Nonlocal Mumford-Shah Model for Image Segmentation

Transcription

1 for Image Segmentaton 1 College of Informaton Engneerng, Qngdao Unversty, Qngdao, ,Chna E-mal:ccluxaoq@163.com ebo e 23 College of Informaton Engneerng, Qngdao Unversty, Qngdao, ,Chna E-mal: njustwwb@163.com Zhenkuan Pan College of Informaton Engneerng, Qngdao Unversty, Qngdao, ,Chna E-mal: zkpan@qdu.edu.cn Jnmng Duan School of Computer Scence, Unversty of Nottngham, Nottngham, NG7 2RD,UK E-mal: duanmujnmng@126.com Guodong ang College of Informaton Engneerng, Qngdao Unversty, Qngdao, ,Chna E-mal: allen_wgd@163.com Mumford-Shah (MS) model s a popular model among mage segmentaton area. A multphase nonlocal Mumford-Shah (NLMS) model whch combnes nonlocal operators and orgnal MS model s proposed n ths paper to segment multphase mages. In order to segment dfferent patterns smultaneously, multple regon partton strategy whch uses n-1 level set functons to segment n regons s adopted. Furthermore, correspondng splt Bregman algorthm s desgned to mprove computatonal effcency. Fnally, numercal experments are shown to valdate the effcency and effectveness of the proposed model. CENet September 2015 Shangha, Chna 1 Speaker 2 Correspongdng Author 3 The work s supported by the Natonal Natural Scence Foundaton of Chna( ). Copyrght owned by the author(s) under the terms of the Creatve Commons Attrbuton-NonCommercal-NoDervatves 4.0 Internatonal Lcense (CC BY-NC-ND 4.0).

2 1. Introducton Image segmentaton s a technology to segment the mage nto regons wth ther own characterstcs and extractng targets wth nterest. It s the fundamental problem of mage processng and analyss[1,2], computer vson and machne recognton. Accordng to dfferent nformaton contaned n the mage, the actve contour model whch has been wdely used n segmentaton can be dvded nto two types: models based on edges and models based on regons. Models based on the edge such as snake model [3], can get deal results n the process of segmentng mages wth obvous edges[4]. However, the contours are often not accurate enough n the process of segmentng mages wth weak edges. To solve the problems, researchers proposed the Mumford-Shah (MS) model based on regons[5]. Ths model equals orgnal mages to pecewse smooth mages and mnmal contours by mnmzng the energy functonal. However, due to the nconsstency n the dmenson of mage and contours, the problem can not be solved. In order to make t computable, Chan and Vese used the level set method to segment the pecewse constant mage[6-8], whch was the Chan-Vese (CV) model. To segment nhomogeneous objects n the mage, Vese and Chan mproved CV model by mnmzng the two-phase pecewse smooth MS energy[9]. In terms of texture mage segmentaton, Sandberg and Chan adopted Gabor transform to convert the orgnal texture mage nto vector mages, and then appled CV model to each layer of vector mages[10]. However, due to the exstence of some redundant and ncorrect layer n vector mages, t leads to wrong segmentaton results. In 2005, Buades et al frst proposed nonlocal method[11]. Ths method can keep the texture well. In 2008, Glboa and Osher defned a new seres of nonlocal operators[12]. Bresson and Chan combned the pecewse smooth Mumford-Shah model wth nonlocal operators to segment two-phase texture mages[9,13]. Ther approaches overcomes the defects [10]. And the approaches get better results. In ths paper, a multphase nonlocal Mumford-Shah model whch combnes nonlocal operators and multple regon partton strategy s proposed for mage segmentaton[13]. The former can cope wth texture nformaton n the mage, whle the latter can uses n-1 label functons to segment n regons. The splt Bregman method s adopted to transform the energy mnmzaton problem nto some subproblems and then solve them respectvely. 2. Two-Phase Nonlocal Mumford-Shah Model and Nonlocal Operators Frst, we wll revew the two-phase nonlocal Mumford-Shah (NLMS) model. For an mage f ( x) : R, x, the mnmzaton problem of NLMS model s as follows (2.1) where u 1 and u 2 s the gray value nsde and outsde of the closed contour denoted by label functon f respectvely. 1, 1, 2, l 2 and g are the penalty parameters balancng these three energy terms. u NL s the gradent operator defned n the nonlocal space. In order to solve (2.1), we need the nonlocal operators. Based on nonlocal means, Glboa and Osher systematcally defned nonlocal gradent operator, dvergence operator and Laplacan etc[12]. Defne u( x) : R be the grayscale mage whch s defned on mage space, nonlocal smlarty of x : x and y : y can be defned as (2.2) where G s s the Gaussan kernel functon, h s the thresholdng value controllng smlarty, s s the standard devaton of Gaussan kernel functon. th the defnton of nonlocal smlarty, at pont x, nonlocal gradent vector operator s shown as 2

