Segmentation by Level Sets and Symmetry

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1 Segmenaion by Level Ses and Symmery Tammy Riklin-Raviv Nahum Kiryai Nir Sochen Tel Aviv Universiy, Tel Aviv 69978, Israel Absrac Shape symmery is an imporan cue for image undersanding. In he absence of more deailed prior shape informaion, segmenaion can be significanly faciliaed by symmery. However, when symmery is disored by perspeciviy, he deecion of symmery becomes non-rivial, hus complicaing symmery-aided segmenaion. We presen an original approach for segmenaion of symmerical objecs accommodaing perspecive disorion. The key idea is he use of he replicaive form induced by he symmery for challenging segmenaion asks. This is accomplished by dynamic exracion of he objec boundaries, based on he image gradiens, gray levels or colors, concurrenly wih regisraion of he image symmerical counerpar (e.g. reflecion) o iself. The symmerical counerpar of he evolving objec conour suppors he segmenaion by resolving possible ambiguiies due o noise, cluer, disorion, shadows, occlusions and assimilaion wih he background. The symmery consrain is inegraed in a comprehensive level-se funcional for segmenaion ha deermines he evoluion of he delineaing conour. The proposed framework is exemplified on various images of skewsymmerical objecs and is superioriy over sae of he ar variaional segmenaion echniques is demonsraed. 1. Inroducion Shape symmery is a useful visual feaure for image undersanding [1, 8, 11, 12, 15, 20, 21, 31, 33, 36, 35]. This research employs symmery for objec deecion and segmenaion. In he presence of noise, cluer, disorion, shadows, occlusions or assimilaion wih he background, segmenaion becomes challenging. In hese cases, objec boundaries do no fully correspond o edges in he image and may no delimi homogeneous image regions. Hence, classic regionbased and edge based segmenaion echniques are no sufficien. We, herefore, sugges a novel approach o faciliae segmenaion of symmerical objecs by using heir replicaive form induced by he symmery. The model presened is applicable for ranslaional-symmery, roaional-symmery and bilaeral symmery (reflecion). In a level se framework for segmenaion [23], images are represened via level-se funcions where he objec regions are assigned o he posiive levels. The resuling parameerizaion-free shape descripion eliminaes he need o relae shape as a collecion of poins or feaures, giving a meaningful inerpreaion o dissimilariy measure beween shapes. Moreover, any ransformaion applied on he image changes he coordinae sysem of is level-se funcion. The represened shape is hus ransformed correspondingly, simplifying he process of shape alignmen. The proposed mehod for symmery-aided segmenaion benefis from he level-se formulaion. Regarding shape as one eniy, we apply a symmery operaor on is levelse represenaion o obain is symmerical counerpar. We hen use he disance beween he shape represenaions o impose a shape consrain ha faciliaes he segmenaion. This approach is, hus, considerably differen from oher mehods ha suppor segmenaion by symmery [9, 16, 18, 34]. When symmery is disored by perspeciviy, he deecion of symmery becomes non-rivial, hus complicaing symmery aided segmenaion. We approach his difficuly by showing ha an image of a symmerical objec, disored by a projecive ransformaion, relaes o is symmerical counerpar (e.g. reflecion) by a projecive ransformaion. The explici form of he homography beween symmerical counerpars is shown and is used o define he limis on he abiliy o recover he disoring