A Region based Active Contour Approach for Liver CT Image Analysis Driven by Local likelihood Image Fitting Energy

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1 Internatonal Journal of Engneerng and Advanced Technology (IJEAT) ISSN: , Volume-6 Issue-5, June 07 A Regon based Actve Contour Approach for Lver CT Image Analyss Drven by Local lkelhood Image Fttng Energy Abstract: Computer tomography mages are wdely used n the dagnoss of lver tumor analyss because of ts faster acquston and compatblty wth most lfe support devces. Accurate mage segmentaton s very senstve n the feld of medcal mage analyss. Actve contours plays an mportant role n the area of medcal mage analyss. It consttute a powerful energy mnmzaton crtera for mage segmentaton. Ths paper presents a regon based actve contour model for lver CT mage segmentaton based on varatonal level set formulaton drven by local lkelhood mage fttng energy. The negh bourng ntenstes of mage pxels are descrbed n terms of Gaussan dstrbuton. The mean and varances of ntenstes n the energy functonal can be estmated durng the energy mnmzaton process. The updaton of mean and varance gude the contour evolvng toward tumor boundares. Also ths model has been compared wth dfferent actve actve contour models. Our results shows that the presented model acheves superor performance n CT lver mage segmentaton. Index Terms: Actve Contours, Chan-Vese model, Level sets I. INTRODUCTION Actve Contours plays an mportant role n the feld of medcal mage segmentaton. Lver tumor segmentaton n CT mages s a key problem n medcal mage processng scenaro. The goal s to exact separaton of the tumor regons from the background organs norder to vsualze and analyze the physcans to predct the bengn and malgnancy condtons.actve contour model proposed by Kass et al. [], has been proved to be an effcent framework for mage segmentaton. These models can be formulated under an energy mnmzaton framework based on the theory of curve evoluton. The solutons to these models can be obtaned by usng the level set method [], whose basc dea s to represent a contour or surface as the zero level set of an mplct functon defned n a hgher dmenson, and to formulate the moton of the contour or surface as the evoluton of the level set functon. The man advantage of ths method s to handle the topologcal changes automatcally. Actve contour models can be generally categorzed nto two groups : the edge-based models[3-8] and regon-based models [9-8]. Edge-based actve contour models utlze local mage gradents to attract contours toward object boundares, whereas regon-based models employ global mage nformaton n each regon, such as the dstrbuton of ntenstes, colors, textures and moton, Revsed Verson Manuscrpt Receved on May 03, 07. Sajth A.G, Research Scholar, Department of Electrcal Engneerng, College of Engneerng, Trvandrum (Kerala), Inda. Dr. Harharan S, Professor, Department of Electrcal Engneerng, College of Engneerng, Trvandrum (Kerala), Inda. Sajth A.G, Harharan S 5 to move contours toward the boundares. In the feld of medcal mage segmentaton, we use an energy functonal from a mathematcal model and mnmzng ths energy functonal to track the tumor regons. The popular pece- wse constant (PC) model s a typcal varatonal level set method, whch ams to mnmze the Mumford_Shah functonal.ths model makes use of nsde and outsde regons global mage propertes wth reference to the evolvng curve nto ther energy functonal as constrants. The PC model fals to segment the ntensty nhomogenty mages because t assumes that the ntenstes n each regon always mantan constant. L et al.[3, 9] proposed an mplct actve contours model based on local bnary fttng energy, whch used the local nformaton as constrant, and works very well on the mages wth ntensty nhomogenetes. Ths model outperformed both PC and PS models n segmentaton accuracy and computatonal effcency. Wang et al. [0] employed Gaussan dstrbutons to model local mage nformaton, and proposed local Gaussan dstrbuton fttng (LGDF) energy. Zhang et al. proposed a novel regon-based actve contour model for mage segmentaton, whch combned the merts of the tradtonal GAC and PC models []. Zhang et al. exploted local mage regon statstcs to present a level set method for segmentng mages wth ntensty nhomogenety. Ths model utlze a Gaussan flterng scheme to regularze the level set functon. The LIF model can acheve smlar segmentaton accuracy to the LBF model wth less computatonal costs.he et al. proposed an mproved regon-scalable fttng model based on the modfyng kernel and local entropy. Two pecewse smooth (PS) actve contour models [7] were developed under the framework of mnmzng the Mumford Shah functonal[].local ntensty crtera has been extensvely used n actve contour models to mprove the tumor boundary accuracy for the mages corrupted by nose and ntensty nhomogenty. In ths paper we makes use of the mage ntenstes wthn the neghborhood of each pxel to construct a mage fttng energy functonal, and s mplemented ths functonal usng varatonal level set formulaton for mage segmentaton. The neghborhood regons can be parttoned nto dsjont set and the local ntenstes n each regon follows. The Gaussan dstrbuton wth varaton n mean and varance. Based on the lkelhood functon of local ntensty functonal a local energy s defned and ntegrated over the entre mage doman to form the local lkelhood mage fttng energy functonal.

