Image Segmentation with Simultaneous Illumination and Reflectance Estimation: An Energy Minimization Approach

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1 Image Segmentaton wth Smultaneous Illumnaton and Reflectance Estmaton: An Energy Mnmzaton Approach Chunmng L 1, Fang L 2, Chu-Yen Kao 3, Chenyang Xu 4 1 Vanderblt Unversty Insttute of Imagng Scence, ashvlle, T 37232, USA 2 Dept. of Mathematcs, East Chna ormal Unversty, Shangha , Chna 3 Dept. of Mathematcs, The Oho State Unversty, Columbus, OH 43210, USA 4 Semens Corporate Research, Prnceton, J 08540, USA lchunmng@gmal.com Abstract Spatal ntensty varatons caused by llumnaton changes have been a challenge for mage segmentaton and many other computer vson tasks. Ths paper presents a novel method for mage segmentaton wth smultaneous estmaton of llumnaton and reflectance mages. The proposed method s based on the composton of an observed scene mage wth an llumnaton component and a reflectance component, known as ntrnsc mages. We defne an energy functonal n terms of an llumnaton mage, the membershp functons of the regons, and the correspondng reflectance constants of the regons n the scene. Ths energy s convex n each of ts varables. By mnmzng the energy, mage segmentaton result s obtaned n the form of the membershp functons of the regons. The llumnaton and reflectance components of the observed mage are estmated smultaneously as the result of energy mnmzaton. Wth llumnaton taken nto account, the proposed method s able to segment mages wth non-unform ntenstes caused by spatal varatons n llumnaton. Comparsons wth the state-of-the-art pecewse smooth model demonstrate the superor performance of our method. 1. Introducton An observed mage of a scene s a composton of llumnaton and reflectance components, whch have been termed as ntrnsc mages n [1]. The reflectance component characterzes a unque physcal property of the object surfaces n the scene the albedo of the object surfaces. The llumnaton component depends on the lght drecton and the orentaton of the object surfaces, and can be used to nfer the geometry of the objects beng vewed. However, spatal varatons n llumnaton cause changes of the ntenstes n the observed mages, whch has been a challenge for many computer vson tasks, such as segmentaton, trackng, and object recognton. In mage segmentaton, a consderable dffculty arses from spatal varatons n llumnaton. The observed mage ntenstes n a sngle object may not be unform due to spatal varatons n llumnaton. In ths case, t s dffcult to defne a regon descrptor n regon-based methods for mage segmentaton. Therefore, many wdely used regonbased methods [15, 5] rely on the unformty of the ntenstes n the regons to be segmented, as t s easy to model the ntenstes wthn the regons of nterest and to defne a regon descrptor. These methods are not applcable to real-world mages wth non-unform ntenstes caused by spatal varatons n llumnaton. The well-known Mumford-Shah model n [11] has provded a theoretcal framework of mage segmentaton, whch can be appled to a wder range of mages, ncludng those wth non-unform ntenstes. The Mumford-Shah model assumes that an mage can be approxmated by a pecewse smooth functon. Based on ths assumpton, the Mumford-Shah model seeks a contour and a number of smooth functons that approxmate the mage ntenstes n dsjont regons separated by the contour. The smoothness of these functons s ensured by mposng a smoothng term n the Mumford-Shah model, whch leads to a set of partal dfferental equatons that have to be solved at each teraton for the evoluton of the contour. Therefore, the computaton n these methods s rather expensve, whch lmts ther practcal applcatons. In addton, due to the non-convexty of the underlyng energy functonals, the correspondng energy mnmzaton algorthms may converge to false local mnma. The pecewse smooth characterzaton of mages n the Mumford-Shah model s too general for real mages. A 2009 IEEE 12th Internatonal Conference on Computer Vson (ICCV) /09/$ IEEE 702

