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entropy Article RiemannianLaplaceDistributionontheSpaceof SymmetricPositiveDefiniteMatrices HatemHajri 1,*,†, IoanaIlea1,2,†,SalemSaid1,†,LionelBombrun1,† andYannickBerthoumieu1,† 1 GroupeSignalet Image,CNRSLaboratoire IMS, InstitutPolytechniquedeBordeaux,Universitéde Bordeaux,UMR5218,Talence33405,France; ioana.ilea@u-bordeaux.fr (I.I.); salem.said@u-bordeaux.fr (S.S.); lionel.bombrun@u-bordeaux.fr (L.B.);Yannick.Berthoumieu@ims-bordeaux.fr (Y.B.) 2 CommunicationsDepartment,TechnicalUniversityofCluj-Napoca,71-73Dorobantilorstreet,Cluj-Napoca 3400,Romania * Correspondence: hatem.hajri@ims-bordeaux.fr;Tel.: +33-5-4000-6540 † Theseauthorscontributedequally to thiswork. AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 19December2015;Accepted: 8March2016;Published: 16March2016 Abstract: The Riemannian geometry of the space Pm, of m×m symmetric positive definite matrices,hasprovidedeffective tools to thefieldsofmedical imaging, computervisionandradar signal processing. Still, an open challenge remains, which consists of extending these tools to correctlyhandle thepresenceofoutliers (orabnormaldata),arisingfromexcessivenoiseor faulty measurements. Thepresentpaper tackles thischallengebyintroducingnewprobabilitydistributions, calledRiemannianLaplacedistributionson the spacePm. First, it shows that thesedistributions provideastatistical foundationfor theconceptof theRiemannianmedian,whichoffers improved robustness indealingwithoutliers (incomparisonto themorepopularconceptof theRiemannian centerofmass). Second, itdescribesanoriginalexpectation-maximizationalgorithm, forestimating mixturesofRiemannianLaplacedistributions. Thisalgorithmisappliedto theproblemof texture classification, incomputervision,which isconsidered in thepresenceofoutliers. It is showntogive significantlybetterperformancewithrespect tootherrecently-proposedapproaches. Keywords: symmetricpositivedefinitematrices;Laplacedistribution; expectation-maximization; Bayesian informationcriterion; textureclassification 1. Introduction Datawithvalues inthespacePm, ofm×msymmetricpositivedefinitematrices,playanessential role inmanyapplications, includingmedical imaging [1,2], computervision [3–7]andradarsignal processing[8,9]. In theseapplications, the locationwhereadataset iscenteredhasaspecial interest. Whileseveraldefinitionsof this locationarepossible, itsmeaningasarepresentativeof theset should be clear. Perhaps, themostknownandwell-usedquantity to represent a centerof adataset is the Fréchetmean.GivenasetofpointsY1, · · · ,Yn inPm, theirFréchetmeanisdefinedtobe: Mean(Y1, · · · ,Yn)=argminY∈Pm n ∑ i=1 d2(Y,Yi) (1) whered isRao’sRiemanniandistanceonPm [10,11]. StatisticsongeneralRiemannianmanifoldshavebeenpoweredbythedevelopmentofdifferent tools forgeometricmeasurements andnewprobabilitydistributionsonmanifolds [12,13]. On the manifold (Pm,d), themajoradvances in thisfieldhavebeenachievedby the recentpapers [14,15], which introduce theRiemannianGaussiandistributionon (Pm,d). Thisdistributiondependsontwo Entropy2016,18, 98 365 www.mdpi.com/journal/entropy