Applying EM Algorithm for Segmentation of Textured Images
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1 Proceedngs of the World Congress on Engneerng 2007 Vol I Applyng EM Algorthm for Segmentaton of Textured Images Dr. K Revathy, Dept. of Computer Scence, Unversty of Kerala, Inda Roshn V. S., ER&DCI Insttute of Technology, CDAC-T, Inda Abstract Texture analyss plays an ncreasngly mportant role n computer vson. Snce the textural propertes of mages appear to carry useful nformaton for dscrmnaton purposes, t s mportant to develop sgnfcant features for texture. Varous texture feature extracton methods nclude those based on gray-level values, transforms, auto correlaton etc. We have chosen the Gray Level Co occurrence Matrx (GLCM) method for extracton of feature values. Image segmentaton s another mportant problem and occurs frequently n many mage processng applcatons. Although, a number of algorthms exst for ths purpose, methods that use the Expectaton-Maxmzaton (EM) algorthm are ganng a growng nterest. The man feature of ths algorthm s that t s capable of estmatng the parameters of mxture dstrbuton. Ths paper presents a novel unsupervsed segmentaton method based on EM algorthm n whch the analyss s appled on vector data rather than the gray level value. Index Terms Texture, Segmentaton, GLCM, EM, Bayes ITRODUCTIO Image segmentaton s a fundamental tas n machne vson and occurs very frequently n many mage-processng applcatons. Texture based segmentaton provdes an mportant cue to the recognton of obects. Texture s one of the vsual features playng an mportant role n scene analyss. Intutvely, t s related to patterned varatons of ntensty across an mage. Texture segmentaton n general s composed of two steps, namely, the extracton of texture based features and secondly the groupng of these features. There are a number of methods for texture feature extracton. These methods can be classfed nto feature-based methods, model-based methods, and structure-based methods. Structure-based methods partton mages under the assumpton that the textures n the mage have detectable prmtve elements, arranged accordng to placement rules. In feature-based methods, regons wth relatvely constant texture characterstcs are sought. Model-based methods hypothesze underlyng processes for textures and segments usng certan parameters of these processes. Model based methods can be consdered as a subclass of feature-based methods snce model parameters are used as texture features. Ths paper descrbes an unsupervsed segmentaton technque wth maxmum-lelhood parameter estmaton problem for parameter estmaton and the Expectaton-Maxmzaton problem for ts soluton. The man feature of ths algorthm s that t s capable of estmatng the parameters of mxed dstrbuton. Texture measures are derved usng the gray-level co occurrence matrces. The EM algorthm s appled to estmate the mean and varance of features for every texture n the mage. At last a Bayesan classfcaton rule s appled to attrbute a label for each pxel by defnng a lelhood functon, whch computes the probablty for a gven pxel as belongng to a gven class. Texture analyss has been used n a range of studes for recognzng synthetc and natural textures. It s very useful n the analyss of aeral mages, bomedcal mages and sesmc mages as well as the automaton of ndustral applcatons. I. IMAGE TEXTURE Texture s an mportant property of many types of mages. Whle there sn t a unversally agreed defnton of what a texture s, features of a texture nclude roughness, granulaton and regularty. Vsual texture contans varatons of ntenstes, whch form certan repeated patterns. Those patterns can be caused by physcal surface propertes, such as roughness, or they could result from reflectance dfferences, such as the color on a surface. Texture s an mportant part of the vsual world of anmals and men and ther vsual systems successfully detect, dscrmnate and segment texture. A class of smple mage propertes that can be used for texture analyss are the frst-order statstcs of local property values,.e., the mean, varance, etc. In partcular, a class of local propertes based on absolute dfferences between pars of gray levels or average gray levels has performed well; Usually dfferent nds of measures are derved from dfference hstograms, such as contrast, angular second moment, entropy, mean, and nverse dfference moment. There s a large need to classfy mages based on textural features n varous felds le scene analyss, medcal mages analyss, etc. Varous texture analyss systems were proposed over the years. Three dfferent types of texture analyss are popular: segmentaton, classfcaton and synthess. Texture segmentaton deals wth detectng the texture boundares n an ISB: WCE 2007