2 more specfc characterzaton of real mages would provde more useful nformaton, whch can be taken nto account n mage segmentaton. In fact, there s a more specfc characterzaton of real mages based on the theory of lghtness percepton: an observed mage I, ts llumnaton mage component S, and reflectance mage component R are related by the followng multplcatve model: I = R S. (1) Furthermore, the llumnaton mage S s assumed to be smooth [6, 8, 2]. The reflectance mage R can be approxmated by a pecewse constant map, as t descrbes a physcal property of the object surfaces n the scene. Therefore, the observed mage can be approxmated by the product of a smooth functon and a pecewse constant functon. In ths paper, we propose a novel energy mnmzaton method for mage segmentaton wth smultaneous estmaton of llumnaton and reflectance mages, based on the above mage composton model n Eq. (1) and the propertes of the llumnaton and reflectance mages. We defne an energy functonal n terms of an llumnaton mage, the membershp functons of the regons, and the correspondng reflectance constants of the regons n the scene. By mnmzng the energy, mage segmentaton result s obtaned n the form of the membershp functons of the regons. Meanwhle, the llumnaton and reflectance components of the observed mage are estmated smultaneously as the result of energy mnmzaton. The proposed method s related to the technques for llumnaton and reflectance estmaton (IRE) from an observed mage, such as those n [7, 14, 9, 12]. However, our method s fundamentally dfferent from these methods for llumnaton and reflectance estmaton. Our method s manly targeted for mage segmentaton. In the formulaton of our method, we ntroduce membershp functons of the object surfaces n the mage, whch drectly represent an mage segmentaton result. The exstng methods for IRE do not provde mage segmentaton results. 2. Problem Formulaton Based on Intrnsc Image Composton For theoretcal completeness, we slghtly extend the mage model (1) wth addtve nose as below: I = R S + n (2) where n s assumed to be zero-mean Gaussan nose. The reflectance mage R s approxmately a constant r n a regon Ω, whch corresponds to an object surface n the scene. The property of the llumnaton mage S and the reflectance mage R mentoned n Secton 1 can be made more specfc as below: (A1) R(x) r for x Ω for =1,,, wth Ω Ω j = for any j and Ω =Ω. (A2) The llumnaton mage S s smooth. The above dsjont regons Ω 1,, Ω form a partton of the mage doman Ω. The goal of mage segmentaton s to determne such a partton. From the above mage model (2), we propose to seek a pecewse constant map ˆR and a smooth functon Ŝ to estmate the ntrnsc mages R and S, respectvely. In general, a pecewse constant map R can be expressed as a lnear combnaton: R = r χ Ω (3) where χ Ω s the characterstc functon of the regon Ω. Each characterstc functon χ Ω s a membershp functon of the regon Ω. Thus, estmaton of the reflectance mage R from an observed mage I can be acheved by seekng membershp functons u assocated wth the regons Ω and the correspondng reflectance constants r,for =1,,. Thus, mage segmentaton result s gven by the obtaned membershp functons. For convenence, the membershp functons u 1,,u and the constants r 1,,r are denoted by a vector valued membershp functon U =(u 1,,u ), and a vector r =(r 1,,r ), respectvely. Thus, the problem of estmatng the ntrnsc mages R and S can be formulated as one that seeks optmal vector membershp functon U, vector r, and a smooth functon S accordng to an optmalty crteron. We wll propose a varatonal formulaton, n whch the optmal U, r, and S are obtaned by mnmzng an energy functonal F(U, r,s). 3. Energy Formulaton 3.1. Statstcal Property of Local Intenstes The varaton n appearance caused by the llumnaton changes has been a challengng problem for vsual tasks, such as segmentaton. However, the smoothness of llumnaton mage, along wth the pecewse constant property of the reflectance mage, mples a useful property of the observed mage ntenstes n local area, whch wll be used n the formulaton of the proposed energy mnmzaton framework for segmentaton wth smultaneous llumnaton and reflectance estmaton. In fact, the observed mage ntenstes n a small local area follow a smple dstrbuton as descrbed below. We consder a crcular neghborhood of each pont x Ω, defned by O x {y : y x ρ}, where ρ s the radus of the neghborhood. The partton {Ω } of the entre doman Ω nduces a partton of the neghborhood O x nto subsets O x Ω, whch form a partton of O x. For a smooth 703