2 A Regon based Actve Contour Approach for Lver CT Image Analyss Drven by Local lkelhood Image Fttng Energy Ths functonal s mplemented usng the varatonal level set formulaton wth a regularzed term. The nformaton of CT mage ntenstes s utlzed to compute the means and varances for mnmzng the energy functonal norder to attract the contour towards the tumor regons. The presented model has been compared wth dfferent actve contour models. II. ACTIVE CONTOUR MODELS A. The Chan-Vese Model Chan and Vese proposed an ACM whch can be seen as a specal case of the Munford-Shah problem, for a gven mage I n doman Ω, the C-V model s formulated by mnmzng the followng energy functonal: CV E I( x) c nsde( C) outsde( C) I( x) c, x () where c and c are two constants whch are the average ntenstes nsde and outsde the contour, respectvely. Wth the level set method, we assume C x : ( x) 0 nsde( C) x : ( x) 0 outsde( C) x : ( x) 0 By mnmzng Eq.(), we solve c and c as follows: I( x). H ( ) c ( ) H ( ) I( x).( H ( )) c( ) 3 ( H ( )) By ncorporatng the length and area energy terms nto Eq.() and mnmzng them, we obtan the correspondng varatonal level set formulaton as follows: ( I c) ( ) t ( I c) 4 where µ 0, ۷ 0, λ >0, λ >0 are fxed parameters, µ controls the smoothness of zero level set, ۷ ncreases the propagaton speed, and λ and λ control the mage data drven force nsde and outsde the contour, respectvely. s the gradent operator. H(Φ) s the Heavsde functon and ( ) s the Drac functon. Generally, the regularzed versons are selected as follows: z H ( z) ( arctan ), z R 5 ( z), z R 6 z B. LBF model The SBGFRLS model cannot make full use of the local mage nformaton, and the precson of the fnal evolvng contour s not hgh. The NL-CV model proposed by Bresson et al. s able to overcome ths lmtaton by ntegratng sem-local and global mage nformaton wth a specfc graph. However how to reasonably defne the graph s the key problem. L et al. proposed the LBF model by embeddng the local mage nformaton. The model can accurately segment mages wth ntensty n homogenetes wth settng the ntal contour approprately. By ntroducng a kernel functon, the LBF model defnes an energy functonal as follows: LBF E K( x y) I( y) f ( x) n( C) K( x y) I( y) f ( x) (7) out ( C) where λ and λ are postve constants, K s a kernel functon of a localzaton property that K(x) decreases and approaches zero wth x ncreasng, and f and f are defned as follows: k k f ( x) average( I ( x ( x) 0 O ) 8 f ( x) average( I ( x ( x) 0 O ) 9 where O k denotes a small neghborhood of the pont x. The energy Eq.(7) can be converted to an equvalent level set formulaton: LBF E K( x y) I( y) f( x) H ( ( y)) K( x y) I( y) f( x) ( H ( ( y))) 0 where H s the Heavsde functon, can be approxmated by Eq.(7). In order to ensure stable evoluton of the level set functon Φ, a dstance regularzng term s added to penalze the devaton of the level set functon Φ from the SDF. On the other hand, a length term s used to regularze the zero level curve of Φ. The total varatonal formulaton of the model s as follows: 6