3 At pont x, the nonlocal gradent module value s as follows (2.3) The nonlocal dvergence at pont x s (2.4) Then the nonlocal Laplace at pont x s (2.6) Model (2.1) can only segment two-phase mages and the optmsaton method (.e. gradent decent flow) used to solve label functon f s dffcult to dscretse to solve. Therefore, t s a nontrval problem proposng multphase segmentaton model. 3. Varatonal Level Set Method and Regon Partton Strategy for Multphase Image Segmentaton 3.1 Varatonal Level Set Method for Multphase Segmentaton Multphase mage segmentaton s usng a pluralty of label functons to dvde an mage nto some mutually adjacent non-overlappng areas accordng to the dfferent characterstc. Assumng f s a gray mage defned on a rectangular area. By ntroducng q label functons, the mage can be classfed nto p areas. The varatonal level set method for multphase segmentaton s gven as follows: (3.1) here c ( f ) s the characterstc functon and p q. Q ( u ) denotes the correspondng parameter estmatng functon for dfferent types of mages., = 1,2, L, p satsfes Characterstc functon c ( f ), = 1, 2, L, p satsfes c ( f ) = 1 x 0 x, 1,2,, p p c ( f ) 1 = 1 (2.5) (3.2) = L (3.3) = (3.4) In order to guarantee that t cannot produce the overlappng or mssng segmentaton results, (3.1) must meet constrant (3.4). Addtonally, n order to let label functons keep the characterstc of sgned dstance functons n the process of evoluton, (3.1) must satsfy (3.5). (3.5) 3.2 Segment n regons wth n-1 label functons hen usng n-1 label functons to segment n regons [14-16], we can learn p = n, q = n - 1. It s (3.6) 3

4 Then constrant (3.4) becomes to n = 1 c ( f ) = 1 (3.7) Fgure 1: Segmentng fve regons wth four label functons Here, we adopted the regon partton strategy proposed by Pan et al [16]. The scheme of characterstc functons can be seen n Fg. 1. Here, the correspondng characterstc functons are 1: c1 ( f ) = H ( f1 )( 1-2: c2 ( f ) = H ( f2 )( 1- H ( f1 ( 1-3: c3 ( f ) = H ( f3 )( 1- H ( f2 ( 1- H ( f1 ( 1- (3.8) 4: c4 ( f ) = H ( f4 )( 1- H ( f3 ( 1- H ( f2 ( 1- H ( f1 ( 1-5: c5 ( f ) = H ( f5 )( 1- H ( f4 ( 1- H ( f3 ( 1- H ( f2 ( 1- H ( f1 ( 1- Then, the general expresson of characterstc functons n Fg. 1 s -1 ( ) c ( f) = H ( f ) 1 - H ( f ) = 1, 2, L, n (3.9) j j= 0 here H ( f 0) = 0 and H ( f n) = 1. Ths scheme also satsfy constrant (3.7). 4. NLMS Model for Multphase Segmentaton and Splt Bregman Algorthm ( ) Here, the multphase nonlocal Mumford-Shah model s proposed as e can solve t wth alternatng optmzaton approach. Frst, fxng f for u, we can get Thus, the detaled form of u can be wrtten as ( ) ( ) (, ) ( y x ) w( x, y) dy ( ) c ( f ) + l c ( f ( + c ( f ( c ( f ) + l c ( f ( + c ( f ( f x y x u y w x y dy u x = = 1,2, L, n Then, we fx u to solve f. Here, splt Bregman algorthm s adopted to solve varable f to mprove the computatonal effcency. By ntroducng auxlary varable parameter b, (4.1) can be transformed nto followng form (4.1) (4.2) (4.3) w and Bregman teratve To mnmze (4.4), we frst fx varable w for f (4.4) 4

