UNSUPERVISED NON STATIONARY IMAGE SEGMENTATION USING TRIPLET MARKOV CHAINS. Pierre Lanchantin and Wojciech Pieczynski

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1 Advaced Cocepts for Itelliget Visio Sstems ACVIS 0 Aug. -Sept. Brussels Belgium 00 SPERVISED O STATIOARY IMAGE SEGMETATIO SIG TRIPLET MARKOV CHAIS Pierre Lachati ad Wojciech Pieczski Pierre.Lachati@it-evr.fr GET/IT Départemet CITI CRS MR 7 9 rue Charles Fourier 9000 Evr Frace ABSTRACT This work deals with the usupervised Baesia hidde Markov chai restoratio eteded to the o statioar case. supervised restoratio based o Epectatio- Maimizatio EM or Stochastic EM SEM estimates cosiderig the Hidde Markov Chai HMC model is quite efficiet whe the hidde chai is statioar. However whe the latter is ot statioar the usupervised restoratio results ca be poor due to a bad match betwee the real ad estimated models. I this paper we preset a more appropriate model for o statioar HMC via recet Triplet Markov Chais TMC model. sig TMC we show that the classical restoratio results ca be sigificatl improved i the case of o statioar data. The latter improvemet is performed i a usupervised wa usig a SEM parameter estimatio method. Some applicatio eamples to usupervised image segmetatio are also provided.. ITRODCTIO The hidde Markov chai HMC model is widel used for various problems icludig sigal ad image processig ecoomical predictio ad health scieces. I such model X = X... X is a stochastic process modelig a uobservable or hidde discrete sigal each X takes its values i a fiite set { } Ω = ω...ω K ad Y = Y... Y is a stochastic process modelig the observatios each Y takes its values i the set of real umbers R. The problem is the to estimate the hidde discrete sigal X = from the observed sigal Y =. The liks betwee X ad Y are the modeled b the followig joit distributio : p = p p... p p... p Such a model is called hidde Markov chai with idepedet oise because the hidde process X is a Markov chai ad the radom variables Y... Y are idepedet coditioall o X. Hereafter it will be deoted b HMC-I. It allows oe to recover the hidde data X = from the observed data Y = usig differet Baesia classificatio techiques like Maimum A Posteriori MAP or Maimal Posterior Mode MPM [-]. These restoratio methods use the distributio of the hidde process coditioal to the observatios which is called its posterior distributio. Whe p is based o ukow parameters θ Θ the latter ca be estimated from Y = cosiderig the statioarit of the hidde process b methods like Epectatio-Maimizatio EM [ ] or Stochastic Epectatio-Maimizatio SEM [ ]. Thus whe the hidde process is a statioar Markov chai parameters are well estimated ad the restoratio based o the estimated parameters gives good results. However whe this is ot the case the estimatio ecessaril gives wrog results which ca impl poor restoratio of X. I this work we propose to model the o statioarit b itroducig a auiliar process goverig the regime switchig of the hidde process. The itroductio of a auiliar process to model the o statioarit has alread bee treated before. I [] the authors itroduce auiliar data to modif the trasitio probabilities of the Markov hidde process. I [] the author preset a model based o the superpositio of two Markov chais. A o-homogeeous discrete process is described b a set of trasitio matrices. At each the choice of the active matri is the govered b a hidde homogeeous Markov chai. evertheless our approach is quite differet : we propose a model of o statioar hidde Markov chai b usig the Triplet Markov Chai TMC model [78]. It is based o the three followig poits : i Whe X is o statioar we ca itroduce a auiliar stochastic process whose aim is to model the regime switchig of X ; ii We directl assume the markoviait ad the statioarit of the pairwise process V = X. The the distributios P X ad P are ot ecessaril Markovia which allows oe to deal with differet kids of stochastic processes; iii The distributio of Y coditioal o V = X is such that the triplet process T = X Y is a Markov chai. Although the margial distributio P of T is X Y