3 Internatonal Journal of Engneerng and Advanced Technology (IJEAT) ISSN: , Volume-6 Issue-5, June 07 ( ) K( y x) I( x) f( y) t ( ) K ( y x) I( x) f ( y) ( ) dv dv where f and f can be obtaned by K( x) H( ) I( x) f( x) () K ( x) H ( ) K( x) ( H( )) I( x) f( x) (3) K ( x) ( H ( )) and δε (Φ) can be obtaned by the dervatve of Eq.(6) ' ( x) H( x) (4) x In Eq.(), the frst and the second terms n the rght hand sde are called the data fttng and the arc length term, respectvely, whch are responsble for drvng the evolvng contour towards the object boundares. The thrd term called the regularzaton term serves to mantan the regularty of the level set functon. The LBF model utlzes the local regon nformaton, and t can obtan auurate segmentaton results, but ts computatonal cost s rather hgh due to the regularzed term and the punshed term correspondng to the second term and thrd term n Eq. (), respectvely. In addton, the model s senstve to the ntal contour. C. LBF model In case of mage contanng a regon of nterest and background of the same ntensty means but wth dfferent varances, snce the LBF model only explots local ntensty means for segmentaton t fals to segment the mage correctly. The LGDF model s developed by ncorporatng more statstcal and local ntenstes nto the energy functonal. For each pxel x, the local ntenstes wthn ts neghborhood are assumed to follow a Gaussan dstrbuton p ( I( y) u ( x), ( x), x ( x) ( I( y) u exp 5 ( x) where u (x) and σ (x) are mean and standard devaton of the ntenstes n each regon. The LGDF energy functonal s defned as E LGDF nsdec nsdec ( C, u, u,, ) K x y p I y u x x ( )log, x( ( ) ( ), ( ) ) K x y p I y u x x ( )log, x( ( ) ( ), ( ) ) (6) Here we consder both the frst and second order statstcs of local ntenstes, the LGDF model can dfferentate regons wth smlar ntensty means but wth dfferent ntensty varances. C. LIF model In ths model the local mage fttng energy functonal s defned by mnmzng the dfference between the ftted mage and the orgnal mage. The formulaton s as follows LIF LFI E ( ) I( x) I ( x), x (7) where m and m are defned as follows m mean( I ( x ( x) 0 Wk ) m mean( I ( x ( x) 0 Wk ) and I LFI the local ftted mage formulaton s defned as follows (8) LFI I m H ( ) m ( H ( )) (9) Where φ s the zero level set of a Lpschtz functon that represents the contour C, H ( ) s the Heavsde functon of the level set φ, and W k s a truncated Gaussan wndow wth the standard devaton σ and radus k. The LIF model utlzes a flterng method wth a Gaussan kernel to regularze the level set functon durng model optmzaton. Ths regularzaton term not only ensures the smoothness of the level set functon, but also enables ths model to avod re-ntalzng the level set functon. As a result, the LIF model s an effcent actve contour model that can acheve smlar segmentaton accuracy to the LBF model wth lower computatonal complexty. Nevertheless, both the LIF and LBF models share the same lmtaton that they may produce less accurate segmentaton results n mages wth strong nose and ntensty n homogenety. III. SUMMARY It should be noted that the CV, LBF and LIF models all use the mean ntensty value to characterze ether the global or local mage nformaton. Because the ntensty mean s statstcally nsuffcent to descrbe a regon, these models may fal to segment mages wth heavy nose and ntensty nhomogenety. To address ths ssue, other actve contour models, such as the LGDF, employ both the frst order and second order statstcs of local ntenstes to defne the energy functonal. Consderng the computatonal effcency of the LIF model, t s promsng to develop a novel actve contour model that has smultaneously hgh segmentaton accuracy and low computatonal cost by ncorporatng the spatally varyng mean and standard devaton of local ntenstes nto the LIF energy functonal. 7

