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A Bayesian framework for image segmentation with spatially varying mixtures

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A Bayesian Framework for Image SegmentationWith Spatially Varying Mixtures
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IEEE Transactions on Image Processing · April 2010
DOI: 10.1109/TIP.2010.2047903 · Source: PubMed
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Christophoros NikouUniversity of Ioannina
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Aristidis LikasUniversity of Ioannina
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Nikolaos P. GalatsanosIllinois Institute of Technology
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2278 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 9, SEPTEMBER 2010
A Bayesian Framework for Image Segmentation WithSpatially Varying Mixtures
Christophoros Nikou
, Member, IEEE
, Aristidis C. Likas
, Senior Member, IEEE
, andNikolaos P. Galatsanos
, Senior Member, IEEE
Abstract—
A new Bayesian model is proposed for image seg-mentation based upon Gaussian mixture models (GMM) withspatial smoothness constraints. This model exploits the Dirichletcompound multinomial (DCM) probability density to model themixing proportions (i.e., the probabilities of class labels) and aGauss–Markov random ﬁeld (MRF) on the Dirichlet parametersto impose smoothness.The main advantagesof this model are two.First, it explicitly models the mixing proportions as probabilityvectors and simultaneously imposes spatial smoothness. Second,it results in closed form parameter updates using a maximum
a posteriori
(MAP) expectation-maximization (EM) algorithm. Pre-vious efforts on this problem used models that did not model themixing proportions explicitly as probability vectors or could notbe solved exactly requiring either time consuming Markov ChainMonte Carlo (MCMC) or inexact variational approximationmethods. Numerical experiments are presented that demonstratethe superiority of the proposed model for image segmentationcompared to other GMM-based approaches. The model is alsosuccessfully compared to state of the art image segmentationmethods in clustering both natural images and images degradedby noise.
IndexTerms—
Bayesianmodel,Dirichletcompoundmultinomialdistribution,Gauss–Markovrandomﬁeldprior,Gaussianmixture,image segmentation, spatially varying ﬁnite mixture model.
I. I
NTRODUCTION
M
ANY approaches have been proposed to solve theimage segmentation problem [1], [2]. Among them,clustering based methods rely on arranging data into groupshaving common characteristics [3], [4]. During the last decade,the main research directions in the relevant literature are fo-cused on graph theoretic approaches [5]–[8], methods based
upon the mean shift algorithm [9], [10] and rate distortion
theory techniques [11], [12].
Modeling the probability density function (pdf) of pixelattributes (e.g., intensity, texture) with ﬁnite mixture models(FMM) [13]–[15] is a natural way to cluster data because it
Manuscript received July 17, 2008; revised September 04, 2009; accepted-March 09, 2010. First published April 08, 2010; current version published Au-gust 18, 2010. The associate editor coordinating the review of this manuscriptand approving it for publication was Dr. Eero P. Simoncelli.C. Nikou and A. C. Likas are with the Department of Computer Science,University of Ioannina, 45110 Ioannina, Greece (e-mail: cnikou@cs.uoi.grarly@cs.uoi.gr).N. P. Galatsanos is with the Department of Electrical and Computer Engi-neering, University of Patras, 26500 Rio, Greece (e-mail: ngalatsanos@upa-tras.gr).Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/TIP.2010.2047903
automatically provides a grouping based upon the componentsof the mixture that generated them. Furthermore, the likelihoodof a FMM is a rigorous metric for clustering performance[14]. FMM based pdf modeling has been used successfully ina number of applications ranging from bioinformatics [16] toimage retrieval [17]. The parameters of the FMM model withGaussian components can be estimated through maximum like-lihood (ML) estimation using the Expectation-Maximization(EM) algorithm [13], [14], [18]. However, it is well-knownthat the EM algorithm ﬁnds, in general, a local maximum of the likelihood. Furthermore, it can be shown that Gaussiancomponents allow efﬁcient representation of a large variety of pdf. Thus, Gaussian mixture models (GMM), are commonlyemployed in image segmentation tasks [14].A drawback of the standard ML approach for image segmen-tation is that commonality of location is not taken into accountwhen grouping the data. In other words, the prior knowledgethat adjacent pixels most likely belong to the same cluster is notused. To overcome this shortcoming, spatial smoothness con-straints have been imposed.Imposing spatial smoothness is key to certain image pro-cessing applications since it is an important
a priori
knownproperty of images [19]. Examples of such applications includedenoising, restoration, inpainting and segmentation problems.In a probabilistic framework, smoothness is expressed througha prior imposed on image features. A common approach is theuse of an MRF. Many MRF variants have been proposed, seefor example [20]. However, determination of the amount of the
imposed smoothness automatically requires knowledge of thenormalization constant of the MRF. Since this is not knownanalytically, learning strategies were proposed [21]–[23].
