Pairwise Markov Random Fields and its Application in Textured Images Segmentation

Transcription

1 Pairwie Markov Random Field and it Application in Textured Image Segmentation Wojciech Pieczynki and Abdel-Naer Tebbache Département Signal et Image Intitut National de Télécommunication, 9, rue Charle Fourier, Evry, France Abtract The ue of random field, which allow one to take into account the patial interaction among random variable in complex ytem, i a frequent tool in numerou problem of tatitical image proceing, like egmentation or edge detection. In tatitical image egmentation, the model i generally defined bhe probability ditribution of the cla field, which i aumed to be a Markov field, and the probability ditribution of the obervation field conditional to the cla field. In uch model the egmentation of textured image i difficult to perform and one ha to reort to ome model approximation. The originality of our contribution i to conider the markovianity of the pair (cla field, obervation field. We obtain a different model; in particular, the cla field i not necearily a Markov field. The model propoed make poible the ue of Bayeian method like MPM or MAP to egment textured image with no model approximation. In addition, the textured image can be corrupted with correlated noie. Some firt imulation to validate the model propoed are alo preented. 1. Introduction We propoe in thi paper a new Pairwie Markov Random Field model (PMRF, which i original with repect to the claical Hidden Markov Random Field model (HMRF [ChJ93]. The main difference i that in a PMRF the cla field i not necearily a Markov field. One advantage, which will be developed in the following, i that textured image can be egmented without any model approximation. To be more precie, let S be the et of pixel, X = (X S the random field of clae, and Y = (Y S the random field of obervation. The problem of egmentation i then the problem of etimating the realization of the field X from the obervation of the realization of the field Y. Prior to conidering antatitical egmentation method, one mut define the probability ditribution P (X,Y of the random field (X,Y. In the claical HMRF model thi ditribution i given by the ditribution of X, which i a Markov field, and the et X=x P Y of the ditribution of Y conditional to X = x. Under ome aumption on P X=x Y, the poterior Y=y ditribution P X of X i a Markov ditribution and different Bayeian egmentation technique like MPM [MMP87], MAP [GeG84], or ICM [Be86] can be applied. Thee aumption can turn out to be difficult to jutify when dealing with textured image, a detailed in [DeE87], [KDH88], [WoD9]. The idea of the PMRF model we propoe i to conider directlhe Markovianity of the pairwie random field Z = (X,Y. The ditribution of X i then the marginal ditribution of P (X,Y and thu it i not necearily a Markovian ditribution. What count i that P Y=y X remain Markovian and o different Bayeian egmentation pecified above can be ued. When the particular problem of egmenting textured image with correlated noie i conidered, the PMRF model i quite uitable becaue different Bayeian egmentation method above can be applied without any model approximation. The organization of the paper i following. In the next ection we pecifhe difference between HMRF and PMRF in the cae of imple Gauian noie field. Section 3 i devoted to ome imulation and ome concluding remark are preented in ection 4.. Pairwie Markov Random Field and textured image egmentation.1 Simple cae of Hidden Markov Field Let u conider the et of pixel S and X = (X S a random field, each random variable X taking it value in the cla et Ω={ω 1,ω }. The field X i Markovian with repect to four nearet neighbor if it ditribution i written