4 A Regon based Actve Contour Approach for Lver CT Image Analyss Drven by Local lkelhood Image Fttng Energy IV. LOCAL LIKELIHOOD IMAGE FITTING (LLIF) MODEL An mage s represented as a real-valued functon I : Ω Ʀ defned over a rectangular doman Ω Ϲ Ʀ. We assume that the ntensty of each pxel x s ndependently sampled from the probablty densty functon p(i(x); θ ), where θ s the assembly of dstrbuton parameters. M Consder the regon of nterest n the mage doman we can defne the followng parameter space Θ x ( x) wth,... M (0) 0 x Accordng to the maxmum lkelhood (ML) crteron, the mage segmentaton problem can be solved by maxmzng the lkelhood probablty of the observed mage w.r.t. the parameters max p( I( x); Applyng the negatve logarthm to the above equaton, maxmzng the functonal s then converted nto the mnmzng functonal mn log p( I( x); we assume that the ntensty of each pxel follows a local Gaussan dstrbuton to explore the local mage nformaton, gven by p( I( x); Sx ( ) ( I( x) U exp 3 Sx ( ) where U(x) s the mean and S(x) the varance are the spatally varyng quanttes. Let the neghborhood of pxel x be defned by a Gaussan wndow W k wth the standard devaton σ and sze (4k + ) x (4k + ), where k s the greatest nteger smaller than σ. The mean and varance can be calculated usng N U ( x) u ( x) L ( x) and N S( x) s ( x) L ( x) (4) where u (x) and s (x) are the mean and varance of the ntenstes n the regon Ώ W k (x) s gven by u ( x) mean( I( y) : y Wk s ( x) var( I( y) : y Wk (5) and L (x) s a label ndcator of pxel x. If x ϵ Ω, L (x) = ; otherwse L (x) = 0. In LLIF model spatally varyng mean and varance s used. Therefore ths model perform very well n segmentng mages wth heavy nose, partcularly n dstngushng regons wth smlar ntensty means but dfferent varances. A. Level Set Formulaton The mage doman Ω conssts of two dsjont regons denoted by Ω and Ω. The level set functon ϕ s used to represent both Ω and Ω by takng postve and negatve sngs, gven by ( x( x) 0) ( x( x) 0) wth the Heavsde functon H, Ω and Ω can be defned as M (ɸ) = H(ɸ) and M (ɸ) = -H(ɸ). Thus, the energy E LLIF can be expressed as a functon of ɸ,U and S LLIF E (, U, S) log p( I( x) U( x), S (7) U(x) and S(x) s gven by U ( x) u( x) M( ) u( x) M ( ) S( x) s( x) M( ) s( x) M ( ) where u ( x) mean( I( y) : y ( y) 0 Wk u( x) mean( I( y) : y ( y) 0 Wk s ( x) var( I( y) : y ( y) 0 Wk s( x) var( I( y) : y ( y) 0 Wk The mean U, varance S and the gradent descent flow equaton s gven by ( y x) I( y) M, ( ( y)) u ( x) 30 ( y x) M ( ( y)), ( y x)( I( y) U M, ( ( y)) s ( x) (3) ( y x) M ( ( y)) In many stuatons, the level set functon wll develop shocks, very sharp and/or flat shape durng the processng of the evoluton, whch n turn makes further computaton hghly naccurate n numercal approxmatons To solve ths problem, t s necessary to keep the level set functon as an approxmate sgned dstance functon. The energy functonal becomes, LLIF L F(, U, S) E (, U, S) E ( ) E ( ) (3) where µ and ν are weghtng constants and P

