Elitist Reconstruction Genetic Algorithm Based on Markov Random Field for Magnetic Resonance Image Segmentation

Transcription

1 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 10, NO. 1, MARCH Eltst Reconstructon Genetc Algorthm Based on Markov Random Feld for Magnetc Resonance Image Segmentaton Xn-Yu Du, Yong-Je L, Cheng Luo, and De-Zhong Yao Abstract In ths paper, eltst reconstructon genetc algorthm (ERGA) based on Markov random feld (MRF) s ntroduced for mage segmentaton. In ths algorthm, a populaton of possble solutons s mantaned at every generaton, and for each soluton a ftness value s calculated accordng to a ftness functon, whch s constructed based on the MRF potental functon accordng to Metropols functon and Bayesan framework. After the mproved selecton, crossover and mutaton, an eltst ndvdual s restructured based on the strategy of restructurng eltst. Ths procedure s processed to select the locaton that denotes the largest MRF potental functon value n the same locaton of all ndvduals. The algorthm s stopped when the change of ftness functons between two sequent generatons s less than a specfed value. Experments show that the performance of the hybrd algorthm s better than that of some tradtonal algorthms. Index Terms Eltst reconstructon, genetc algorthm, mage segmentaton, Markov random feld. 1. Introducton In the computer vson feld, there are many algorthms usng statstcal technques for modelng and processng mage data. In these works, Markov random feld (MRF) s often modeled to mages segmentaton problems [1],[2]. The am of mage segmentaton s to partte an nput mage nto some smoothng and non-overlappng regons and each regon s homogeneous, connected and smoothng. Therefore, t s mportant to consder the dependence among mage pxels. And the dependence can be descrbed perfectly by the MRF model [3]. The detaled ntroducton of MRF s presented n L s works [1]. In the MRF model, the mage segmentaton problem s converted to the problem of fndng an optmal nstantaton of the label feld gven the Manuscrpt receved October 18, 2011; revsed January 8, X.-Y. Du, Y.-J. L, C. Luo, and D.-Z. Yao are wth the School of Lfe Scence and Technology, Unversty of Electronc Scence and Technology of Chna, Chengdu , Chna (e-mal: duxnyu126@126.com; lyj@uestc.edu.cn; hl7roch@yahoo.com.cn; dyao@uestc.edu.cn). Dgtal Object Identfer: /j.ssn X mage pxels. There have been already a lot of lteratures about how to embed natural computaton algorthms nto MRF models. For example, n the work of Wang et al. [4], an evolutonary algorthm was mplemented wth a knd of MRF as a pror n segmentaton. Ther results showed that the algorthm worked well n texture segmentaton and natural mages segmentaton. Tseng et al. adopted a genetc algorthm (GA) for the MRF-based segmentaton [5]. Bhandarkar used the smulated-genetc (SA-GA) algorthm for mage segmentaton, but the algorthm s not based on MRF [6]. In one of our early works, we proposed a knd of SA-GA algorthm based on MRF for mage segmentaton [7]. In ths paper, based on our early works about MR mages segmentaton [7],[8], we propose an eltst reconstructon genetc algorthm (ERGA) wth MRF pror to segment MR mages. Based on GA and Metropols rules, ths algorthm s processed to select the locaton whch denotes the largest MRF potental functon value at the same locaton n all ndvduals to restructure an eltst ndvdual after an mproved selecton, crossover, and mutaton n every generaton. The algorthm can be stopped when the change of ftness functons between two successve generatons s less than a threshold, so that unnecessary teratve would be avoded. The eltst of the last generaton denotes the labeled mage as the segmentaton result. In addton, MRF potental functon values are stretched so as to avod premature when usng the roulette wheel selecton n early generatons and accelerate convergence n later generatons. The crossover rate and mutaton rate are also declned exponentally wth the evoluton number ncreasng. In the codng stage, we use two dmensonal (2D) chromosome codng scheme. As for the populaton ntalzaton, we dsturb the threshold to generate dfferent ndvduals. In the recombnaton and mutaton stage, we combne the MRF clque wth 2-dmenson wndows n an ndvdual. Smlar to many lteratures about MRF, the ftness functon s adopted as the MRF potental functon. The rest of the paper s organzed as follows. Secton 2 descrbes the ERGA and ts applcaton to MRF-based mages segmentaton. In Secton 3 we present the experments results, and fnally a dscusson and concluson s drawn n Secton 4.