projecive ransformaion. The paper conains wo fundamenal conribuions. The firs is he use of inrinsic shape propery - symmery - as a prior for image segmenaion. The second is a heoreical resul concerns wih symmerical objecs disored by projecive ransformaions which has significan implicaions relaed also o 3D objec reconsrucion. We hus presen unified framework for segmenaion of symmerical objecs disored by perspeciviy, inegraing region-based, edgebased, smoohness and symmery consrains. The proposed mehod is exemplified and verified on various images of roughly symmerical objecs in he presence of projecive disorion. 1

2 2. Level ses for segmenaion In his secion we review sae of he ar segmenaion wih level ses. We describe and generalize fundamenal conceps of he wo-phase region based segmenaion approach of Vese and Chan [3, 32]. We hen recall he basics of geomeric acive conours as inroduced in [2] exended in [14] and represened in [13]. These are incorporaed wihin he Chan-Vese level-se formulaion. In he subsequen secions we show how he symmery cue can be inegraed, o esablish a unified level se framework ha efficienly explois all image feaures essenial for segmenaion Region-based erm Le I : R + denoe a gray level image, where R 2 is he image domain. Le ω be open subse of. In he spiri of he Mumford-Shah model [22], we define a boundary C, C = ω ha delimis homogeneous regions in I. Thus, for a general feaure G(I), in he paricular wo phase case, we look for a curve C ha maximizes he difference beween wo scalars u + and u defined as follows: u + = A + G + (I(x))dx u = A G (I(x))dx ω \ω where x (x, y), A + =1/ ω dx and A =1/ \ω dx. The feaure chosen depends on he image homogeneiy. In he work of Chan and Vese [3] he image is approximaed by a piecewise consan funcion whose values are given by G + (I(x)) = G (I(x)) = I(x). Hence u + = I in and u = I ou are he average gray levels in he objec regions and in he background regions respecively. In his sudy we use he average gray level and he variance: (1) G + (I) =(I(x) I in ) 2 G (I) =(I(x) I ou ) 2 (2) This was considered by [32] and by [19] in he pas. In [27] a probabilisic formulaion o G has been proposed. In he level se framework for curve evoluion [23], an evolving curve C() is defined as he zero level of a level se funcion φ: R a ime : C() ={x φ(x,)=0}. (3) Following [3], he Heaviside funcion of φ H(φ()) = { 1 φ() 0 0 oherwise is used o indicae he objec-background regions in he image ha correspond he non-negaive and negaive levels in (4) φ, respecively. Pracically, a smooh approximaion of he Heaviside funcion H ɛ is used [3]: H ɛ (φ) = 1 2 (1 + 2 π arcan(φ )) (5) ɛ The above formulaion enables he consrucion of a region based cos funcional wih a well defined inegraion domain: [ E RB (φ) = (G + (I(x)) u + ) 2 H ɛ (φ) + (G (I(x)) u ) 2 (1 H ɛ (φ)) ] dx (6) where he subscrip RB sands for region-based. We use boldface o denoe vecors. The opimal curve would bes separae is inerior from is exerior wih respec o heir relaive characerisic scalars (or vecors) u + and u. Noe, ha he funcional minimizer is φ. The evolving boundary C() is derived from φ() using (3). The level se funcion φ is updaed according o: φ RB =δ ɛ (φ) [ (G (I(x)) u ) 2 G + (I(x)) u + ) 2] (7) The evoluion of φ a each ime seps is weighed by he derivaive of he regularized form of he Heaviside funcion: δ ɛ (φ) = dh ɛ(φ) dφ = 1 π 2.2. Geodesic acive conour erm ɛ ɛ 2 + φ 2 Edge based segmenaion approaches usually define he objec boundaries by he local maxima of he image gradiens. Le I(x, y) =(I x,i y ) T ) T = ( I(x,y) x, I(x,y) y denoe he vecor field of he image gradiens. The inverse edge indicaor funcion is defined by; g(x) =1/(1 + I 2 ) (8) Le s be an arc-lengh parameer. The geodesic acive conour funcional L g(c(s))ds inegraes he inverse edge 0 indicaor along he curve. A minimizer C will be obained when g(c(s)) vanishes, ha is when he conour C is aligned wih he image edges. The corresponding erm for φ akes he form: E GAC = g( I ) H ɛ (φ(x)) dx, (9) and he evoluion of φ is deermined by: φ GAC = δ ɛ (φ)div ( g( I ) φ ). (10) φ

