZY|Z|=1m(Z) + |Y |m(Y ) (4)3Here we work on classical power-set, but DSmP can be dened also forworkingwithother fusionspaces, hyper-power setsor super-power setsifnecessary.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4344where |XY | and |Y | denote the cardinals of the sets XYandYrespectively; 0 is a small number which allows toincreasethePICvalueoftheapproximationof m(.)intoasubjectiveprobabilitymeasure. Usually =0, but insomeparticular degenerate cases, when the DSmP=0(.) valuescannot be derived, theDSmP>0values can however alwaysbederivedbychoosingasaverysmall positivenumber,say=1/1000forexampleinordertobeascloseaswewant to the highest value of the PIC. The smaller , thebetter/bigger PICvalue one gets. When =1andwhenthe masses of all elements Zhaving|Z| = 1 are zero,DSmP=1(.)=BetP(.),wherethepignistictransformationBetP(.) is dened by [19]:BetP{X} =

Y 2|Y X||Y |m(Y ) (5)with convention ||/|| = 1.C. DS combination ruleDempster-Shafer (DS) rule of combination is the mainhistorical (and still widely used) rule proposed by GlennShafer inhis milestonebook[16]. Verypassionatedebateshaveemergedintheliteratureaboutthejusticationandthebehavior of this rulefromthefamous Zadehs criticismin[22]. Wedont plantoreopenthis endless debateandjustwant torecall brieyherehowit ismathematicallydened.Lets consider a given discrete and nite frame of discernment= {1, 2, . . . , n} of exclusive and exhaustive hypotheses(a.k.asatisfyingShafersmodel)andtwoindependent bbasm1(.) andm2(.) dened on2, then DS rule of combinationis dened bymDS() = 0 and X = andX 2:mDS(X) =11 K12

X1,X22X1X2=Xm1(X1)m2(X2) (6)where K12

X1,X22X1X2=m1(X1)m2(X2) represents thetotal conict betweensources. If K12=1, thesources ofevidenceareinfull conict andDSrulecannot beapplied.DS rule is commutative and associative and can be extented forthe fusion ofs > 2 sources as well. The main criticism aboutsuchsuchconcernsitsunexpected/counter-intuitivebehaviorassoonasthedegreeof conict betweensourcesbecomeshigh(see[17], Vol.1, Chapter 5andreferences thereinfordetails and examples).D. Decision-making supportDecisions are achieved by computing the expected utilitiesof the acts using either the subjective/pignistic BetP{.} (usu-ally adopted in DST framework) orDSmP(.) (as suggestedinDSmTframework) astheprobabilityfunctionneededtocompute expectations. Usually, one uses the maximum of thepignisticprobabilityasdecisioncriterion. ThemaximumofBetP{.}is oftenconsideredas a prudent bettingdecisioncriterionbetweenthetwoother decisionstrategies(maxofplausibility or max. of credibility which appears to be respec-tively too optimistic or too pessimistic). It is easy to show thatBetP{.}isindeedaprobabilityfunction(see[19], [20])aswell asDSmP(.)(see[17], Vol.2). Themaxof DSmP(.)is considered as more efcient for practical applications sinceDSmP(.) is more informative (it has a higher PIC value) thanBetP(.) transformation.III. EDGE DETECTION BASED ON DSMT AND FUSIONIn this work, we use the most common RGB (Red-Green-Blue) representationof thedigital color imagewhereeachlayer(channel)R, GandBconsistsinamatrixof ni njpixels. The discrete value of each pixel in a given color channelis assumed in a given absolute interval of color intensity[cmin, cmax]. TheprincipleofournewEdgedetectorbasedon DSmT is very simple and consists in the following steps:A. Step 1: Construction of bbasLetsconsideragivenchannel (colorlayer)anddenoteitasLwhichcanrepresenteithertheRed(R)colorlayer, theGreen (G) color layer or the Blue (B) color layer, or any otherchannel in a more general case for multispectral images. Forsimplicity, we focus our work and presentation here on colorimages only.Apply an edge detector algorithm for each color channelLtoget foreachpixel xLij, i =1, 2, . . . , ni, j =1, 2, . . . , njanassociatedbbamLij(.)expressingthelocalbeliefthatthispixel belongs or not to an edge. The frame of discernmentused to dene the bbas is very simple and is dened as = {1 Pixel Edge, 2 Pixel / Edge} (7)isassumedtosatisfyShafersmodel (i.e. 1 2= ).It isclear that many(binary) edgedetectionalgorithmsareavailable in the image processing literature but here we wanta smooth algorithm able to provide both the belief of eachpixel to belong or not to an edge and also the uncertainty onehas on the classication of this pixel. In the this subsection, wepresent a very simple algorithm for accomplishing this task atthe color channel level. Obviously the quality of the algorithmusedinthisrst stepwill haveastrongimpact ofthenalresult and therefore it is important to focus research efforts onthe development of efcient algorithms for realizing this stepas best as possible.AsinSobel method[9], two3 3kernelsareconvolvedwiththe original image ALfor eachlayer Ltocalculateapproximations of the derivatives - one for horizontal changes,andone for vertical. We thenobtaintwogradient imagesGLxandGLyfor eachlayer Lrepresent thehorizontal andverticalderivativeapproximationsforeachpixel xLij. Thex-coordinate is dened as increasing in the right-direction, andthey-coordinateis as increasinginthedown-direction. Ateachpixel xLijofthecolorlayer L, thegradient magnitudegLijcanbeestimatedbythecombinationofthetwogradientapproximations as:gLij=12(GLx(i, j)2+GLy(i, j)2)1/2(8)Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4345whereGLx=181 0 12 0 21 0 1 AL;GLy=181 2 10 0 01 2 1 AL;and where denotes the 2-dimensional convolution operation.In Sobels detection method, the edge detection for a pixelxijof a gray image is declared based on a hard thresholdingofgijvalue. SuchSobeldetectorissensitivetonoiseanditcangeneratefalsealarms. Inthiswork, gLijvaluesareusedonlytodenethemassfunction(bba)ofeachpixelineachlayer over the power-set of dened in (7). If the valuegLijvalue of a pixel is big, it implies that this pixel is more likelyto belong to an edge. IfgLijvalue of the pixelxLijis low thenourbeliefthat it belongstoanedgemust belowtoo. Suchverysimpleandintuitivemodelingcanbeobtaineddirectlyfromthe sigmoid functions commonly used as activationfunctioninneural networks, orasfuzzymembershipinthefuzzy subsets theory as explained below.Lets consider the sigmoid function dened asf,t(g) 11 +e(gt)(9)gis the gradient magnitude of the pixel under consideration.t is the abscissa of the inection point of the sigmoid whichcanbeselectedbyt =p max(g)wherepisaproportionparameter and is the scalar product operator. When workingwith noisy images,p always increases with the level of noise. is the slope of the tangent at the inection point.It canbeeasilyveriedthat thebbamLij(.|gLij)satisfyingtheexpectedbehaviorcanbeobtainedbythefusion4ofthetwo following simple bbas dened by:focal element m1(.) m2(.)1f,te(g) 020 f,tn(g)1 21 f,te(g) 1 f,tn(g)with0 < tn< te< 255, > 0.teis thelower thresholdfor theedgedetection, andtnisthe upper threshold for the non edge detection. Thus, [tn, te]corresponds to our uncertainty decision zone and the gLij valueslying in this interval correspond to the unknown decision state.The bounds (thresholds)tnandtecan be tuned based on theaverage gradients values of the image, and the lengthtetndependsonthelevel of thenoise. If thetheimageisverynoisy, it means the informationis veryuncertain, andthelengthof theinterval [tn, te] canbecomelarge. Otherwise,it issmall. Becauseof structureof thesetwosimplebbas,4with DS, PCR5 or even with DSmH rule [17].the fusion obtained with PCR5, DS of even with DSm hybrid(DSmH) rules of combination provide globally similar resultsand therefore the choice of the fusion rule here does not reallymatter to buildmLij(.|gLij) as shown on the gures 1-3. PCR5,which is the most specic fusion rule (it reduces the level ofbeliefcommittedtotheuncertainty), isusedinthisworktogeneratemLij(.|gLij).Figure 1. Computation of mLij(.|gLij) from m1(.) and m2(.) with [tn, te] =[60, 100] and = 0.09.Figure 2. Computation of mLij(.|gLij) from m1(.) and m2(.) with [tn, te] =[50, 80] and and = 0.06.Figure 3. Computation of mLij(.