2 Advaced Cocepts for Itelliget Visio Sstems ACVIS 0 Aug. -Sept. Brussels Belgium 00 ot ecessaril a Markov distributio it ca be used to perform Baesia restoratios T beig a particular case of Triplet Markov chai [7]. We provide differet simulatio studies showig that such a itroductio of a auiliar process ca improve the results obtaied with the classical usupervised Baesia restoratio. We provide a complete derivatio of the Forward-Backward MPM ad SEM algorithms for the ew model ad give computable versios of these oes. The paper is orgaized i the followig wa : The et sectio is devoted to a brief descriptio of the Triplet Markov Chai model. The particular model used ad the related processig method is described i sectio three ad differet simulatio results described i sectio four.. TRIPLET MARKOV CHAIS Let X = X ad Y = Y be two stochastic processes. X is hidde each X takes its values i a fiite set Ω = { } ω...ω K ad Y is observed each Y takes its values i the set of real umbers R. The problem is the to estimate X = from Y =. Defiitio. The model cosidered is called a Triplet Markov Chai if there eists a stochastic process = with λ M takig its values i a set Λ = { λ... } each such T = X Y = X Y is a Markov chai. that The idea of TMC is to cosider the distributio of Z = X Y which models the iteractios betwee the observed ad the searched process as a margial distributio of a Markov chai T = X Y. The distributio s p which are used i MPM restoratio are u Λ u the computable. I fact let V = X. As T is Markovia the process V Y is a Pairwise Markov Chai PMC ad we ca formulate ad calculate the distributio of V Y usig Forward-Backward recursios [9]. This meas that the distributios p = p u = p v also are computable. Fiall although the distributio of X Y is ot ecessaril a Markov oe the margial distributios p are computable. which i particular eables the use of the Baesia MPM restoratio method. We said above that Z = X Y i ot ecessaril a Markov chai. More precisel the followig result specifies locall o what the greater geeralit of TMC with respect to PMC cosists [8] : Λ Propositio. Let T = X Y be a TMC verifig : a p t t + does ot deped o ; ad b p t t = + p t + t. The the three followig coditios: i Z = X Y is a Markov chai ; iifor each p u z z = p u z ; iiifor each p u z = p u z are equivalet. The Pairwise Markov Chai PMC [90] model i which Z is assumed to be a Markov chai is the a particular case of the TMC obtaied b takig Λ = Ω ad = X. Furthermore accordig to the Propositio. above TMC is strictl more geeral tha PMC the latter beig strictl more geeral tha classical HMC i that there eist Markov chais T such that Z are ot Markov chais.. THE PARTICLAR TMC SED AD RELATED PROCESSIG METHOD.. TMC used model Let X be a o statioar hidde process X. Let us itroduce =... each takes its values i a fiite set Λ = { λ... λ M } goverig the regime switchig of X. The pairwise process V = X is assumed to be a Markov chai. Further we assume that Y...Y are idepedet coditioall o X ad p u = p. The T = V Y is a particular TMC i which V is a Markov chai ad which will be called TMC i the followig... MPM restoratio i TMC Accordig to the geeral Baesia theor the MPM restoratio is give b s... = ˆ... ˆ with ˆ MPM ˆ = arg ma p Ω Thus we eed to calculate p. As T is a TMC the process V Y is a PMC ad we ca write the distributio of V Y as p v = α v β v with. α v = p v... Forward probabilities ad v = p... + v β Backward probabilities.