2 Proceedngs of the World Congress on Engneerng 2007 Vol I Fgure : Images consstng of dfferent textured regons mage to obtan a boundary map. Texture classfcaton deals wth the recognton of mage regons usng texture propertes. Each regon n an mage s assgned a texture class. The goal of texture synthess s to extract three-dmensonal nformaton from texture propertes. II. IMAGE SEGMETATIO A central problem, called segmentaton, s to dstngush obects from bacground. An mage conssts of a number of natural obects defned by dstnct regons. The mage bacground n tself s a dstnct regon. These regons are defned by ther boundares that separate them from other regons. Image segmentaton ams at dentfyng these boundares and as to whch pxel comes from whch regon. The segmentaton problem can be nformally descrbed as the tas of parttonng an mage nto homogeneous regons. For textured mages one of the man conceptual dffcultes s the defnton of a homogenety measure n mathematcal terms. A texture method s a process that can be appled to a pxel of a gven mage n order to generate a measure (feature) related to the texture pattern to whch that pxel and ts neghbors belong. The performance of the dfferent famles of texture methods bascally depends on the type of processng they apply, the neghborhood of pxels over whch they are evaluated (evaluaton wndow) and the texture content. Texture methods used can be categorzed as: statstcal, geometrcal, structural, model-based and sgnal processng features []. Van Gool et al. [2] and Reed and Buf [3] present a detaled survey of the varous texture methods used n mage analyss studes. Randen and Husoy [4] conclude that most studes deal wth statstcal, model-based and sgnal processng technques. Wesza et al. [5] compared the Fourer spectrum; second order gray level statstcs, co-occurrence statstcs and gray level run length statstcs and found the co-occurrence were the best. Smlarly, Ohanan and Dubes [6] compare Marov Random Feld parameters, mult-channel flterng features, fractal based features and co-occurrence matrces features, and the co-occurrence method performed the best. The same concluson was also drawn by Conners and Harlow [7] when comparng run-length dfference, gray level dfference densty and power spectrum. Buf et al. [8] however report that several texture features have roughly the same performance when evaluatng co-occurrence features, fractal dmenson, transform and flter ban features, number of gray level extrema per unt area and curvlnear ntegraton features. Compared to flterng features [9], co occurrence based features were found better as reported by Strand and Taxt [0], however, some other studes have supported exactly the reverse. Pchler et al. [] compare wavelet transforms wth adaptve Gabor flterng feature extracton and report superor results usng Gabor technque. However, the computatonal requrements are much larger than needed for wavelet transform, and n certan applcatons accuracy may be compromsed for a faster algorthm. Oala et al. [2] compared a range of texture methods usng nearest neghbour classfers ncludng gray level dfference method, Law's measures, center-symmetrc covarance measures and local bnary patterns applyng them to Brodatz mages. The best performance was acheved for the gray level dfference method. Law's measures are crtczed for not beng rotatonally nvarant, for whch reason other methods performed better. III. PROPOSED METHOD Ths secton descrbes n detal the proposed technque for feature extracton and classfcaton. The overall bloc dagram of texture feature extracton and classfcaton and hence segmentaton of the nput mage s presented n Fgure 2. The proposed technque s dvded nto three stages. Stage deals wth the quantzaton and feature extracton from the texture mages, stage 2 deals wth the estmaton of the Gaussan parameters by applyng the EM algorthm and fnally stage 3 mplements the labelng process and thereby the segmentaton of the texture regons. A. Feature Extracton The feature extracton algorthms analyze the spatal dstrbuton of pxels n grey scale mages. The dfferent methods capture how coarse or fne a texture s. The textural character of an mage depends on the spatal sze of texture prmtves. Large prmtves gve rse to coarse texture and small prmtves fne texture. To model these characterstcs, spatal methods are found to be superor to spectral methods. The most commonly used texture measures are those derved from the Grey Level Co-occurrence Matrx (GLCM). Haralc suggested the use of grey level co-occurrence matrces (GLCM) to extract second order statstcs from an mage. GLCMs have been used very successfully for texture segmentaton. The GLCM s a tabulaton of how often dfferent combnatons of pxel brghtness values (grey levels) occur n an mage. Fgure 2. Bloc Dagram ISB: WCE 2007