3 functon S, the values S(y) for all y n the crcular neghborhood O x can be well approxmated by S(x). Therefore, the ntenstes S(y)R(y) n each subregon O x Ω are approxmately the constant S(x)r. Thus, we have the followng approxmaton S(y)R(y) S(x)r for y O x Ω (4) From the mage model (2), we have I(y) S(x)r + n(y) for y O x Ω where n(y) s addtve zero-mean Gaussan nose. The above arguments show that the ntenstes wthn the neghborhood O x are samples from Gaussan dstrbutons wth dstnct means S(x)r, = 1,,. In the other word, the ntenstes n the neghborhood O x form clusters: {I(y) :y O x Ω }, each wth a cluster center m S(x)r. As shown above, the cluster center m can be approxmated by the product of a regon dependent reflectance constants r and an llumnaton factor S(x), whch approxmates the llumnaton wthn the neghborhood O x. Ths property of local ntenstes s fully exploted n the proposed method for mage segmentaton and estmaton of ntrnsc mages Clusterng Crteron for Local Intenstes In vew of the above clusterng characterzaton of local ntenstes n terms of llumnaton and reflectance, we ntroduce a clusterng-based approach for mage segmentaton wth smultaneous llumnaton and reflectance estmaton. Lets frst revew a basc clusterng crteron that has been used to derve the well known fuzzy C-means and K-means algorthms. A standard clusterng algorthm seeks the membershp functons u and the cluster centers c, such that the followng clusterng crteron functon s mnmzed: J FCM = u q (x) I(x) c 2 dx (5) where q 1 s the fuzzfer, I s the nput mage, u (x) s the membershp value at pont x for class such that u (x) =1, and c s the cluster center of class. Ths clusterng crteron has been used to derve a smple yet powerful algorthm to classfy data that conssts of separable clusters wth dstnct cluster centers and determne the cluster centers at the same tme. The correspondng mnmzaton process s the well-known fuzzy C-means (FCM) algorthm. As a specal case when the fuzzfer q s 1, the mnmzaton of the above clusterng crteron s acheved by the K-means algorthm. The above FCM clusterng crteron s n general not vald for mage I wth a spatally varyng llumnaton mage component S. However, from the statstcs of local ntenstes n Secton 3.1, the above standard clusterng crteron can be appled for local ntensty classfcaton. The analyss n Secton 3.1 has shown that the ntenstes n the neghborhood O x form clusters, each wth a dstnct cluster center m S(x)r. Therefore, we propose to mnmze the followng clusterng crteron for local ntenstes n the neghborhood O x : J loc x (U, r,s(x)) O x u q (y)k(x y) (6) I(y) S(x)r 2 dy where u s the membershp functon of the regon Ω, and K(x y) s a weghtng functon, whch decreases as the dstance from y to the neghborhood center x ncreases, and becomes zero when x y >ρ,.e., y / O x. In ths paper, the notaton a represents the Eucld norm for a vector a or the absolute value for a scalar a. The above crteron functon can be used for both gray level mages and color mages. The above clusterng crteron functon Jx loc can be wrtten as J loc x (U, r,s(x)) = u q (y)k(x y) (7) I(y) S(x)r 2 dy due to the fact that K(x y) =0for y / O x Integraton of Local Clusterng Crteron The optmal U, r, and S are defned as those that mnmze the clusterng crteron functon Jx loc (U, r, S(x)) for all the neghborhood O x. Mnmzaton of Jx loc for all x Ω can be acheved by mnmzng the ntegral of Jx loc over Ω. Therefore, we defne an energy J (u, r,s) J loc x (u, r,s(x))dx,.e. J (U, r,s) u q (y)k(x y) (8) I(y) S(x)r 2 dydx By changng the order of ntegraton, the above energy J (u, r,s) can be wrtten n the form: J (U, r,s)= u q (y)d (I(y))dy (9) where d (I(y)) K(x y) I(y) S(x)r 2 dx (10) 704

4 Image segmentaton and the estmaton of llumnaton and reflectance can be acheved by solvng the followng constraned energy mnmzaton problem: mn J (U, r, S) subject to U U, (11) where U s the space of all the membershp functons,.e. U {(u 1,,u ) T :0 u (x) 1,,,, and u (x) =1, for all x Ω} (12) 3.4. Energy Formulaton wth Membershp Regularzaton For mages wth hgh level nose, t s necessary to regularze membershp functon by addng a regularzaton term R(U) to the above clusterng energy J (U, r,s). Therefore, we defne J reg (U, r,s)=j (U, r,s)+γr(u) (13) In ths work, we use total varaton of the membershp functons as the membershp regularzaton term,.e. Then we defne R(U) = u (x) dx (14) J reg (U, r,s) J (U, r,s)+γr(u), (15) and solve the followng constraned energy mnmzaton problem: mn J reg (U, r,s) subject to U U, (16) It s worth notng that ths energy s convex n the varables r and S, whch mples a unque global mnmum n each of these varables gven the others. The mnmzaton of the energy functonals J (U, r,s) and J reg (U, r,s) s descrbed n the next secton. 