5 Internatonal Journal of Engneerng and Advanced Technology (IJEAT) ISSN: , Volume-6 Issue-5, June 07 P E ( ) ( x) L and E ( ) H ( 33 Implementng H ϵ s a smooth functon used to approxmate the Heavsde functon H, δ ϵ s the dervatve of H ϵ, and ϵ s a postve constant. Thus the energy the energy functonal F(ϕ,U,S) can be approxmated by LLIF L F (, U, S) E (, U, S) E ( ) where E P x H ( x) arctan ( x) x ( ) (34) Mnmzng the energy functonal F ϵ w.r.t. ϕ can be acheved by solvng the gradent descent flow equaton. F t ( s s ) I U u u ( ) I U ss S B. Implementaton ( ) dv dv Implementaton of our model for the level set formulaton can be summarzed as follows. Step : Intalze the level set functon ϕ. Step : Update local means U and varances S usng Eqs. (30) and (3). Step 3: Update the level set functon ϕ accordng to Eq. (36). Step 4: Return to Step untl the convergence crtera met. V. RESULTS AND DISCUSSION Ths secton presents the mage segmentaton results obtaned by applyng the proposed LLIF model to CT lver tumor mages. In our experments, the level set functon ϕ was smply ntalzed to a bnary step functon, whch took a negatve value -c 0 nsde the tumor regon and a postve value c 0 outsde t. Unless otherwse specfed, parameters nvolved n the LLIF model were set as follows: σ = 3.0, tme step Δt = 0.05, μ =.0, ν = 0.0 x55 and e =.0. Fg. demonstrates that the level set approach wthout re-ntalzaton by L et al. outperforms the orgnal level set method by Osher and Sethan n tumor detecton. Although the approach by L et al. has acheved a great success, evdence shows that ths algorthm requres further mprovements. For example, t has been observed that n a nosy mage wth ambguous boundares ths approach cannot deally locate the tumor boundares (Fg. ). Fg. 3 llustrates ndvdual performance of usng the orgnal level set algorthm and the LLIF model approach, where the former s shown on the frst row whle the latter s demonstrated on the second row. The orgnal level set method s dstracted by the near edges of the lver tumors. LLIF method gets around ths problem, and fnally addresses on the deal tumor boundary. Ths ndcates that our approach s able to gnore the nterference of neghborng organs edges, and fast approach to the tumor boundary. Segmentaton of multple tumors s shown n Fg. 4. Multple tumors exst n the mage, and dfferent tumors have dfferent ntenstes. As seen n ths fgure, the orgnal level set scheme cannot locate the edges of the tumor regons n the lver CT mage. The edges have not been correctly outlned. These edges are vague somehow so that the energy mnmzaton n the classcal level set functon cannot be deally acheved. The model proposed by L et al. has a better outcome of tumor detecton. The accuracy of the edge detecton stll needs to be mproved.. Ths may need more efforts to optmze the contour settlement. On the other hand, t s clear that the LLIF model approach has a better performance than these two methods n terms of edge detecton ( mages on the thrd row). Ths experment utlzes a CT lver tumor mage where the tumor regon s much brghter. The target s to outlne the correct tumor regon usng the avalable methods. Fg. 5 demonstrates that the LLIF level set scheme has the best performance n tumor detecton, where L s method leads to errors n detectng the exact segmentaton of the tumor regon. Meanwhle, the orgnal level set method cannot correctly locate the tumor boundary, resultng from the sde-effects rased by the mage nose. Fg.6 demonstrate how these methods cope wth the nosy background. The orgnal level set method faled to segment the tumor regon. Meanwhle, L s model and the LLIF method have been successfully segment the tumor regon.. Interestngly, we observe that at teraton 50 fractonal order scheme seems reluctant to pck up a concave area. However, t recovers very soon and successfully addresses on the exact tumor regon at teraton 350. Ths may be due to the oscllaton of the energy terms durng the evoluton (before teraton 350). Takng a closer look at the results from L s model, we observe that ths model has less detectng accuracy on the block corners of the tumor regon than the LLIF model approach. 9