2 84 2. ERGA for MRF 2.1 Image Segmentaton Based on MRF As we know, mage segmentaton s the task to assgn labels to every pxel n the mage. Specally, the nput mage s y = ( y, I), where s the locaton of a pxel y n the mage and I s the locaton set. The segmentaton result s denoted as x = ( x, I). Obvously, x = k ndcates that the class label k s assgned to the pxel ste. We assume that the number of class s K and each class s represented by a label k K. Therefore, mage segmentaton would be transformed to optmze the potental functons of MRF models. A MRF potental functon can be presented as [9] U ( x = k y, x, j G \ ) = j 1 y μ k, 2 σ k, 2 + log( σ ) + φ ( x) (1) k, c,, j c, j,, j C where μ k, and σ k, represent the mean mage ntensty and the standard devaton of the nose of the class k at ste, respectvely. The local energy at ste s U ( ) and G \j ndcates the neghborhood G, excludng the ste. y s the value of the pxel at ste. C s the Markov clque confguraton and the 3 3 regon wth the center s defaulted. Accordng to mult-level logstc model (MLL), two pont potental functons on the knd of clque C are JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 10, NO. 1, MARCH 2012 operatons at every generaton (detals n secton B), the locaton, whch denotes the largest MRF potental functon value n the same locaton of all ndvduals, s selected and an eltst ndvdual could be restructured by ths means. The algorthm can be stopped when the change of ftness functons between two sequent generatons s less than a specfed value ε, so that unnecessary teraton would be avoded and the eltst of the last generaton denotes the labeled mage as the segmentaton result. In GA, selecton, crossover and mutaton can cause the genetc nformaton of the best ndvdual dsappearng. Here the eltst ndvdual could be restructured wthout nfluencng selecton, crossover, and mutaton operatons. The dfference between restructure eltst strategy and roulette wheel selecton s that the restructure eltst strategy s to create a best ndvdual of the current generaton but roulette wheel selecton s to elmnate some worst ones. Therefore, the restructured eltst strategy can control generatons through adjustng to decrease unnecessary teratve and the best segmentaton of current generaton can be constructed when the algorthm needs to accelerate convergence. B. Flow of ERGA The flowchart of ERGA s shown n Fg. 1. The detals of the ERGA are presented below. 1) In codng stage, 2-dmensonal ndvdual codng scheme s adopted and the alleles k Λ = {1, 2,, K} and the space of alleles s K Intalzaton I. φ φ ( x, x ) = β, x = x c,, j j j ( x, x ) =+ β, x x (2) c,, j j j where β s the Gbbs parameter and has a postve value. The meanng of β s the penalty of dfferent pxels n the neghborhood of the wanted pxel. It s also the potental measurement between the center and ts neghbors. Obvously, the object energy functon s U( x) = U ( x). (3) The problem of the segmentaton s expressed as the mnmzaton of the energy functon. In ths work, t s equalzed to mnmze (1) nstead of (3). Ths task s usually a hard optmzaton problem because the number of possble label confguratons s generally very large and moreover, the energy functon (1) may contan local mnma. In the followng, ths paper shows a novel GA for the mnmzaton of the energy functon (1). 2.2 ERGA A. Strategy of Restructurng Eltst After the mproved selecton, crossover, and mutaton Fuzzy clusterng Intalzng populatons Updatng accordng metropols functon at T Stretchng MRF U Roulette wheel selecton Recombnaton Mutaton Eltst restructo U n U n 1 < ε Yes Declnng temperature Segmentaton end Fg. 1. Flowchart of ERGA. Temperature not low enough stretchng MRF U