3 Advaced Cocepts for Itelliget Visio Sstems ACVIS 0 Aug. -Sept. Brussels Belgium 00 These probabilities ca the be recursivel calculated for b α v = p v p ; v = + α v p v+ v p + + v Ω Λ v = α β ; β v β v p v v p = v Ω Λ + The margial posterior distributios of the hidde state ca be calculated b p v α v β v ad the trasitio of the posterior Markov distributio p v are calculated b p v v... + p v v p β v Havig calculated p v we ca the calculate p = p v ad to perform MPM. u Λ.. Learig TMC We assume that p are Gaussia. For K classes { } Ω = ω...ω K we have to estimate K meas µ µ K ad K variaces σ σ K of the K Gaussia desities p = p = K. Further the distributio of the statioar Markov chai V = X K M K M is give b parameters p ij = p v = i v = which is a probabilit o [ Λ] j Ω. The SEM method we use rus as follows : i cosider a iitial values p µ σ K Λ 0 i j ad k K ; ii for each q * : -simulate q V = v accordig to v which is possible due to ; q+ q+ q+ q+ θ = p µ σ with -calculate ij k k q+ q+ p ij = [ v = i v + = j ] k = q+ + = µ k q [ = k ] k = = [ = k ] ij k k θ = for p based o θ q µ = = [ = k ] = [ = k ] σ 7. SPERVISED IMAGE SEGMETATIO SIG TMC I this sectio we compare the HMC-I ad the proposed TMC models i the field of image segmetatio. The HMC-I model has alread bee successfull applied to this problem : to do so the bi-dimesioal set of piels is trasformed ito a moo-dimesioal sequece through the Hilbert-Peao sca [0]. Parameters are the estimated with SEM algorithm ad the MPM is applied to perform the segmetatio. We preset two series of eperimets. I the first oe two sthetic hidde o statioar images are cosidered. We give differet usupervised restoratio results cosiderig the HMC ad the TMC models ad show that the latter is more appropriate whe the hidde process presets differet regimes. I the secod oe we compare the two models cosiderig a real image usupervised segmetatio... Sthetic images We cosider two HMC X Y verifig with { } Ω = ω ω ad = 8 8. I the two HMC the hidde processes are o-statioar i differet wa. We defie three trasitio matrices : M = M = ad M = ad two auiliar processes =... ad =... each ad Λ = λ λ λ. takes its values i the fiite set { } * is defie b :... / = λ = λ = λ. / +... / ad / +... * is a Markov chai defie as follow : i the distributio of is /// ii the trasitio matri of is give b M = I the first eperimet the Markov chai X = X... is o statioar i the followig X wa. i the distributio of X is ii M p is the trasitio matri i with p = λ X whe p

4 Advaced Cocepts for Itelliget Visio Sstems ACVIS 0 Aug. -Sept. Brussels Belgium 00 I the secod eperimet the Markov chai X is obtai i the same wa b replacig b. We will deote = ω = ω = ω λ λ ω λ ω λ ω λ = = =. λ The processes X X are the coverted via a Hilbert-Peao sca ito a bi-dimesioal set as described i [] ad their realizatios are preseted i Figure. X ˆ = ˆ TMC τ = % X ˆ = ˆ TMC τ =.% = u X = Y = X ˆ = ˆ HMC τ = % = u X = Y = X ˆ = ˆ HMC τ = 9%. ˆ = uˆ TMC ˆ = uˆ TMC Figure : Differet images ad their usupervised HMC ad TMC based restoratios. I the two eperimets a realizatio X = is simulated ad Y = is sampled accordig to p p where p = is Gaussia with mea ad variace ad p = is Gaussia with mea ad variace which gives Y = ad Y = i Figure. The realizatio X = is the estimated b the Baesia MPM method from Y = i two differet was. The first restoratio is obtaied usig the parameters estimated with the SEM algorithm ad cosiderig that X Y is a HMC-I the hidde chai is assumed statioar which gives X ˆ = ˆ HMC τ = % ad X ˆ = ˆ HMC τ = 9% i Figure τ is the error ratio. The secod restoratio is obtaied usig the parameters estimated with the SEM algorithm assumig that T = X Y is the TMC cosidered above which gives X ˆ = ˆ TMC τ = % ad X ˆ = ˆ TMC τ =.% i Figure. We also show i Figure the estimates ˆ = uˆ ad ˆ = uˆ of the of the auiliar processes = u ad = u. We clearl see how the use of TMC ca improve the results obtaied with the classical HMC-I model. Furthermore differet parameters are well estimated Table ad which show a good behavior of the SEM cosidered.