3 Proceedngs of the World Congress on Engneerng 2007 Vol I Haralc defned the GLCM as a matrx of frequences at whch two pxels, separated by a certan vector, occur n the mage. The dstrbuton n the matrx wll depend on the angular and dstance relatonshp between pxels. Varyng the vector used allows the capturng of dfferent texture characterstcs. Once the GLCM has been created, varous features can be computed from t. These have been classfed nto four groups: vsual texture characterstcs, statstcs, nformaton theory and nformaton measures of correlaton. To reduce the computatonal tme requred for extractng features for 256 pxel values the nput mage s quantzed before applyng the feature extracton process. Quantzaton can be done ether to the pxel values or to the spatal coordnates. Operaton on pxel values s referred to as gray-level reducton and operatng on the spatal coordnates s called spatal reducton. Texture feature extracton s performed on the quantzed mage by usng Gray level co-occurrence matrx (GLCM) method, one of the most nown texture analyss method whch estmates mage propertes related to second-order statstcs. A GLCM or SGLD matrx s the ont probablty occurrence of gray levels and for two pxels wth a defned spatal relatonshp n an mage. The spatal relatonshp s defned n terms of dstance d and angle θ. If the texture s coarse and dstance d s small compared to the sze of the texture elements, the pars of ponts at dstance d should have smlar gray levels. Conversely, for a fne texture, f dstance d s comparable to the texture sze, then the gray levels of ponts separated by dstance d should often be qute dfferent, so that the values n the SGLD matrx should be spread out relatvely unformly. From SGLD matrces, a varety of features may be extracted. From each matrx, 4 statstcal measures are extracted ncludng: angular second moment, contrast, correlaton, varance, nverse dfferent moment, sum average, sum varance, sum entropy, dfference varance, dfference entropy, nformaton measure of correlaton I, nformaton measure of correlaton II, and maxmal correlaton coeffcent. The measurements average the feature values n all four drectons. The results may be combned by averagng the GLCM for each angle before calculatng the features or by averagng the features calculated for each GLCM. To reduce the computatonal complexty, only some of the features were selected. In our experments gray level co occurrence matrces were calculated for each pxel usng a neghborhood of sze 8 x 8 at angles of 0, 45, 90, 35 etc. Algorthm for creatng a symmetrcal normalzed GLCM. Create a framewor matrx 2. Decde on the spatal relaton between the reference and neghbour pxel 3. Count the occurrences and fll n the framewor matrx 4. Add the matrx to ts transpose to mae t symmetrcal 5. ormalze the matrx to turn t nto probabltes 6. Change the value of angle and offset to get matrces n the other drectons. B. Features calculated from a normalzed GLCM based on textures Varance: -, (),=0 Contrast= P ( ) -,,=0 (2) Dssmlarty= P P Homogenety= (3) -, P, 2,=0 +(-) -, 2 (4),=0 Energy (P ) -, P, (5),=0 Entropy = P ( ln ) Mean: µ = P ( ), = 0 (, ) µ = P ( ) (6), = 0 (, ) = P, ( ),=0 (7) σ µ = P, ( ),=0 (8) σ µ Correlaton = ( µ )( µ ) - P, (9) 2 2,=0 σσ We had calculated the features from the co occurrence matrces obtaned. The average of the features for each case gves the feature value for the partcular case. Typcal values obtaned for one of the test mages s shown n Table. C. Maxmum Lelhood Estmaton We have a densty functon p (x θ) that s governed by the set of parameters θ (e.g., p mght be a set of Gaussans and θ could be the means and covarances). ISB: WCE 2007