3 DU et al.: Eltst Reconstructon Genetc Algorthm Based on Markov Random Feld for Magnetc Resonance Image Segmentaton 85 2) The declnng temperature table and the ended temperature T e control the generatons f the dfference of last two sequent eltsts ftness s less than ε. The declnng temperature table s gven by c0 T = ln( c + n) 1 where c 0 and c 1 are constants and n s the current generaton. 3) Durng the ntalzaton of populaton, the threshold s dsturbed n order to generate dfferent ndvduals (4) T = T rand T con (5) k k k where con s a small swng value and rand s a random value between 0 and 1. In ths algorthm, a fuzzy clusterng algorthm s adopted as the ntal processng to determne the ntal value of μ k, and σ k,. So the T k of (5) can be consdered as the membershp of the center of the kth class. 4) The new soluton s determned by Metropols rule, whch s shown as where 1, U( x ) < U( x) p = U( x ) U( x) exp, U( x ) U( x) T x (6) s the soluton of the prevous generaton (teratons) that mnmzes (1). The probablty of x s replaced by x whch s determned by the value computed by (6). 5) The ftness functon s stretched, so that premature can be avoded usng roulette wheel selecton n earler stage of evoluton, and contrast can be ncreased n later stage. U () e ( x) = e ( ) U ( x)/ T M U ( ) ( x )/ T = 1 where U () s the ftness value of the th chromosome computed by (3) and M s the populaton sze. T s the current temperature. U () s the stretched ftness functon U (). 6) The crossover rate P cross and mutaton P mute rate are declned exponentally wth the evoluton: ( ( n 1)/(Gen 1)) P = e P cross P = e P cross (7) ( ( n 1)/(Gen 1)) (8) mute mute where Gen s the total teraton, and n s the current teraton number. When the crossover rate and the mutaton rate are larger n the frst several generatons, the ERGA can get larger soluton space so that the ERGA can get faster convergence. When the crossover rate and the mutaton rate are smaller n the last several generatons, the ERGA can get better clmbng ablty and can segment more precsely. 7) Restructurng the eltst of the current generaton. 8) When the change of ftness functon values between two sequent generatons s less than ε, the algorthm s stopped and goes to 10, else goes to 4. 9) If the current temperature s below T e, the evoluton should be ended. 10) The segmented mage s constructed by the eltst ndvdual of the last generaton. 3. Experments Two groups of experments are presented. There are not true segmentaton results for real data. To valdate the advantages of ERGA, synthetcal data wth manual groud-truth are used n the frst group of experments. Then real data are adopted to show that the ERGA can be mplemented well to segment real MR mages wth bas feld effect [10] n the second group. 3.1 Synthetcal Data The synthetcal MR mages are processed by the tradtonal optmzaton algorthms, such as teratve condtonal model (ICM), smulated annealng (SA), GA, and ERGA shown n ths artcle, respectvely. The parameters of all algorthms are the same and presented n Table 1. To denose and segment an mage n mult-part (>2) smultaneously are challengng. The volume data s from the web ste: mcgll.ca/branweb/, and the orgnal mage (Fg. 2 ) we used s the 72nd bran slce of ths data. Fg. 2 s the mage wth Gaussan noses 5%. Fg. 3 (e) are the denosng and segmented mages usng ICM (50 teratve cycles), SA (50 teratve cycles), SA (1500 teratve cycles), GA (50 teratve cycles), and ERGA (50 teratve cycles). And the average error rates of those algorthms to mplement 10 tmes to the mages wth Gaussan noses from 