3 Thus, he zero level of φ is consrained o follow he image gradiens. When g =1Eq. 10 reduces o: E LEN = H ɛ (φ(x)) dx (11) This funcional measures he curve lengh C and usually indicaes he curve smoohness [3]. Minimizing (11) wih respec o φ, we obain he following evoluion equaion: φ LEN 2.3. Alignmen erm = δ ɛ (φ)div ( φ φ ). (12) The geodesic acive conour erm (9) deermines he locaion of he zero level of φ. In [14] i is suggesed o incorporae he direcional edge informaion o refine he segmenaion. This is done by aligning he level se normal direcion, n = φ φ wih he image gradiens direcion I. E RA = φ I, φ H ɛ(φ) dx. (13) where RA refer o robus alignmen, as defined in [13]. The associaed gradien descen equaion: 2.4. Shape erm φ RA = δ ɛ (φ)sign( φ, I ) I (14) The image daa by iself is no sufficien for accurae objec exracion in he presence of noise, occlusions or assimilaion wih he background. Recen variaional approaches for segmenaion sugges o incorporae a prior shape consrain o faciliae segmenaion [4, 5, 6, 7, 17, 24, 28, 30]. When only a single image is given, such prior is no available. Neverheless, if an objec is known o be symmerical, is replicaive form, induced by he symmery, can be used. Secion 4 considers he incorporaion of he symmery consrain wihin a level-se framework for segmenaion. The formulaion is esablished based on a resul shown in secion Symmery and Projeciviy 3.1. Symmery An objec is symmerical wih respec o a given operaion if i remains invarian under ha operaion. In 2D geomery hese operaions relae o he basic Euclidean plane isomeries: reflecion, inversion, roaion and ranslaion. We denoe a symmery operaion by S. S is an isomery operaing on homogeneous vecors x =(x, y, 1) T represened as S = sr 0 T 1 (15) where is a ranslaion 2D vecor, 0 is a null 2D vecor, R is he 2 2 roaion marix and s is he diagonal marix diag(±1, ±1). Specifically, we relae o eiher of he following ransformaions: 1. S is ranslaion if 0 and s = R = diag(1, 1). 2. S is roaion if = 0 and s = diag(1, 1). 3. S is reflecion if = 0, R = diag(1, 1) and s is eiher diag( 1, 1) for lef-righ reflecion or diag(1, 1) for up-down reflecion. In he case of reflecion, he symmery operaion reverses orienaion, oherwise (ranslaion, roaion and inversion) i is orienaion preserving. Definiion 1 Le I denoe an image defined w.l.o.g. on he symmerical domain =[ a, a] [ b, b], I is symmerical wih respec o S,if I(x) =I(Sx) for each x. (16) Î(x) I(Sx) is he symmerical counerpar of I wih respec o S. An image is idenical o is symmerical counerpar, if and only if i is symmerical. We claim ha he image of a symmerical objec disored by a projeciviy is relaed o is symmerical counerpar by projecive ransformaion differen from he defining symmery. Before we proceed proving his claim we recall he definiion of projecive ransformaion Projeciviy This subsecion follows he definiions in [10]. Definiion 2 A planar projecive ransformaion (projeciviy) is a linear ransformaion represened by a non-singular 3 3 marix H operaing on homogeneous vecors, x = Hx, where, H = h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 (17) Imporan specializaions of he group formed by projecive ransformaion are he affine group and he similariy group which is a subgroup of he affine group. These groups form a hierarchy of ransformaions. A similariy ransformaion is represened by [ ] κr H S = 0 T (18) 1 where R is a 2 2 roaion marix and κ is an isoropic scaling. When κ =1, H S is he Euclidean ransformaion