|gLij) from m1(.) and m2(.) with [tn, te] =[30, 40] and = 0.04.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4346In summary, mLij(.|gLij) can be easily constructed from thechoiceof thresholdingparameters te, tndeningtheuncer-taintyzoneofthegradient values, theslopeparameter ofsigmoids, and of course from the gradient magnitude gLij. Thisapproach is very easy to implement and very exible since itdepends on the parameters which are totally under the controlof the user.B. Step 2: Fusion of bbasmLij(.)Many combination rules like DS rule, Dubois & Prade ruleYagersrule, andsooncanbeusedwithour approach. Inthis work, we just make investigations based on the two mostwell-knownrules(DSandPCR5ruleproposedinDSTandDSmTrespectively). Soweuseeither DSor PCR5ruletocombinethethreebbasmRij(.), mGij(.)andmBij(.)foreachpixel xijinorder toget theglobal bba mij(.)toestimatethe degree of belief of the belonging ofxijto an edge in thegiven image. Since PCR5 is not associative, we must apply thegeneral PCR5 formula for combining the 3 sources (channels)altogether5asexplainedindetailsin[17], Vol.2, Chap. 1&2. A suboptimal approach requiring less computations wouldconsist in applying a PCR5 sequential fusion of these bbas insuch a way that the two least conicting bbas are combined atrst by PCR5 and then combine again with PCR5 the resultingbbaswiththethirdoneaccordingto(3). ThemoresimplePCR6 rule could also be used instead of PCR5 as well - see[17], Vol. 2.C. Step 3: Decision-makingTheoutput of step2istheset of Ni Njbbas mij(.)associatedtoeachpixel xijoftheimageinthewholecolorspace (R,G,B). mij(.) commits some degree of belief to1Pixel Edge, to2Pixel / Edgeandalsototheuncertainty 1 2. The binary decision-making processconsists in declaring if the pixel xijunder considerationbelongs or not to an edge fromthe bba mij(.), or in amorecomplicatedmanner frommij(.)andthebbasof itsneighbours. In this paper, we just recall the principal methodsbased on the use ofmij(.).Based on mij(.) only, howto decide 1or 2? Manyapproaches have been proposed in the literature for answeringthisquestionwhenworkingwithan-Dframe. Thepes-simistic approach consists in declaring the hypothesisi which has the maximum of credibility, whereas the optimisticapproachconsistsindeclaringthehypothesiswhichhasthemaximumofplausibility. Whenthecardinalityoftheframeisgreater thantwo, thesetwoapproachescanyieldtoadifferent nal decision. In our particular application and sinceour framehasonlytwoelements, thenal decisionwillbethesameifweusethemaxofcredibilityorthemaxofplausibility criterion. Other decision-making methods suggest,as agoodbalancebetweenaforementionedpessimisticandoptimisticapproaches, toapproximatethebbaat rst intoa5i.e. a generalization of the PCR5 formula described in section II-A.subjectiveprobabilitymeasurefromasuitableprobabilistictransformation, andthentochoosetheelement of whichhasthehighest probability. Inpractice, onesuggeststotakeasnal decisiontheargument ofthemaxof BetP(.)orofthe max of DSmP(.). In our binary frame case however thesetwoapproachesalsoprovidethesamenaldecisionaswiththemaxof credibilityapproach. ThiscanbeeasilyprovedfromBetP(.)or DSmP(.)formulas. Indeed, letsconsiderm(1) > m(2) > 0 withm(1) + m(2) + m(1 2) = 1(which means that1is taken as nal decision because it hasahigher credibilitythan2), thenonegets as approximatesubjective probabilities:BetP(1) = m(1) +m(1 2)/2 m(1) +KBetP(2) = m(2) +m(1 2)/2 m(2) +KDSmP(1) = m(1)[1 +m(1 2)m(1) +m(2)] m(1)[1 +K

]DSmP(2) = m(2)[1 +m(1 2)m(1) +m(2)] m(2)[1 +K

]where Kand K