5%, 7% and 9% are shown n Table 2. The consumng tme of ICM_50 and SA_50 s about 15 s, and SA_1500 and GA_50 s consumng tme s about 4 mn. The ERGA_50 s consumng tme s less than 4 mn as ts teratve cycles are determned by ε. The algorthms are executed n the condton of Inter 2.8G, 512M DDR, and Vsual C Table 1: Parameters of all algorthms ICM SA Iteratve N 50 T e 0.25 or MLL β 0.1 C C N 50 or 1500 MLL β 0.1 GA ERGA Populaton sze P m 20 T e 0.25 Crossover P c 0.5 C Mutaton P m C Generaton N 50 P m 20 MLL β 0.1 P c 0.5 P m End value ε MLL β 0.1

4 86 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 10, NO. 1, MARCH 2012 Table 2: Average error rates of those algorthms to mplement 10 tmes to the mage wth noses 5%, 7%, and 9% Noses ICM_50 SA_50 SA_1500 GA_50 ERGA 5% % % Fg. 2. The 72nd bran slce: orgnal mage of the 72 nd slce and mage wth noses 5%. (c) (d) (e) Fg. 3. Segmentaton results: denosng and segmented mages usng ICM (50 teratve cycles), denosng and segmented mages usng SA (50 teratve cycles), (c) denosng and segmented mages usng SA (1500 teratve cycles), (d) denosng and segmented mages usng GA (50 teratve cycles), and (e) denosng and segmented mages usng ERGA (50 teratve cycles). 3.2 Real Data The real data are obtaned from Huax Hosptal, Chengdu, Chna. The whole volumes are , 16 bts, and analyze format (Fg. 4 ). We choose the 100th slce as the orgnal mage (Fg. 4 ). For real MR mage segmentaton, bas feld effect must also be consdered. Bas feld effects can cause MR mage s gray values unevenness. The fuzzy clusterng algorthm (Fg. 4 (c)) and ERGA (Fg. 4 (d)) are mplemented to extract the whte matter of ths (c) (d) Fg. 4. Real data and ts segmentaton result: 3-D real data, 100th slce, (c) segmentaton result by fuzzy clusterng, and (d) ERGA result. slce. As there s no groud-truth for real MR mage s segmentaton, the crteron adopted n ths work s human s (often doctor s) subjectve evaluaton. The parameters of ERGA algorthm are the same as n Secton 3.1. Obvously, the segmentaton result of ERGA s better than the result of the fuzzy clusterng algorthm. 4. Dscusson and Conclusons In general mage processng experments, the mages for experments are natural mages that look much dfferent each other. In ths paper, however, our algorthm s tested on human bran MR mages that possess nner smlartes. Therefore, t s reasonable that one representatve slce s selected to process n the volume of MR mages. Certanly, the selected slces have the most complcated structures n the volumes and they are typcal mages for mage segmentaton. In the frst group of experments, as presented n Table 2, ERGA obtans the lowest classfcaton error rates and shows better results than ICM, SA, and GA n the condton of noses. The reason s that fuzzy clusterng used as pre-segmentaton procedure of all those methods can not set the ntal soluton to approach the best soluton n a nosy mage. ICM uses greedy strategy sngly to determne new solutons so that t can not reach the global optmzaton n lmted steps because of some local solutons. SA can accept worse solutons probably to clmb hlls and reach the global solutons, but t needs much more teratve cycles. GA s crossover and mutaton may destroy a better soluton n ntal steps because GA also adopts greedy strategy to determne new chromosomes after crossover and mutaton. Not only does