4 Proceedngs of the World Congress on Engneerng 2007 Vol I the unnown data Y gven the observed data X and the current parameter estmates. That s, we defne: (-) ( ) Q( θθ, ) = E[log p( X, Y θ) X, θ ] Fgure 3: Maxmum-lelhood estmaton and relatve-frequency estmaton We also have a data set of sze, supposedly drawn from ths dstrbuton,.e., X = {x,., x }. That s, we assume that these data vectors are ndependent and dentcally dstrbuted wth dstrbuton p. Therefore, the resultng densty for the samples s p( X / θ) = p( x θ) = L( θ X) = Ths functon L (θ X) s called the lelhood of the parameters gven the data, or ust the lelhood functon. The lelhood s thought of as a functon of the parameter θ where the data X s fxed. In the maxmum lelhood problem, our goal s to fnd the θ that maxmzes L. That s, we wsh to fnd θ * where * θ = arg max L( θ X) D. Expectaton-Maxmzaton (EM) algorthm The EM algorthm s a general method of fndng the maxmum-lelhood estmate of the parameters of an underlyng dstrbuton from a gven data set when the data s ncomplete or has mssng values. There are two man applcatons of the EM algorthm. The frst occurs when the data ndeed has mssng values, due to problems wth or lmtatons of the observaton process. The second occurs when optmzng the lelhood functon s analytcally ntractable but when the lelhood functon can be smplfed by assumng the exstence of and values for addtonal but mssng (or hdden) parameters. The latter applcaton s more common n the computatonal pattern recognton communty [3,4]. Let X be the observed data generated by some dstrbuton and suppose a complete data set exsts Z = ( X,Y ). We also have a ont densty functon p ( z θ ) = p ( x,y θ ) = p ( y x, θ ) p(x θ ) We can defne a lelhood functon L ( θ Z ) = L ( θ X,Y ) = p ( X, Y θ ), called the complete data lelhood. The orgnal lelhood L (θ X) s referred to as the ncomplete data lelhood functon. The EM algorthm frst fnds the expected value of the complete data log lelhood log p ( X, Y θ ) wth respect to θ ( ) s where θ the current parameter estmate used to evaluate the expectaton and θ s the new parameter optmzed to ncrease Q. The evaluaton of ths expectaton s called the E-step of the algorthm. The second step of the EM algorthm s to maxmze the expectaton computed n the frst step. θ = arg max Q( θ, θ ) () ( ) θ These two steps are repeated as necessary. Each teraton s guaranteed to ncrease the log- lelhood and the algorthm s guaranteed to converge to a local maxmum of the lelhood functon. In the case of a Gaussan dstrbuton, the teratve EM algorthm for densty functon parameter estmaton s gven by: ω ( x ) = L π + = π π ( x \ θ ) f = f ( x \ θ ( )) l = / ω ( x ) µ ω + ( ) = ( + ) π = ( x )( x ) [( σ ) ] ( )[ ] where ω 2 ( + ) ( ) ( + ) 2 = ω ( ) x x + µ π = ( ) x s an ntermedate functon. A valdaton procedure s also added to ad the EM process. The procedure s termnated f the dfference between the estmated parameters of two consecutve EM steps s nferor to a fxed threshold ε. Fgure 4: Input and output of the EM algorthm ISB: WCE 2007
5 Proceedngs of the World Congress on Engneerng 2007 Vol I where P( x ω ) s the probablty densty functon of class ω n the feature space and P( ω ) s the apror probablty, whch tells the probablty of the class before measurng any features. If pror probabltes are not actually nown, they can be estmated by the class proportons n the tranng set. The dvsor Fgure 5: Procedure of the EM algorthm Algorthm. Intalze the value for the mxture parameters 2. Calculate the ntermedate functon for the gven set of ntal values 3. Calculate the new mxture parameters by usng the ntermedate functon and the prevous mxture values. 4. Contnue the calculaton untl the threshold or valdaton condton s met. 5. Repeat for each class of texture n the mage. E. Bayesan Classfcaton Classfcaton s a basc tas n data analyss and pattern recognton that requres the constructon of a classfer that s a functon that assgns a class label to nstances descrbed by a set of attrbutes. The classfer taes an unlabeled example and assgns t to a class. umerous approaches to ths problem are based on varous functonal representatons such as decson trees, decson lsts, neural networs, decson graphs, and rules. One of the most effectve classfers, n the sense that ts predctve performance s compettve wth state-of-the-art classfers, s the Bayesan classfers. Ths classfer learns from tranng data the condtonal probablty of each attrbute A gven the class label C. Classfcaton s then done by applyng Bayes rule to compute the probablty of C gven the partcular nstance of A..A n, and then predctng the class wth the hghest posteror probablty. So Bayesan classfcaton and decson mang s based on probablty theory and the prncple of choosng the most probable or the lowest rs (expected cost) opton. Assume that there s a classfcaton tas to classfy feature vectors (samples) to K dfferent classes. A feature vector s denoted as x = [x, x 2,..,x D ] T where D s the dmenson of a vector. The probablty that a feature vector x belongs to class ω s P( ω x). The classfcaton of the vector s done accordng to posteror probabltes or decson rss calculated from the probabltes. The posteror probabltes can be computed wth the Bayes formula Px ( ω) P( ω) P( ω x) = Px ( ) K Px ( ) = Px ( ω) P( ω) = s merely a scalng factor to assure