4. Energy Mnmzaton We have defned two constraned energy mnmzaton problems n Eqs. (11) and (16) n the prevous secton. The energy mnmzaton for the varables r and S are the same for both cases, as the addtonal membershp regularzaton term γr(u) n (16) s fxed as a constant when mnmzng the energy functonals J (U, r,s) and J reg (U, r,s)wth respect to r and S Mnmzaton of J (U, r,s) Case q>1 Mnmzaton of the energy J (U, r,s) can be performed by nterleavng the mnmzaton wth respect to the varables U, r, and S. ote that the energy J (U, r,s) s convex n each varable, and t s a quadratc form n terms of the varables r and S. The mnmzaton wth respect to each varable, gven the other two fxed, can be acheved by a closed form soluton as below: For fxed U and S, there s a unque mnmzer of the energy J (U, r,s) wth respect to r, whch s n the form denoted by ˆr =(ˆr 1,, ˆr ), s gven by (S K)Iu q ˆr = dx (S2 K)u q, =1,,. (17) dx where s the convoluton operaton. For fxed r and S, there s a unque mnmzer of the energy J (U, r,s) wth respect to U, whch s gven by û (y) = 1 k=1 ( d(i(y)) d k (I(y)) ) 1 q 1 (18) Gven U and r, there s a unque mnmzer of the energy J (U, r,s) wth respect to S, whch s gven by Ŝ = (IR(1) ) K (19) R (2) K where R (1) = r u q and R(2) = r2 uq. ote that the convolutons wth a kernel K n the expresson of Ŝ n Eq. (19) confrms the smoothness of the derved optmal llumnaton mage Ŝ, whch has been explaned from the defnton of the clusterng crteron functon Jx loc n the prevous secton Case q =1 For the case of q =1, the optmzaton of the varables r and S n the energy J (U, r,s) s the same as (17) and (19) wth q =1. However, the optmal membershp functon U cannot be computed by (18) n ths case. The optmzaton of r for the case of q =1s gven below. Let u 1,,u be postve functons such that u (x) =1for all x Ω, then the energy d (I(x))u (x)dx (20) s mnmzed f for each x Ω, the values of u 1 (x),,u (x) are gven by { 1, = mn (x); û (x) = 0, mn (x). (21) 705

5 where mn (x) =argmn{d (I(x))} Mnmzaton of J reg (U, r,s) wth Respect to U Usng the same technque as n [10]. The frst constrant can be relaxed by addng exact penalty convex functons ν(u )=max (0, 2u 1 1). Usng Lagrange multpler method, the second constrant can be removed by addng a term α ( u 1) 2 dx n (15) where α s a postve Lagrange multpler. Then the equvalent unconstraned problem s to mnmze J reg (U, r,s)= ( γ u dx + u d dx + ν(u ) ) + α 2 ( u 1) 2 dx. (22) Observng that the energy (22) s convex n varables u,we choose to follow [3] and take use of the fast dualty projecton method n [4]. So we add auxlary varables v and mnmze the followng approxmate energy J reg (U, V, r,s)= (γ v dx + 1 2θ (v u ) 2 dx + u d dx + ν(u )) + α 2 ( u 1) 2 dx (23) where θ>0s chosen to be small enough so that u and v are almost dentcal wth respect to the L 2 norm. Snce (23) s convex n u and v, we can use alternate mnmzaton method to fnd the mnmzer of (23). Decompose the problem nto four sub-problems as follows For fxed U, r,s,solvev by mnmzng γ v dx + 1 (v u ) 2 dx 2θ Ths problem can be effcently solved by fast dualty projecton algorthm. The soluton s gven by v = u θdv p,,..., K where the vector p can be solved by fxed pont method: Intalzng p 0 =0and teratng p n+1 = pn + τ (dv pn u /θ) 1+τ (dv p n u /θ). wth τ 1/8 to ensure convergence. For fxed V,r,S,wesolveu by mnmzng 1 2θ (v u ) 2 dx + u d dx +ν(u )+ α 2 ( u 1) 2 dx. Remark that the presence of the ν(u ) term n the energy s equvalent to cuttng off each u at 0 and at 1. On the other hand, f u [0, 1], then ν(u )=0, and the Euler-Lagrange equaton of ths problem s 1 θ (u v )+d + α( u 1) = 0 Then the mnmzaton s gven by u = mn(max( v θd θα( j u j 1), 0), 1). 1+θα For fxed U and V, r and S are gven by Eqs. (17) and (19) respectvely. 