are two positive constants. Fromtheseexpressions, oneseesthatifm(1)>m(2)>0, thenalsoBetP(1)>BetP(2)andDSmP(1)>DSmP(2)andthusthenal decisionbasedonmaxof BetP(.)ormaxofDSmP(.) is nally the same. Note that when m(1) = m(2),no rational decision can be drawn fromm(.) and only arandom decision procedure or ad-hoc method can be used insuch particular case.Insummary, one sees that whenworkingwitha binaryframe,allcommondecision-makingstrategiesprovidethesamenal decisionandthereforethereisnointerest touseacomplexdecision-makingprocedureinthatcaseandthatswhywecanadopt herethemaxofbeliefasnal decision-making criterion in our simulations. Note that aside the naldecision and because we havem(12), we are able (if wewant)alsotoplot thelevel ofuncertaintyrelatedwithsuchdecision (not presented in this paper).IV. SIMULATIONS RESULTSIn this section we present the results of our newedgedetectionalgorithmtestedontwocolorimagesfordifferentparameter settings.A. Test on original Lenas pictureLenaSoderbergpictureisoneofthemostusedimagefortestingimageprocessingalgorithmsintheliterature[4]andthereforeweproposetotest ouralgorithmonthisreferenceimage. This imagecanbefoundas part of theUSCSIPIImageDatabaseintheirmiscellaneouscollectionavailableat http://sipi.usc.edu/database/index.php. Theoriginal Lenaspicture scan is shown on Fig. 4-(a). The gure 5-(a)(c) showsthe edge detection on each channel (layer) based on the bbasmLij(.|gLij) in section III-A. One sees that the edges in differentchannels are different, and the task of our proposed algorithmAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4347Figure 4. Lenas picture before and after noiseFigure 5. Edge detections in each channel.Figure 6. Cannys edge detector on Lenas gray image.Figure 7. Sobels edge detector on Lenas gray image.Figure 8. DS-based edge detector on Lenas color image.Figure 9. PCR5 edge detector on Lenas color image.is to combine efciently the underlying bbasmLij(.|gLij) gen-erating the subgures 5-(a)(c).Sobel [9] and Canny [2] edge detectors are commonly usedin image processing community and thats why we makecomparisonofournewedgedetectorw.r.t. CannysandSo-bels approaches. Canny and Sobel edge detectors are applieddirectlytothegrayimageconvertedfromtheoriginal Lenacolor image Fig. 4-(a). The gures 69 show the results of thedifferent edge detectors on Lenas picture. In our simulations,we took=0.06, andtgdened ast=p max(g) in eachlayer, was taken with pn= 0.17 and pe= 0.19, correspondingtogradient thresholds[tRn, tRe ] =[15, 17], [tGn, tGe ] =[13, 14]and[tBn, tBe ] =[11, 13]. Themaxof credibility, plausibility,DSmPorBetPfor decision-making to generate nal resultprovidethesamedecisionasexplainedinthesectionIII-Cwhich is normal in this binary frame case.One sees that nally on the clean (noise-free) Lenas picture,our edge detector provides close performances to Sobelsdetector applied on Lenas grey image. Cannys detector seemstoprovide a better abilitytodetect some edges inLenaspicture than our method, but it also generates much more falseAdvances and Applications of DSmT for Information Fusion. Collected Works. Volume 4348alarms too. It is worth noting that the results provided by DS-basedorPCR5-basededgedetectorsshowacoarselocationof the edges. Soit is quite difcult todrawna clear andfairconclusionbetweentheseedgedetectorssinceit highlydependsonwhat wewant, i.e. thereductionoffalsealarmsor the reduction of miss-detections.B. Test on Lenas picture with noiseInthissimulation, weshowhowouredgedetectorworksonanoisyimage. Samplingof independent GaussiannoiseN(0, 2) is added to each pixels of each layer of the originalLenaspictureasseenonFig. 4-(b). Inthepresentedsim-ulation, 2=1100whichcorrespondapproximativelytothevalue of the variance of the blue channel and half the varianceoftheothers. Local edgedetectionforeachlayerbasedonmLij(.|gLij)isshownonFig. 