Research efforts in imposing spatial smoothness for imagesegmentation can be grouped into two categories. In themethods of the ﬁrst category, spatial smoothness is imposed onthe discrete
hidden
variables of the FMM that represent classlabels, see for example [7], [24]–[26]. These approaches may
be categorized in a more general area involving simultaneousimage recovery and segmentation which is better known as
image modeling
[27]–[30]. More speciﬁcally, spatial regular-
ization is achieved by imposing a discrete Markov random ﬁeld(DMRF) on the classiﬁcation labels of neighboring pixels thatpenalizes solutions where neighboring pixels belong to dif-ferent classes. Another method in this category is proposed in[7] which is based upon the optimization of an energy functionhaving a term for the quality of the clustering and a term forthe spatial tightness. Minimization of the energy function isaccomplished using graph cuts [31].
1057-7149/$26.00 © 2010 IEEE
NIKOU
et al.
: A BAYESIAN FRAMEWORK FOR IMAGE SEGMENTATION WITH SPATIALLY VARYING MIXTURES 2279
The Gaussian scale mixtures (GSM) and their extension of mixtures of projected GSM (MPGSM) and the ﬁelds of GSM(FoGSM) were also used in image denoising in the wavelet do-main in [32]–[34]. In GSM denoising [32], clusters of wavelet
coefﬁcients are modeled as the product of a Gaussian randomvector and a positive scaling variable. In MPGS denoising [34],the model is extended to handle different local image character-istics and incorporates dimensionality reduction through linearprojections. By these means, the number of model parametersis reduced and fast model training is obtained. In the case of FoGSM [33], multiscale subbands are modeled by a productof an exponentiated homogeneous Gaussian Markov randomﬁeld (hGMRF) and a second independent hGMRF. In [33], itis demonstrated that samples drawn from a FoGSM model havemarginal and joint statistics similar to subband coefﬁcients of photographic images.To estimate the smoothness parameters, Woolrich
et al.
pro-posed in [35] and [36] a model based upon a logistic transform
that approximates the previously mentioned DMRF with a
con-tinuous
GaussianMarkovrandomﬁeld.However,forthismodelinference of the
contextual mixing proportions
(posterior classlabel probabilities) of each pixel cannot be obtained in closedform. Thus, in [35], inference based upon Markov Chain MonteCarlo (MCMC) is proposed, while in [36] inference based uponVariationalBayes(VB)isemployed.AlthoughMCMCmethodshave been studied in statistics for a long time and several gen-eral criteria have been proposed to determine their convergence[37], [38], inference based upon them may be notoriously time
consuming. On the other hand, VB-based inference is approxi-mate and there is no easy way to assert the tightness of the vari-ational bound. Moreover, similar in spirit approaches to avoidlocal maxima of the likelihood, which is a drawback of the MLsolution, rely on the stochastic EM and its variants [39], [40].In the second category of methods, the MRF-based smooth-nessconstraintisnotimposedonthelabelsbutonthecontextualmixing proportions. This model is called spatially variant ﬁnitemixture model (SVFMM) [41] and avoids the inference prob-lemsofDMRFs.Inthismodelmaximum
aposteriori
(MAP)es-timation of the contextual mixing proportions via the MAP-EMalgorithm is possible. However, the main disadvantage of thismodel is that the M-step of the proposed algorithm cannot beobtained in closed form and is formulated as a constrained opti-mization problem that requires a projection of the solution ontothe unit simplex (positive and summing up to one components)[41], [42]. Consequently, the parameters that control the spatial
smoothness cannot be estimated automatically from the data.In [43], a new family of smoothness priors was pro-posed for the contextual mixing proportions based upon theGauss–Markov random ﬁelds that takes into account clusterstatistics, thus, enforcing different smoothness strength foreach cluster. The model was also reﬁned to capture informationin different spatial directions. Moreover, all the parameterscontrolling the degree of smoothness for each cluster, as wellas the label probabilities for the pixels, are estimated in closedform via the maximum
a posteriori
(MAP) methodology. Theadvantage of this family of models is that inference is obtainedusing an EM algorithm with closed form update equations.However, the implied model still does not take into accountexplicitly that the mixing proportions are probabilities, thus,the constraint that they are positive and must sum to one is notguaranteed by the update equations. As a result, the M-stepof this EM algorithm also requires a
reparatory
projectionstep which is ad-hoc and not an implicit part of the assumedBayesian model. A synergy between this category of priors andline processes, to account for edge preservation, was presentedin [44].In this paper, we present a new hierarchical Bayesian modelfor mixture model-based image segmentation with spatial con-straints. This model assumes the
contextual mixing proportions
to follow a Dirichlet compound multinomial (DCM) distribu-tion. More precisely, the class to which a pixel belongs is mod-eled by a discrete multinomial distribution whose parametersfollow a Dirichlet law [45]. Furthermore, spatial smoothness isimposed by assuming a Gauss–Markov random ﬁeld (GMRF)prior for the parameters of the Dirichlet. The parameters of themultinomial distribution are integrated out in a fully Bayesianframework and the updates of the parameters of the Dirichletare computed in closed form through the EM algorithm.The Dirichlet distribution has been previously proposed asa prior for text categorization [46], [47], object recognition