5. Implementaton and Expermental Results 5.1. Implementaton and Parameter Settng The energy mnmzaton s performed by an nterleaved optmzaton of each varable of the energy J (U, r,s) n an teratve process. In every teraton, we mnmze the energy wth respect to one of the three varables gven the other two obtaned from the prevous teraton. Before the teraton, the three varables U, r, and S are ntalzed as follows. In our mplementaton, we always ntalze the llumnaton mage S as a constant map for smplcty. To reduce the teraton number and save computaton tme, we apply the effcent K-means algorthm as a prelmnary segmentaton for the ntalzaton of the membershp functon U and the reflectance constant r. After the optmal Û, ˆr, and Ŝ are obtaned as the result of energy mnmzaton, the segmentaton result s gven by the membershp functons n Û =(û 1,, û ). The llumnaton and reflectance mages are gven by Ŝ and ˆR(x) = ˆr û (x) (24) To estmate the llumnaton mage more accurately, t s preferable to choose a small scale parameter σ n the Gaussan kernel K. In ths case, we use σ =1for the estmaton of llumnaton and reflectance mages. A larger scale parameter σ would blur the resultng llumnaton mage. If one s only nterested n the segmentaton result, a large scale parameter σ can be used n our method. In addton, the fuzzfer q s set to q =1.2 n most of our experments Expermental Results We frst demonstrate the effectveness of the proposed method for three mages n Column 1 of Fg. 1. The spatal llumnaton varaton s obvous n these mages. The segmentaton results and llumnaton mages are shown n Columns 2 and 3, respectvely. The segmentaton results 706

6 are hghly consstent wth the orgnal mages. The estmated reflectance mages exhbt desrable homogenety n each regon, whle ntensty varaton due to the llumnaton changes s extracted to the estmated llumnaton mages n Column 3, whch characterze the shapes of the surfaces n the scenes and the lghtng condton. Fgure 2. Result for two face mages. Column 1: Orgnal mage; Column 2: Segmentaton result; Column 3: Estmated llumnaton mage. Fgure 1. Results for three color mages. Column 1: Orgnal mage; Column 2: Segmentaton result; Column 3: Estmated llumnaton mage. We have also appled our method to face mages, as shown n Fg. 2. The varaton n appearance caused by spatal llumnaton changes often poses addtonal challenge n face recognton. Therefore t s n general desrable to obtan an llumnaton free mage,.e., a pure reflectance mage, as the nput to face recognton systems to mprove robustness and recognton rate. Fg. 2 shows the results of our method for two face mages n Column 1. The segmentaton results are shown n Column 2. ote that the skns on both faces and necks are correctly dentfed as a sngle regon. As a result, the correspondng estmaton of the reflectance mages exhbt such homogenety n the faces and necks. The spatal varaton n llumnaton, characterzng the geometry of the face, s extracted n the llumnaton mages, as shown n Column 3. The proposed method s by nature able to deal wth ntensty nhomogenetes n mage segmentaton. Fg. 3 shows the segmentaton results of our method for four mages. There are fast local ntensty varatons, such as the ntensty varatons wthn the grass or the squrrel n the frst mage. For such mages, t would be helpful to smooth the mage to sgnfcantly suppress fast local ntensty varatons. Our method s then appled to the smoothed mages, whch gves desrable segmentaton results as shown n Fg. 3. As mentoned n Secton 1, the Mumford-Shah model s able to deal wth ntensty n mage segmentaton. aturally, we compare our method wth the pecewse smooth (PS) model proposed n [13], whch s a well-known method that Fgure 3. Applcaton of our method for mage segmentaton. Column 1: The nput mages; Columns 2 and 3 show the segmentaton results n the forms of regons and boundares, respectvely. mplements the Mumford-Shah model usng a level set approach. We have mplemented the PS model for gray level mages n a two-phase formulaton. For a far comparson, we chose a synthetc gray level mage n Fg. 4 that can be segmented by the PS model n a two-phase formulaton. For ths mage, we appled our model wth the membershp regularzaton term. We used dfferent ntalzatons to test the robustness of the two methods. The results show that our method s robust to ntalzaton. Ths can be seen from the results n Fgs. 4(b) and 4(c) for two dfferent ntalzatons. The values of the ntal membershp functons are generated as random numbers for the frst ntalzaton (for the result n Fg. 4(b)), and a crcular regon s used to defne the membershp functons for the second ntalzaton (for the result n Fg. 4(c)). It s clear that these two results are almost the same, whch demonstrates the robustness of our method to ntalzaton. The PS model often produces qute dfferent results for even slghtly dfferent ntalzatons n our mplementaton. The results of the PS model wth two 707