10-(a)(c), wheretheredpointsrepresent the ignorant pixel which commits the most belief tothe ignorance 12. As shown in Fig 10, the edge detection ineach channel is very noisy. Our method allows to commit auto-matically highest belief value to uncertainty for most of pixelsassociatedtoanedgewhichactuallycorrespondtonoises.6Theedgedetectionbasedonfusionresult areinterestingasshown by Fig.11 and Fig. 12 because it shows the ability of ouredgedetectortosuppressthenoiseeffects. Forcomparison,we give on Fig. 13 and Fig.14, the performance of Canny andSobel edgedetectors appliedclassicallyonthenoisygray-level Lenaspicture. Inthissimulation, wetook=0.06,andtusingpn=0.22 max(g)andpe=0.39 max(g)ineachlayer with[tRn, tRe ] =[36, 20], [tGn, tGe ] =[35, 19] and[tBn, tBe ] = [31, 18]. The decision-making is still based on maxof credibility.The visual comparisonandanalysis of results shownofgures1112clearlyindicatesthat ouredgedetectorbasedon the fusion of belief constructed on each layer works muchbetter than the edge detection applied separately on eachlayer. Thereisnoignorant pixel correspondingtoredcoloraccording to the fusion results, since the fusion process of DSor PCR5 rule effectively decrease the uncertainty. Our resultsshow also clearly that Canny and Sobel edge detectors appliedtonoisygray-level Lenas pictureareverysensitivetothenoise perturbations. Our proposed method (based on DS ruleor onPCR5rule) ismorerobust tothenoiseperturbationsand provides better results than Sobel or Canny edge detectorfor suchnoisyimage. For thistestedimage, it appearsthatthe results using DS and PCR5 rules are very close, becausethere is not too much conict actually between bbas of layersand one know that insuch case PCR5 rule behavior is closetoDSrulebehavior. DSruleisusuallygoodenoughinthelowconict case, whereas PCR5rule is preferredfor thecombinationofhighconictingsourcesofevidence. Sothepreference of PCR5 with respect to DS rule for edge detectionmust be guided by the level of conict which appears in thelayers of the color image that we need to process.6So we are also able at layer level to lter these pixels (false alarms) beforeapplying the fusion. This has not yet be done in this work.Figure 10. Edge detections in each channel on noisy image.Figure 11. DS edge detector on noisy Lenas color image.V. CONCLUSIONS AND PERSPECTIVESAnewunsupervisededgedetector for color imagebasedon belief functions has been proposed in this work. The basicbelief assignment (bba) associatedwiththeedgeof apixelin each channel of the image is dened according to itsgradient magnitude, and one can easily model the uncertaintyabout our belief it belong or not to an edge. PCR5 andDSrules havebeenappliedinthis worktocombinethesebbastoget theglobal bbafornal decision-making. Otherrules of combination of bbas could also have been usedinsteadbut theyareknowntobeless efcient thanPCR5orDSrulesinhighandlowconictcasesrespectively. TheFigure 12. PCR5 edge detector on noisy Lenas color image.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4349Figure 13. Sobels edge detector on noisy Lenas gray image.Figure 14. Cannys edge detector on noisy Lenas gray image.fusionprocessisabletoreducenoiseperturbationsbecausethenoisesareassumedtobeindependentbetweenchannels.Thenal decisionmakingontheedgecanbemadeeitheron the maximum of credibility, plausibility,DSmPorBetPvaluesaswell. Therst simulationdoneonoriginal Lenaspictureshows that our edgedetector works as well as theclassical Sobels edge detector and it provides less false alarmsthan with Cannys detector, but seems to generate more miss-detections. In our second simulation based onnoisy Lenaimage, theresultsshowthat ournewedgedetectorismorerobust to the noise perturbations than Sobel or Canny classicaledge detectors. As possible improvement of this algorithm andfor further research, we would like to include some morpho-logical or connexity constraints at a higher level of processingand develop automatic technique for threshold selection. Theapplication of this new approach of edge detection to satellitemultispectral images is under investigations.REFERENCES[1] S. BenChaabane, F. Fnaiech, M. Sayadi, E. Brassart, RelevanceoftheDempster-Shaferevidencetheoryforimagesegmentation, 3rdInterna-tional conference on Signals, Circuits and Systems, Nov. 6-8th, Medenine,Tunisia, 2009.[2] J.F. 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Zadeh, Onthevalidityof Dempstersruleof combination, MemoM79/24, Univ. of California, Berkeley, USA, 1979.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 4350