6 A Regon based Actve Contour Approach for Lver CT Image Analyss Drven by Local lkelhood Image Fttng Energy D E F Fg.. Performance comparson between the orgnal level set method (frst row) and the level set scheme wthout re-ntalzaton (second row). λ=.5,µ=0.4,ν= and tme step = 3. (a) Intal, (b) teraton 00, (c) teraton 350, (d) ntal, (e) teraton 00, (f) teraton 300. Fg.. Performance of the level set scheme wthout re-ntalzaton. λ=.5, µ= 0:4, ν=, and tme step = 3. (a) Intal, (b) teraton 00, (c) teraton 300. D E F Fg. 3. Performance comparson of the orgnal level set scheme and the LLIF model approach.λ=3,µ=0.3, ν=, and tme step =.5. (a) Intal, (b) teraton 50, (c) teraton 300, (d) ntal, (e) teraton 50, (f) teraton 50. 0

7 Internatonal Journal of Engneerng and Advanced Technology (IJEAT) ISSN: , Volume-6 Issue-5, June 07 Fg. 4. Performance comparson between the orgnal level set method (frst row), the level set scheme wthout re-ntalzaton (second row), and the LLIF model approach (thrd row). λ=.5,µ=0.4, ν=, and tme step=. 5 (a) Intal, (b) teraton 50, (c) teraton300. Fg. 5. Performance comparson between the orgnal level set method (frst row), the level set scheme wthout re-ntalzaton (second row), and the LLIF model approach (thrd row). λ=.5, µ=0.4, ν=, and tme step =. 5. (a) Intal, (b) teraton 50, (c) teraton 350.

8 A Regon based Actve Contour Approach for Lver CT Image Analyss Drven by Local lkelhood Image Fttng Energy a a Fg. 6. Performance comparson between the orgnal level set method (frst row), the level set scheme wthout re-ntalzaton (second row), and LLIF model approach (thrd row). λ=.5, µ=0.4, ν=, and tme step =. 5. (a) Intal, (b) teraton 350, (c) teraton 400 VI. CONCLUSION An effectve segmentaton tool for lver CT mages based on actve contour model wth local lkelhood mage fttng term s presented. Level set scheme s used for the analyss. A local lkelhood fttng term s ncluded n the energy functonal model. By ncorporatng the local lkelhood fttng term, the novel fttng term can descrbe the orgnal mage more accurately, and be robustness to nose. Ths model s effcent for CT lver mages, and ntensty nhomogeneous mages. REFERENCES. Kass, M., Wtkn, A., and Terzopoulos, D.: Snakes: Actve contour models, Internatonal journal of computer vson, 988,, (4), pp Osher, S., and Sethan, J.A.: Fronts propagatng wth curvature-dependent speed: algorthms based on Hamlton-Jacob formulatons, Journal of computatonal physcs, 988, 79, (), pp Caselles, V., Kmmel, R., and Sapro, G.: Geodesc actve contours, Internatonal journal of computer vson, 997,, (), pp Kmmel, R., Amr, A., and Brucksten, A.M.: Fndng shortest paths on surfaces usng level sets propagaton, IEEE Transactons on Pattern Analyss and Machne Intellgence, 995, 7, (6), pp L, C., Xu, C., Gu, C., and Fox, M.D.: Dstance regularzed level set evoluton and ts applcaton to mage segmentaton, IEEE transactons on mage processng, 00, 9, (), pp Mallad, R., Sethan, J.A., and Vemur, B.C.: Shape modelng wth front propagaton: A level set approach, IEEE transactons on pattern analyss and machne