4 denoed by H E. An affine ransformaion is obained by muliplying he marix H S wih H A = [ K 0 0 T 1 ]. (19) K is an upper-riangular marix normalized as K =1.The marix H P defines he essence of he projecive ransformaion and akes he form: [ ] I 0 H P = v T. (20) v A projecive ransformaion can be decomposed ino a chain of ransformaions of a descending (or ascending) hierarchy order, [ ] A H = H S H A H P = v T (21) v where v 0and A = κrk +v T is a non-singular marix Relaion beween symmerical counerpars In his subsecion we consider he relaion beween an image of symmerical objec disored by planar projecive ransformaion H and is symmerical counerpar. Theorem 1 Le I S denoe a symmerical image as defined by (16). The image I A is obained from he symmerical image I S by applying he planar projecive ransformaion H: I A (x) =I S (Hx). Le ÎA(x) =I A (Sx) denoes he symmeric counerpar of I A wih respec o a symmery operaion S. The images I A and ÎA are relaed by planar projecive ransformaion, represened by a 3 3 marix of he form M = S 1 H 1 SH. (22) Proof 1 The symmerical counerpar ÎA can be generaed from I A eiher by he symmery operaion S or by a projecive ransformaion M. I A (x) = I A (H 1 Hx) =I A (H 1 y) = I S (y) =I S (Sy) = I S (HH 1 Sy) = I A (H 1 Sy) =I A (H 1 SHx) = I A (SS 1 H 1 SHx) = ÎA(S 1 H 1 SHx) =ÎA(Mx) (23) The chain of equaliies in (23) is equivalen o he following sequence of operaions: 1. Apply he inverse of he projecive (disoring) ransformaion, H 1 on I A o generae a symmerical image I S. 2. Apply he symmery operaion S on I S, under which i remains invarian. 3. Muliply I S by he projecive ransformaion marix H o obain back he image I A. 4. Apply again he symmery operaion S on I A o obain is symmerical counerpar ÎA. Le N = M 1 = H 1 S 1 HS. N and hus M are projecive ransformaions since HN = S 1 HS is a projecive ransformaion according o: [ ] [ S 1 HS = S 1 A A ] v T S = v v T v = H (24) The claim is exemplified for wo paricular cases. Consider he image of he symmerical objec and is lef-righ reflecion shown in Fig. (1)a-b. Suppose ha he image symmery has been disored by an Euclidean ransformaion of he form: cos θ sin θ x H E = sin θ cos θ y Noe ha he Euclidean ransformaion is an isomery and hus preserves he objec symmery. However, i draws he symmery axis of he objec away from he symmery axis of he image, roaing i by angle θ and ranslaing i by x. Fig. (1)a relaes o is symmerical counerpar by: M = S 1 H 1 E SH E = cos 2θ sin 2θ 2 x cos θ sin 2θ cos 2θ 2 x sin θ 0 0 1, where S = diag( 1, 1, 1). Fig. (1)a can hus be obained from Fig. (1)b by a ranslaion by 2R(θ)[ x, 0] T and a roaion by 2θ. Noe ha any ranslaion parallel o he symmery axis (in his case y ) canno be recovered from M. Consider, nex he images shown in Fig. (1)c-d. The objec is disored by a projecive ransformaion H P : H P = v 1 v 2 1 The relaion beween he wo images can be described by: M = S 1 H 1 P SH P = v When v 2 0he objec shape is disored bu is symmery is preserved, hus v 2 canno be recovered from M. In general, any operaion ha does no disor he objec symmery can no be recovered from M. Refer o [26] for he complee proof.