5 Advaced Cocepts for Itelliget Visio Sstems ACVIS 0 Aug. -Sept. Brussels Belgium 00 Model µ σ µ σ p HMC TMC p HMC TMC Table : estimated desities parameters. The best model is the oe with the lowest BIC. Results are preseted i Table. a e e e b Table : estimated a p v + v ad b p cosiderig the TMC model. a e- 0 9e b Table : estimated a p v + v ad b p cosiderig the TMC model... Real Image Our eample here is a size satellite image of Toko which is preseted i Figure. We restore the image cosiderig differet umber of real ad auiliar states. Comparisos betwee models are carried out usig the Baes Iformatio Criterio BIC []. This Criterio is defied as BIC = LL + p log where LL is the log-likelihood of the model p its umber of idepedet parameters ad the umber of data. We do ot take ito accout parameters estimated to be zeros accordig to the covetio established i []. v 0 v 0 Figure : Image Toko O Figure we preset the segmetatio result b MPM ad SEM parameters estimatio based o the statioar HMC model with classes. O Figure we preset the segmetatio result b MPM ad SEM parameters estimatio based o the TMC model with classes ad states auiliar process. umber of real states umber of auiliar states LLumber of parameters BIC Table : Comparisos betwee models. Accordig to the BIC values we ca sa that for each umber of states studied the TMC model with the largest umber of auiliar states is more appropriate. Furthermore we ca see b comparig segmetatio results o Figure ad Figure that the segmetatio result cosiderig the TMC model preset fier features.

6 Advaced Cocepts for Itelliget Visio Sstems ACVIS 0 Aug. -Sept. Brussels Belgium 00 precisel the o statioar prior distributio of the hidde Markov chai was modeled b a auiliar process goverig the switchig of the trasitio matri. Baesia segmetatio techiques are the redered applicable ad differet eperimets show its iterest i simulated ad real images.. REFERECES Figure : MPM segmetatio after SEM parameters estimatio based o the statioar HMC mode with real states. BIC=80. Figure : MPM segmetatio after SEM parameters estimatio based o the TMC model with real states ad auiliar states. BIC=0.. COCLSIOS AD PERSPECTIVES I this paper we have provided a usupervised method for restorig o statioar hidde Markov chais with potetial applicatios to usupervised image segmetatio. The mai cotributio was to tackle the lack of statioarit usig the Triplet Markov Chai model. More. L. E. Baum T. Petrie G. Soules. Weiss. A maimizatio techique occurig i the statistical aalsis of probabilistic fuctios of Markov chais A. Math. Statistic. pp G. D. Fora The Viterbi algorithm Proceedigs of the IEEE Vol. o. pp G. J. McLachla ad T. Krisha EM Algorithm ad Etesios Wile G. Celeu ad J. Diebolt The SEM algorithm: a probabilistic teacher derived from the EM algorithm for the miture problem Comput. Statist. Quarter. pp J. P. Hugues P. Guttorp ad S. P. Charles A ohomogeeous hidde Markov model for precipitatio occurrece Appl.stat vol. 8 pp A. Berchtold The double chai Markov model Commu. Statist. Theor Methods W. Pieczski Chaîes de Markov Triplet Triplet Markov Chais Comptes Redus de l Académie des Scieces Mathématiques Paris Ser. I pp W. Pieczski Triplet Markov Chais ad Theor of Evidece Submitted to Iteratioal Joural of Approimate Reasoig September W. Pieczski Pairwise Markov chais IEEE Tras. o PAMI Vol. o pp S. Derrode ad W. Pieczski SAR image segmetatio usig geeralized Pairwise Markov Chais SPIE s Iteratioal Smposium o Remote Sesig September -7 Crete Greece 00.. S. Derrode ad W. Pieczski Sigal ad image restoratio usig Pairwise Markov Chais IEEE Tras. o Sigal Processig scheduled for September Giordaa ad W. Pieczski Estimatio of Geeralized Multisesor Hidde Markov Chais ad supervised Image Segmetatio IEEE Tras. o PAMI Vol. 9 o. pp R. W. Katz O some criteria for estimatig the order of a Markov chai Techometrics Y. M. Bishop S.E. Fieberg P. W Hollad Discrete Multivariate Aalsis MIT Press Cambridge 97.