2 P[X = x] = λ exp ϕ 1 ϕ (,tneighbor (.1 where " (,t neighbor" mean that the pixel and t are neighbor and lie either on a common row or on a common column. The random field Y = (Y S i the field of obervation and we aume that each Y take it value in R. The ditribution of (X,Y i then defined by (.1 and the ditribution of Y conditional on X = x. Auming that the random variable (Y are independent conditionall to X and that the ditribution of each Y conditional to X = x i equal to it ditribution conditional to X = x, we have : P[Y = yx= x] = f x ( (. where f x i the denity of the ditribution of Y conditional to X = x. Thu : = λ exp[ ϕ 1 [ϕ + Logf x ( ]] (.3 (,tneighbor So the pairwie field (X,Y ditribution i Markovian and the ditribution of X conditional to Y = y i till Markovian. It i then poible to imulate realization of X according to it ditribution conditional to Y = y, which afford the ue of Bayeian egmentation technique like MPM or MAP. In practice, the random variable (Y are not, in general, independent conditionally on X. In particular, (. i too imple to allow one to take texture into account. For intance, if we conider that texture i a Gauian Markov random field realization [CrJ83], (. hould be replaced with : P[Y = yx= x] = = λ (x exp[ a x x t 1 (,tneighbor [a x x + b x ]] (.4 The field Y i then Markovian conditionally on X, which model texture. The drawback i that the product of (.1 with (.4 i not, in general, a Markov ditribution. In fact, for the covariance matrix Γ(x of the Gauian ditribution of Y = (Y S (conditional to X = x, we have : Markovianity invalidate the rigorou application of MPM or MAP.. Simple cae of Pairwie Markov Field To circumvent the difficultie above we propoe to conider the Markovianity of (X,Y. Specifically, we put = λ exp[ ϕ[,,(x t, ] ϕ *[, ]] = (,tneighbor = λ exp[ [ϕ 1 + a x x t + b x x t + (.6 (,tneighbor +c x x t ] [ϕ + a x x + b x ]] The Markovianity of the pairwie field (X,Y implie the Markovianity of Y conditionally on X, and the Markovianity of X conditionally on Y. The firt property allow one to model texture, a in (.4, and the econd one make poible to imulate X according to it poterior ditribution, which allow u to ue Bayeian egmentation method like MPM or MAP. Let u brieflpecify how to imulate realization of the pair (X,Y. The pair (X,Y being Markovian, we can pecifhe ditribution of each (X,Y conditionally on it neighbor. Let u conider the calculu of the ditribution of (X,Y conditional to the four nearet neighbor : [(X t1,y t1,(x t,y t,(x t3,y t3,(x t4,y t4 ] = [(x t1, 1,(x t,,(x t3, 3,(x t4, 4 ] Thi ditribution can be written a (.7 h, = p f x ( (.8 where p i a probability on the et of clae and, for each cla x, f x i the denity of the ditribution of Y conditional to X = x ( p and f x alo depend on (x t1, 1,(x t,,(x t3, 3,(x t4, 4, which are fixed in the following and o will be omitted. (.8 make the ampling of (X,Y quite eay: one ample x according to p, and then according to f x. We thu have: P{( X,Y =, λ (x = [(π N det(γ(x] 1/ (.5 which i not, in general, a Markov ditribution with repect to x. Finally, X i Markovian, Y i Markovian conditionally on X, but neither (X,Y, nor X conditionally on Y, are Markovian in general. Thi lack of the poterior [(X t1,y t1,...,(x t4,y t4 ] = [(x t1, 1,...,(x t4, 4 ]} exp[ ϕ[,,(x ti, i ] ϕ *[, ] = = exp[ [ϕ 1 i + a x x ti i + b x x ti + +c x x ti i ] [ϕ + a x x y + b x ]] (.9

3 Thi can be written h, = p f x (, where f x i a Gauian denity defined bhe following mean M x and variance σ x : M x = b x + (a x x ti i + b x x ti a x x, σ x = 1 a x x (.10 and p the probability given on the et of clae with : p (a x x 1 exp[ (b x + a x x ti i + b x x ti 4a x x ϕ (ϕ 1 i + c x x ti i ] (.11 Finally, the main difference between the claical HMRF model and the new PMRF we propoe are : (i The ditribution of X (it prior ditribution i Markovian in HMRF and i not necearily Markovian in PMRF; (ii The poterior ditribution of X i not necearily Markovian in HMRF and i Markovian in PMRF; (iii In the cae of cae of image which are textured, and poibly corrupted with correlated noie, the PMRF allow one to applhe Bayeian MPM or MAP method without any model approximation. Remark 1. The claical HMRF can alo be applied in the cae of multienor image [YaG95]. For m enor the obervation on each pixel S are then aumed to be a realization of a random vector Y = Y 1 m [,...,Y ]. It i poible to conider a multienor PRMF. For example, a multienor PMRF would be obtained by replacing in (.6 and by φ 1 [(y 1,...,y m,(y 1 t,...,y m t ] and φ [(y 1,...,y m ].. In ome particular cae the claical model allow one to take correlated noie into account [Guy93], [Lee98]. The difficultie arie when wihing to conider the cae when correlation vary with clae..3 General cae of PMRF Generalizing of the ditribution given by (.6 doe not poe any problem. Let u conider k clae Ω={ω 1,...,ω k }, m enor (each Y = (Y 1,...,Y m take it value in R m, and C a et of clique defined bome neighborhood ytem. The random field Z = (Z S, with