and detection [48] and scene classiﬁcation [49]. The differ-
ence of the proposed model with respect to existing methodsis twofold. At ﬁrst, text, scene or object categorization aresupervised learning problems while the proposed segmentationmethod is unsupervised. Also, in the existing studies, estima-tion of the parameters of the Dirichlet distribution is generallyaccomplished by variational inference or by simpliﬁed logisticmodels. The advantage of the herein proposed model is that,not only the E-step can be expressed in closed form, but alsoour model explicitly assumes that the contextual mixing pro-portions are probability vectors. Inference through the EMalgorithm leads to a third degree polynomial equation for theparameters of the Dirichlet distribution. Therefore, the closedform M-step yields parameter values automatically satisfyingthe necessary probability constraints.Another approach to handle non stationary images and re-lying on MRF is the triplet Markov ﬁeld (TMF) model [50]which was also applied to image segmentation [51], [52]. The
main difference of TMF with respect to our model is that, inTMF, the random ﬁeld is imposed jointly on the hidden vari-ables, the observation and a set of auxiliary variables which de-terminethetypeofthestationarity.Incontrast,inourmodel,therandom ﬁeld is imposed on the
contextual mixing proportions
.Numerical experiments are presented to assess the perfor-mance of the proposed model both with simulated data wherethe ground truth is known and real natural images where theperformance is assessed both visually and quantitatively.The remainder of the manuscript is organized as follows:background for the spatially variant ﬁnite mixture model isgiven in Section II. The proposed modeling of probabilitiesof the pixel labels with a DCM distribution is presented inSection III. In Section IV, the MAP-EM algorithm for theestimation of the proposed model parameters is developed.Experimental results of the application of our model to naturaland artiﬁcial images are presented in Section V and conclusionsand directions for future research are given in Section VI.
NIKOU
et al.
: A BAYESIAN FRAMEWORK FOR IMAGE SEGMENTATION WITH SPATIALLY VARYING MIXTURES 2281
Fig. 1. Graphical model for the spatially variant ﬁnite mixture model(SVFMM).
It is easily veriﬁed that (9) has always a real nonnegativesolution for . However, the main drawback of the SVFMMis that it imposes spatial smoothness on without explicitlytaking into account that it is a probability vector ( ,, ). For this purpose,
reparatory
computations were introduced in the M-step to enforce the vari-ables to satisfy these constraints. A gradient projection al-gorithm was used in [41] and quadratic programming was pro-posed in [42]. This approach was shown to improve both thecriterion function (7) and the performance of the model. How-ever,
reparatory
projections compromise the assumed Bayesianmodel.III. D
IRICHLET
C
OMPOUND
M
ULTINOMIAL
M
ODELING OF
C
ONTEXTUAL
M
IXING
P
ROPORTIONS
To overcome the limitations of SVFMM, we propose in thissection,anewBayesianmodelformixture-basedtheimageseg-mentation problem based upon a hierarchical prior for the thecontextual mixing proportions , which are assumed to followa DCM distribution. The DCM distribution is a multinomialwhose parameters are generated by a Dirichlet distribution[45],thus are probability vectors.Similarinspirit priorshavebeenproposed, in the totally different context of text modeling [47]where the DCM parameters are estimated through an iterativegradient descent optimization method. Also in a recent work [53], a new family of exponential distributions is proposed ca-pable of approximating the DCM probability law in order tomake its parameter estimation faster than [47]. In what follows,we describe how to compute them in closed form. Furthermore,spatialsmoothnessisimposedontheparametersoftheDirichletdistributionswhicharecomputedinclosedformthroughacubicequation having always one real non negative solution that sat-isﬁes the constraints of the Dirichlet parameters.
A. Dirichlet Compound Multinomial Distribution
More precisely, for the th pixel, , theclass label is considered to be a random variable fol-lowing a multinomial distribution with probability vectorwith being the number of classes.Let also to be the set of pa-rameters for the whole image. By the multinomial deﬁnition itholds that(10)with(11)The model described by (10) represents the probabilitythat pixel belongs to class , as one of the possible out-comes of a multinomial process with realizations. Each of the outcomes of the process appears with probability ,. Generally speaking, this is a generative modelfor the image. When the multinomial distribution is used togenerate a clustered image, the distribution of the number of emissions(i.e.,counts)ofan individualclassfollowsa binomiallaw.The DCM distribution assumes that parameters of the multinomial follow a Dirichlet distribution param-eterized by where, , is the vector of Dirichlet parameters for(12)where , , and is theGamma function.Under the Bayesian framework, the probability label for theth image pixel is obtained by marginalizing the parameters(13)Substituting (10) and (12) into (13) we obtain , with some easymanipulation, the following expression for the label probabili-ties:(14)with .
B. Hierarchical Image Model
We now assume a generative model for the image where thedetermination of component generating the th pixel is anoutcome of a DCM process with only one realization. Conse-quently, the vector of the hidden variables (6) has the th

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