5 DU et al.: Eltst Reconstructon Genetc Algorthm Based on Markov Random Feld for Magnetc Resonance Image Segmentaton 87 ERGA use Metropols rule to accept worse solutons n a knd of varatonal probablty so as to jump out local solutons, but also ERGA makes the most superor propertes reserved n the restructured eltst of the current generaton act as the coheson n the selecton operaton of the next generaton so that segmentaton precson s ncreased n lmted steps. From the second group of experments, we can see that ERGA can perform better than fuzzy clusterng n the case of real MR mages wth bas feld effects. Acknowledgment The authors would lke to thank Ph.D. Yu-Qng Wang for the preparaton of real MR data. References [1] S.-Z. L, Markov Random Feld Modelng n Image Analyss, 3rd ed. London: Sprnger, 2009, pp [2] H.-X. Xu, G.-X. Zhu, J.-W. Tan, et al., Image segmentaton based on support vector machne, Journal of Electronc Scence and Technology of Chna, vol. 3, no. 3, , [3] S. Geman and D. Geman, Stochastc relaxaton, Gbbs dstrbutons, and the Bayesan restoraton of mages, IEEE Trans. on Pattern Analyss and Machne Intellgence, vol. 6, no. 6, pp , [4] X. Wang and H. Wang, Evolutonary optmzaton wth Markov random feld pror, IEEE Trans. on Evolutonary. Computaton, vol. 8, no. 6, pp , [5] D. Tseng and C. La, A genetc algorthm for MRF-based segmentaton of mult-spectral textured mages, Pattern. Recogn. Lett., vol. 20, no. 14, pp , [6] S. Bhandarkar and H. Zhang, Image segmentaton usng evolutonary computaton, IEEE Trans. on Evolutonary. Computaton, vol. 3, no. 1, pp. 1 21, [7] X. Du, Y. L, W. Chen, et al., A Markov random feld based hybrd algorthm wth smulated annealng and genetc algorthm for mage segmentaton, n Proc. of the Second Internatonal Conf. on Advances n Natural Computaton, X an, 2006, do: / _95. [8] X. Du, Y. L, and D. Yao, A support vector machne based algorthm for magnetc resonance mage segmentaton, n Proc. of the 4th Natural Computaton, Jnan, 2008, pp [9] J. Rajapakse, J. Gedd, and J. Rapoport, Statstcal approach to segmentaton of sngle-channel cerebral MR mages, IEEE Trans. on Medcal Imagng, vol. 16, no. 2, pp , [10] J. Luo, Y. Zhu, P. Clarysse, and I. Magnn, Correcton of bas feld n MR mages usng sngularty functon analyss, IEEE Trans. on Medcal Imagng, vol. 24, no. 8, pp, , Xn-Yu Du was born n Gansu Provnce, Chna, n He receved the B.E. degree and M.E. degree from the Unversty of Electronc Scence and Technology of Chna (UESTC), Chengdu, n 1999 and 2007, both n bomedcal engneerng. He s currently pursung the Ph.D. degree wth the School of Lfe Scence and Technology, UESTC. Hs research nterest s mage processng. Yong-Je L was born n Hebe, Chna, n He receved Ph.D. n bomedcal engneerng from UESTC n He s now a professor wth the School of Lfe Scence and Technology, UESTC. Hs research nterest s vsual nformaton processng. Cheng Luo was born n Schuan Provnce, Chna, n He receved the B.M. degree from North Schuan Medcal College, Nanchong, n 1999, M.E. and Ph.D. degrees from UESTC, n 2006 and 2010, respectvely. Hs research nterests nclude smultaneous EEG and fmri nvestgaton n eplepsy. De-Zhong Yao receved the Ph.D. degree n appled geophyscs from Chengdu Unversty of Technology, Chengdu, Chna, n 1991, and completed the postdoctoral fellowshp n electromagnetc feld from UESTC, Chengdu, n He has been a Faculty Member wth UESTC snce 1993, a professor snce 1995, and the Dean of the School of Lfe Scence and Technology, UESTC, snce He was a vstng scholar at Unversty of Illnos at Chcago, IL, from September 1997 to August 1998, and a vstng professor at McMaster Unversty, Hamlton, ON, Canada, from November 2000 to May 2001, and at Aalborg Unversty, Aalborg, Denmark, from November 2003 to February He s the author or coauthor of more than 80 papers and 5 books. Hs current nterests nclude EEG, fmri and ther applcatons n cogntve scence and neurologcal problems.