Z = (X,Y, i a Pairwie Markov Random Field if it ditribution may be written a P[Z = z] = λ exp[ ϕ c (z c ] (.1 c C Let u note that the exitence of the ditribution (.1 i not enured in a general cae. In particular, three enor PMRF can be ued to egment colour image. Remark A mentioned above, X i not necearily Markovian in a PMRF. Thi could be felt a a drawback, becaue the ditribution of X model the "prior", i.e., without any obervation, knowledge we have about the cla image. Of coure, thi "prior" ditribution of X alo exit in PMRF (it i the marginal ditribution of P (X,Y, but it i not Markovian. The gravity of thi "drawback" i undoubtedly difficult to dicu in the general cae. However, uppoing that it really i a drawback, let u mention that it alo exit in the claical HMRF model. In fact, even in the verimple cae defined by (.3, the obervation field Y i not a Markov field. So, one could conider that the non Markovianity of X in the PMRF model i not tranger that the non Markovianity of Y in the claical HMRF model. 3. Viual example We preent in thi ection two imulation and two egmentation bhe MPM. We conider rather noiy cae : one can hardly ditinguih the cla image in the noiy one. One can notice that although the cla image are not Markov field realization, they look like. The two realization of PMRF preented in Fig.1 are Markovian with repect to four nearet neighbor; the ditribution of (X,Y i written : = λ exp[ ϕ[,,(x t, ] ϕ *[, ]] (3.1 (,tneighbor with ϕ[,,(x t, ] = 1 (a x x t + b x x t + c x x t + d x x t (3. ϕ *[, ] = 1 (α x y + β x + γ x x t The different coefficient in (3. are given in Tab.1. Let u notice that it i intereting, in order to have an idea about the noie level, to dipoe of ome information about the ditribution of Y conditional to X = x. In fact, thee are

4 Gauian ditribution and knowing ome parameter like mean and variance can provide ome information about the noie level. Of coure, the noie level alo depend on different correlation and the prior ditribution of X. We may note that ome of the coefficient in (3. are imply linked with mean or variance of the ditribution of Y conditional to X = x. In fact, denoting by Σ x the covariance matrix of the Gauian ditribution of Y conditional to X = x and putting Q x = [q x t ],t S =Σ 1 x, we have : P[Y = yx= x] exp (y m x t Q x (y m x (3.3 Developing (3.3 and identifying to (3. we obtain Image 3 Image 6 Fig. 1. Two realization of Pairwie Markov Field (Image 1, Image, (Image 4, Image 5, and the MPM egmentation of Image (giving Image 3, and Image 5 (giving Image 6, repectively. m x = β x, σ x = 1 (3.4 α x α x Image 1,, 3 Image 4, 5, 6 α x So, all other parameter being fixed, one can ue (3.4 to varhe noie level. For intance, keeping the ame variance the noie level increae when one make the mean approach each other. Otherwie, there are no imple link between correlation of the random variable (Y (conditionally on X = x and the coefficient in (3.. The correlation in Table 1, whoe variation make appear different texture, are etimated one. The value of the mean how that the level of the noie i rather trong, which i confirmed viually. Image 1 Image 4 Image Image 5 β x m x m x γ x x t m x m x a x x t 0, 4 0,1 b x x t 0, 4m xt 0,1m xt c x x t 0, 4m x 0,1m x d x x t 0, 4m x m xt + ϕ 0,1m x m xt + ϕ m 1 0,3 1 m 0,3 1,5 σ 1 σ ρ 11 0,6 0,05 ρ 0,6 0,07 τ 13,1% 07, 9% Nb Tab.1 α x,..., d x x t : function in (3., the function ϕ being defined by ϕ = 1 if x = x t and ϕ = 1 if x x t. m 1, m, σ 1, σ : the mean and the variance in (3.3. ρ 11, ρ : the etimated covariance inter-cla (neighboring pixel. τ : the error rate of wrongly claified pixel with MPM. Nb = n 1 n : the number of iteration in MPM (the poterior marginal etimated from n 1 realization, each realization obtained after n iteration of the Gibb Sampler. 4 Concluion We propoed in thi paper an novel model called Pairwie Markov Random Field (PMRF. A random field of clae X and a random field of obervation Y form a PMRF when the pairwie random field Z = (X,Y i a Markov