5 (4.5) where (4.5) can be solved by one teraton of Gauss-Sedel. Then varable f can be solved effcently by projecton formula f = Max Mn ( ( f,1),0) After obtanng f w, the followng soft thresholdng equaton s used to solve wth the conventon that 0 0 = Numercal Experments In ths secton, we apply ths model on synthetc and real mages to verfy the performance of the proposed model. e would compare our model wth the two-phase Chan-Vese model [17]. In fgure 2 and fgure 3, we apply our model on three-phase mage segmentaton (n=2) and four-phase mage segmentaton (n=3). It can be seen that the segmentaton performance s satsfactory. Here, we lst last few rows to better demonstrate the fnal label functons. It s obvous that those label functons are very close to the bnary mages (0 and 1). Contours gradually approach edges of objects accordng to dfferent nformaton. The parameters for experment shown n the frst column of fgure 2 s a = 1, l = 1, g = 2000, q = Those for the second column s a = 1, l = 1, g = 5000, q = And those for fgure 3 s a = 1, l = 1, g = 2000, q = (4.6) (4.7) (4.8) Fgure 2 : Three-phase mages segmentaton. Frst row: ntalzaton; Second row: segmentaton results; Thrd and fourth row: two label functons f 1 and f 2 ; Ffth row: segmentaton results by two-phase Chan-Vese model 5

6 Fgure 3: Four-phase mages segmentaton. Frst mage: ntalzaton; Second mage: segmentaton results; Thrd, fourth and ffth mages: three label functons f 1, f 2 and f3 Fgure 4 s the multphase segmentaton results (n=2) for real bran MR mage. It shows that proposed model and algorthm can stll effcently segment bran MR mages, even though the gray values of edges and area nsde bran are very smlar. The parameters for ths experment s a = 1, l = 1, g = 5000, q = Fgure 5 s the multphase segmentaton results (n=2) for CT mages. It can be seen that proposed nonlocal Munford-Shah model and algorthm can segment CT mages whch contan many tssues effcently. It s worth mentonng that the regon partton strategy cannot produce the overlappng or mssng segmentaton results as t can automatcally satsfy the unque partton condton (.e., one pxel pont can only belong to one segmented regon). The parameters for ths experment s a = 1, l = 1, g = 6000, q = 1e4. Fgure 4 : Multphase segmentaton results for pecewse smooth bran MR mages. Frst column: dfferent orgnal mages; Second column: segmentaton results; Thrd column: segmentaton results by two-phase Chan-Vese model 6

7 Fgure 5 : Multphase segmentaton results for pecewse constant CT mages. Frst column: the contour ntalzaton for dfferent mages; Second column: segmentaton results; Last column: segmentaton of the lver 6. Concluson In ths paper. nonlocal Mumford-Shah model for multphase mage segmentaton s proposed. Takng advantage of orgnal Mumford-Shah model, ths model combnes nonlocal operators wth multple regon partton strategy whch uses n-1 level set functons to segment n regons. So t has deal performance n multphase mage segmentaton. Splt Bregman algorthm s desgned to ncrease the computatonal effcency. The dea n the paper can be extended to other mathematcal problems n mage processng. References [1] G. Aubert, P. Kornprobst. Mathematcal Problems n Image Processng: Partal Dfferental Equatons and the Calculus of Varatons. In: AMS, vol Sprnger, Berln (2002) [2] F. T. Chan, J. Shen. Image processng and analyss: varatonal, PDE. avelet, and Stochastc methods. SIAM, Phladelpha (2005) [3] M. Kass, A. trw, D. Terzopoulos. Snakes: Actve contour models. Internatonal Journal of Computer Vson. 1(4): (1988) [4] V. Caselles, R. Kmmel, G. Sapro. Geodesc actve contour. IJCV. 22(1): 61-79(1997) [5] D. Mumford, J. Shah. Optmal approxmatons of pecewse smooth functons and assocated varatonal problems. Comm. Pur. Appl. Math. 42(5): (1989) [6] T. F. Chan, L. A. Vese. Actve Contours thout Edges. IEEE T. Image Process. 10(2): (2001) [7] S. Osher, N. Paragos. Geometrc Level Set Methods n Imagng, Vson, and Graphcs. Sprnger- Verlag, New York (2003) [8] S. Osher, R. Fedkw. Level Set Methods and Dynamc Implct Surfaces. Sprnger-Verlag, New York 1-288(2002) 7

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