that posteror probabltes are really probabltes,.e., ther sum s one. After estmatng the Gaussan parameters that correspond to each regon a labelsaton process s requred n order to attrbute a label to each pxel. Ths s carred out by usng the Bayesan classfcaton method. Algorthm. Calculate the probablty for each pxel value n the feature vector 2. ormalze the value by multplyng weght and dvdng by the sum total 3. Fnd the product of the probablty matrx for dfferent texture features 4. Calculate the probablty for each pxel to belong to a partcular class 5. Assgn label to each class. IV. RESULTS The method has been tested on a number of 256x256 grey level mages from the Brodatz textures. Usng GLCM method wth an offset of and for 4 dfferent angles the dfferent texture features were extracted. The parameters of the Gaussan mxture model were calculated for each texture n the mage usng EM algorthm. A lelhood functon s calculated whch gves the probablty of a pxel as belongng to a partcular class whch forms the bass of labelng of the pxel. For the test mage taen, the amount of local varatons n the mage s consderable as ndcated by the hgh contrast value obtaned. It also ndcates that most elements do not le on the dagonal of the GLCM. The dfference of each element from other elements of the co-occurrence matrx s ndcated by the dssmlarty feature. Homogenety s measure for unformty of TABLE I: TYPICAL VALUES FOR THE TEXTURE FEATURES OBTAIED FOR THE SAMPLE TEST IMAGE Features Value obtaned Contrast Dssmlarty Homogenety Energy Entropy 2.09 ISB: WCE 2007
6 Proceedngs of the World Congress on Engneerng 2007 Vol I Fgure 6. Test Image Fgure 6.3 Energy feature Fgure 6.2 Dssmlarty feature Fgure 6.4 Homogenety feature [5] J.S. Wesza, C. R. Dyer and A. Rosenfeld, "A comparatve study of texture measures for terran classfcaton", IEEE Transactons on Systems, man and Cybernetcs, vol. 6, pp , 976. [6] P.P. Ohanan and R.C. Dubes, "Performance evaluaton for four class of texture features", Pattern Recognton, vol. 25, no. 8, pp , 992. [7] R.W. Conners and C.A. Harlow, "A theoretcal comparson of texture algorthms", IEEE Transactons on Pattern Analyss and Machne Intellgence, vol. 2, no. 3, pp , 980. [8] J.M.H. Buf, M. Kardan and M. Spann, "Texture feature performance for mage segmentaton", Pattern Recognton, vol. 23, no. 3/4, pp , 990. [9] T. Randen and J.H. Husoy, "Flterng for texture classfcaton: A comparatve study", IEEE Transactons on Pattern Analyss and Machne Intellgence, vol. 2, no. 4, pp , 999. [0] J. Strand and T. Taxt, "Local frequency features for texture classfcaton", Pattern Recognton, vol. 27, no. 0, pp , 994. [] O. Pchler, A. Teuner and B.J. Hostca, "A comparson of texture feature extracton usng adaptve Gabor flter, pyramdal and tree structured wavelet transforms", Pattern Recognton, vol. 29, no. 5, pp , 996. [2] T. Oala, M. Petanen, "A comparatve study of texture measures wth classfcaton based on feature dstrbutons, Pattern Recognton, vol. 29, no., pp. 5-59, 996. [3] Jeff A. Blmes - A gentle tutoral of the EM algorthm and ts applcaton to parameter estmaton for Gaussan mxture and hdden Marov models. [4] Fran Dellaert - The Expectaton Maxmsaton algorthm Fgure 6.5 Entropy feature Fgure 6.6 Segmented Image co-occurrence matrx, and f most elements le on the man dagonal, ts value wll be large, compared to other case. Energy s the measure of the textural unformty of the mage. It reaches a hgh value when gray level dstrbuton has ether a constant or a perodc form. Entropy measures randomness and t wll be very large when all elements of the co- occurrence matrx are same. When the mage s not texturally unform, the GLCM elements wll have small values and therefore entropy s large. The above values can be represented pctorally as shown above. The segmented mage shows reasonably good accuracy. Accuracy can be mproved by usng addtonal features from the GLCM. REFERECES [] M. Tuceyran and A.K. Jan, "Texture analyss", Handboo of Pattern Recognton and Computer Vson, C.H. Chen, L.F. Pau and P.S.P. Wang (Eds.), chapter 2, pp , World Scentfc, Sngapore, 993. [2] L. vangool, P. Dewaele and A. Oosterlnc, "Texture analyss", Computer Vson, Graphcs and Image Processng, vol. 29, pp , 985. [3] T. R. Reed and J.M.H. Buf, "A revew of recent texture segmentaton and feature extracton technques", Computer Vson, Image Processng and Graphcs, vol. 57, no. 3, pp , 993. [4] T. Randen and J.H. Husoy, "Flterng for texture classfcaton: A comparatve study", IEEE Transactons on Pattern Analyss and Machne Intellgence, vol. 2, no. 4, pp. 29 ISB: WCE 2007
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