5 field. Such a model i different from the claical Hidden Markov Random Field (HMRF; in particular, in PMRF the random field X i not necearily a Markov field. The PMRF allow one to deal with the tatitical egmentation of textured image which can be, in addition, corrupted with correlated noie. On the contraro the ue of hierarchical model [DeE87], thi can be done in the framework of the model, without any approximation. Roughlpeaking, in the Hierarchical HMRF the prior ditribution of X i Markovian and it poterior ditribution i not Markovian; and in PMRF the prior ditribution of X i not Markovian and it poterior ditribution i Markovian. When uing a Bayeian method of egmentation like MPM or MAP we have to make ome approximation when uing Hierarchical HMRF, and we have not when uing PMRF. Furthermore, the ditribution of Y conditional to X, which model different texture and different poibly correlated noie, can be trictlhe ame in the both Hierarchical HMRF and PMRF model. We have preented two imulation of PMRF and two reult of the Bayeian MPM egmentation of the obervation field. The two cae preented are rather noiy and the reult how that the well known efficiency of the HMRF can alo occur when uing PMRF. A perpective for further work, let u mention two important point. Firt, the exitence of the ditribution given by (.1 i not enured in the general cae. Even in the imple Gauian cae given by (.6 we hould verify that all Gauian ditribution of Y conditionally on X exit. There exit ome condition of exitence of Gauian field [Guy95] and thu one poible way of verifying the exitence of PMRF could be the verification of the exiting condition "uniformly" with repect to X. The econd problem i the parameter etimation one. One could view applying the general Iterative Condition Etimation (ICE [Pie9], which give atifying reult in ome claical ituation [DMP97], [GiP97], [SaP97]. Uing ICE requet conidering an etimator from complete data (X,Y: one poible way of eeking uch an etimator could be conidering the tochatic gradient [You88], applied to (X,Y intead of X. Acknowledgment. We thank Alain Hillion, Directeur Scientifique de l'école Nationale Supérieure de Télécommunication de Bretagne, for numerou dicuion which greatly helped the writing of thi paper. 5. Reference [Be86] J. Beag, On the tatitical analyi of dirty picture, Journal of the Royal Statitical Society, Serie B, 48, pp , [ChJ93] R. Chellapa, A. Jain Ed., Markov Random Field, Theory and Application, Academic Pre, San Diego, [CrJ83] G. R. Cro and A. K. Jain, Markov Random Field Texture Model, IEEE Tran. on PAMI, Vol. 5, No. 1, pp. 5-39, [DMP97] Y. Delignon, A. Marzouki, and W. Pieczynki, Etimation of Generalized Mixture and It Application in Image Segmentation, IEEE Tranaction on Image Proceing, Vol. 6, No. 10, pp , [DeE87] H. Derin and H. Elliot, Modelling and egmentation of noiy and textured image uing Gibb random field, IEEE Tran. on PAMI, Vol. 9, No. 1, pp , [GeG84] S. Geman, G. Geman, Stochatic relaxation, Gibb ditribution and the Bayeian retoration of image, IEEE Tran. on PAMI, Vol. 6, No. 6, pp , [GiP97] N. Giordana and W. Pieczynki, Etimation of Generalized Multienor Hidden Markov Chain and Unupervied Image Segmentation, IEEE Tranaction on Pattern Analyi and Machine Intelligence, Vol. 19, No. 5, pp , [Guy95] X. Guyon, Random Field on Network. Modeling, Statiticz, and Application. Springer-Verlag, Probabily and it Application, New York, [KDH88] P. A. Kelly, H. Derin, and K.D. Hart, Adaptive egmentation of Speckled image uing a hierarchical random field model, IEEE Tran. on ASSP, Vol. 36, No 10, pp , [LaD89] S. Lakhmanan, H. Derin, Simultaneou parameter etimation and egmentation of Gibb random field, IEEE Tranaction on PAMI, Vol. 11, pp , [Lee98] T. C. M. Lee, Segmenting image corrupted by correlated noie, IEEE Tran. on PAMI, Vol. 0, No. 5, pp , [MMP87] J. Marroquin, S. Mitter, T. Poggio, Probabilitic olution of ill-poed problem in computational viion, Journal of the American Statitical Aociation, 8, pp , [Pie9] W. Pieczynki, Statitical image egmentation, Machine Graphic and Viion, Vol. 1, No. 1/, pp , 199. [SaP97] F. Salzentein and W. Pieczynki, Parameter Etimation in Hidden Fuzzy Markov Random Field and Image Segmentation, Graphical Model and Image Proceing, Vol. 59, No. 4, pp. 05-0, [WoD9] C. S. Won and H. Derin, Unupervied egmentation of noiy and textured image uing Markov random field, CVGIP: Graphical Model and Image Proceing, Vol. 54, No 4, 199, pp [YaG95] T. Yamazaki and D. Gingra, Image claification uing pectral and patial information baed on MRF model, IEEE Tran. on IP, Vol. 4, No. 9, pp , [You88] L. Youne, Etimation and annealing for Gibbian field, Annale de l'intitut Henri Poincaré, Vol. 4, No., 1988, pp