ntellgence, 995, 7, (), pp Vaslevsky, A., and Sddq, K.: Flux maxmzng geometrc flows, IEEE transactons on pattern analyss and machne ntellgence, 00, 4, (), pp Xu, C., and Prnce, J.L.: Snakes, shapes, and gradent vector flow, IEEE Transactons on mage processng, 998, 7, (3), pp Chan, T.F., and Vese, L.A.: Actve contours wthout edges, IEEE Transactons on mage processng, 00, 0, (), pp Cremers, D., Rousson, M., and Derche, R.: A revew of statstcal approaches to level set segmentaton: ntegratng color, texture, moton and shape, Internatonal journal of computer vson, 007, 7, (), pp He, L., Peng, Z., Everdng, B., Wang, X., Han, C.Y., Wess, K.L., and Wee, W.G.: A comparatve stu of deformable contour methods on medcal mage segmentaton, Image and Vson Computng, 008, 6, (), pp L, C., Huang, R., Dng, Z., Gatenby, J.C., Metaxas, D.N., and Gore, J.C.: A level set method for mage segmentaton n the presence of ntensty nhomogenetes wth applcaton to MRI, IEEE Transactons on Image Processng, 0, 0, (7), pp L, C., Kao, C.-Y., Gore, J.C., and Dng, Z.: Mnmzaton of regon-scalable fttng energy for mage segmentaton, IEEE transactons on mage processng, 008, 7, (0), pp Paragos, N., and Derche, R.: Geodesc actve regons and level set methods for supervsed texture segmentaton, Internatonal Journal of Computer Vson, 00, 46, (3), pp Ronfard, R.: Regon-based strateges for actve contour models, Internatonal journal of computer vson, 994, 3, (), pp Samson, C., Blanc-Féraud, L., Aubert, G., and Zeruba, J.: A varatonal model for mage classfcaton and restoraton, IEEE Transactons on Pattern Analyss and Machne Intellgence, 000,, (5), pp Tsa, A., Yezz, A., and Wllsky, A.S.: Curve evoluton mplementaton of the Mumford-Shah functonal for mage segmentaton, denosng, nterpolaton, and magnfcaton, IEEE transactons on Image Processng, 00, 0, (8), pp

9 Internatonal Journal of Engneerng and Advanced Technology (IJEAT) ISSN: , Volume-6 Issue-5, June Vese, L.A., and Chan, T.F.: A multphase level set framework for mage segmentaton usng the Mumford and Shah model, Internatonal journal of computer vson, 00, 50, (3), pp L, C., Kao, C.-Y., Gore, J.C., and Dng, Z.: Implct actve contours drven by local bnary fttng energy, n Edtor (Ed.)^(Eds.): Book Implct actve contours drven by local bnary fttng energy (IEEE, 007, edn.), pp Wang, L., He, L., Mshra, A., and L, C.: Actve contours drven by local Gaussan dstrbuton fttng energy, Sgnal Processng, 009, 89, (), pp Zhang, K., Song, H., and Zhang, L.: Actve contours drven by local mage fttng energy, Pattern recognton, 00, 43, (4), pp Mumford, D., and Shah, J.: Optmal approxmatons by pecewse smooth functons and assocated varatonal problems, Communcatons on pure and appled mathematcs, 989, 4, (5), pp Sajth A. G, receved the M.Tech degree n Electrcal Engneerng from College of Engneerng, Trvandrum, Kerala, Inda n 008, where he s currently pursung the PhD n Electrcal Engneerng. He has got many Natonal and Internatonal publcatons. Hs current research nterests nclude varatonal methods for medcal mage processng and effcent algorthms to solve them. Dr. Harharan S receved the M.Tech degree n bomedcal engneerng from IIT Bombay,Inda,n 99 and the Ph.D. degree n electrcal engneerng from IIT Kharagpur, Inda, n 003.He has got many Natonal and Internatonal publcatons. Hs current research nterests nclude varatonal methods for medcal mage processng and effcent algorthms to solve them. 3