7 method s able to segment mages wth non-unform ntenstes caused by spatal varatons n llumnaton. Comparsons wth the state-of-the-art pecewse smooth model demonstrate the superor performance of our method. (a) Orgnal mage. (b) Result 1 of our (c) Result 2 of our method. method. (d) Result 1 of the PS (e) Result 2 of the PS model. model. Fgure 4. Comparson of our method wth the pecewse smooth model. ntalzatons are shown n Fgs. 4(d) and 4(e). The ntalzaton of the level set functons n the PS model are obtaned by two smlar crcular regons of dfferent szes. We also compared the CPU tmes of the two methods wth the same crcular regon for the defnton of the ntal membershp functon n our method (for the result n Fg. 4(c))) and the level set functon n the PS model (for the result n Fg. 4(e)). The CPU tmed consumed by our method and the PS model are 7.83 and seconds respectvely. These CPU tmes are recorded n runnng our Matlab programs on a Lenovo ThnkPad notebook wth Intel (R) Core (TM)2 Duo CPU, 2.40 GHz, 2 GB RAM, wth Matlab 7.4 on Wndows Vsta. Ths experment demonstrates the advantage of our method n terms robustness and computatonal effcency over the PS model. 6. Concluson We have presented a novel method for mage segmentaton wth smultaneous estmaton of llumnaton and reflectance mages. The proposed method s based on the composton of an observed scene mage as the product of an llumnaton component and a reflectance component, known as ntrnsc mages. We defne an energy functonal n terms of an llumnaton mage, the membershp functons of the regons, and the correspondng reflectance constants of the regons n the scene. Ths energy s convex n each of ts varables. By mnmzng the energy, mage segmentaton result s obtaned n the form of the membershp functons of the regons. Meanwhle, the llumnaton and reflectance components of the observed mage are estmated smultaneously as the result of energy mnmzaton. Wth llumnaton taken nto account, the proposed References [1] H. Barrow and J. Tenenbaum. Recoverng ntrnsc scene characterstcs from mages. In A. Hanson and E. Rseman, edtors, Computer Vson Systems, pages Academc Press, [2] A. Blake. Boundary condtons for lghtness computaton n Mondran world. Computer Vson, Graphcs and Image Processng, 32(3): , December [3] X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P. Thran, and S. Osher. Fast global mnmzaton of the actve contour/snake model. J. Math. Imagng Vs., 28(2): , [4] A. Chambolle. An algorthm for total varaton mnmzaton and applcatons. J. Math. Imagng Vs., 20(1-2):89 97, January [5] T. Chan and L. Vese. Actve contours wthout edges. IEEE Trans. Imag. Proc., 10(2): , February [6] B. Horn. Determnng lghtness from an mage. Comput. Graphcs, Image Processng, 3(4): , December [7] R. Kmmel, M. Elad, D. Shaked, R. Keshet, and I. Sobel. A varatonal framework for retnex. Int l J. Comp. Vs., 52(1):7 23, Aprl [8] E. Land and J. McCann. Lghtness and retnex theory. J. Opt. Soc. of Amerca, 61(1):1 11, [9] Y. Matsushta, K. shno, K. Ikeuch, and M. Sakauch. Illumnaton normalzaton wth tme-dependent ntrnsc mages for vdeo survellance. IEEE Trans. Pattern Anal. Mach. Intell., 26(10): , October [10] S. E. Mla kolova and T. F. Chan. Algorthms for fndng global mnmzers of mage segmentaton and denosng models. SIAM J. Appl. Math., 66(5): , [11] D. Mumford and J. Shah. Optmal approxmatons by pecewse smooth functons and assocated varatonal problems. Commun. Pure Appl. Math., 42(5): , [12] M. Tappen, E. Adelson, and W. Freeman. Estmatng ntrnsc component mages usng non-lnear regresson. In Proceedngs of IEEE Conference on Computer Vson and Pattern Recognton (CVPR), volume 2, pages , [13] L. Vese and T. Chan. A multphase level set framework for mage segmentaton usng the Mumford and Shah model. Int l J. Comp. Vs., 50(3): , December [14] Y. Wess. Dervng ntrnsc mages from mage sequences. In Proceedngs of the Eghth Internatonal Conference on Computer Vson (ICCV), volume II, pages 68 75, [15] S.-C. Zhu and A. Yulle. Regon competton: Unfyng snakes, regon growng, and Bayes/MDL for multband mage segmentaton. IEEE Trans. Patt. Anal. Mach. Intell., 18(9): , September