5 (a) (b) (c) (d) Figure 1. (a) An image of a symmerical objec ransformed by an Euclidean ransformaion. The objec s symmery axis deviaes by x andbyangleθ from he symmery axis of he image. (b) The symmerical counerpar of he image in (a). (c) An image of a symmerical objec disored by projecive ransformaion. (d) The symmerical counerpar of he image in (c). 4. Symmery based segmenaion 4.1. Symmery consrain The discussion in he previous secion relaed o images. We will now refer o he dynamic objec indicaor funcions represened by H ɛ (φ()). Leˆφ: R denoe he symmerical counerpar of φ wih respec o a symmery operaion S. Specifically, ˆφ(x) =φ(sx) where S is eiher a reflecion or roaion or ranslaion. We assume ha he S is known. We denoe by T p he alignmen funcion beween H ɛ (φ) and H ɛ ( ˆφ S ). T p capures he deviaion of he objec symmery axis from ha of he image and he projecive ransformaion ha disors is symmery. Noe, however, ha his informaion is no known in advance. T p is recovered by a regisraion process held concurrenly wih he segmenaion, deailed in subsecion 4.3. Le D = D(H ɛ (φ), T p H ɛ ( ˆφ)) denoe a dissimilariy measure beween he evolving shape represenaion and is symmerical counerpar. Noe ha if T p is correcly recovered and φ capures a perfecly symmerical objec (up o projeciviy) hen D =0. D hus quanifies he disorions of objec symmery which are no caused by he projeciviy. Whenever hese disorions are due o false deecion of he objec boundaries (caused by noise, occlusions, cluer, ec.) and no feaures of he objec shape, D defines an appropriae symmery consrain. In [24], D measures he noneoverlapping objec-background regions beween he evolving segmenaion and a well-defined prior ˆφ: D(φ, ˆφ) = [ H ɛ (φ(x)) T p H ɛ ( ˆφ(x))] 2 dx. (25) Neverheless, since in our case H ɛ ( ˆφ) is idenical o H ɛ (φ) up o an isomery, i is subjec o he same disorions and hus canno replace a well defined prior. A differen formulaion is hen needed, o be described in he following subsecion Biased shape dissimilariy measure Consider, for example, he approximaely bilaeral symmerical (up o projeciviy) images shown in Fig. 2a,d. The objecs symmery is disored by eiher deficiencies or excess pars. We would like o use he symmery o overcome hese shape disorions. Neverheless, incorporaing he unbiased shape consrain (according o Eq. 25) in he cos funcional for segmenaion, resuls in he undesired segmenaion shown in Fig. 2b,e. The symmerical counerpar of a level-se funcion φ is as imperfec as φ. To suppor a correc evoluion of φ by ˆφ, we have o accoun for he specific ype of corrupion. Refer again o he dissimilariy measure in Eq. (25). The cos funcional inegraes he non-overlapping objecbackground regions in boh images indicaed by H ɛ (φ) and H ɛ ( ˆφ). This is equivalen o a poinwise exclusive-or (xor) operaion inegraed over he image domain. We may hus rewrie he funcional as follows: D(φ, ˆφ) = [ ( H ɛ (φ) 1 H ɛ ( ˆφ ) T ) + (1 H ɛ (φ)) T p H ɛ ( ˆφ) ] (26) dx Noe ha he expressions (25) and (26) are approximaely idenical, since H ɛ (φ) (H ɛ (φ)) 2 (equaliy is obained for ɛ 0). There are wo ypes of disagreemen beween he labeling of H ɛ (φ) and T p H ɛ ( ˆφ). The firs addiive erm in he righ hand side of (26) does no vanish, if here exis image regions labeled as objec by φ and labeled as background by is symmerical counerpar ˆφ. The second addiive erm of (26) does no vanish if here exis image regions labeled as background by φ and labeled as objec by ˆφ. We can change he relaive conribuion of each erm by a relaive weigh parameer µ 0: E S (φ, ˆφ) = [ ( µh ɛ (φ) 1 T p H ɛ ( ˆφ) ) + (1 H ɛ (φ)) T p H ɛ ( ˆφ) ] (27) dx The associaed gradien equaion for φ is hen: φ S = δ ɛ (φ)[t p H ɛ ( ˆφ) µ(1 T p H ɛ ( ˆφ))] (28) Now, if excess pars are assumed, he lef penaly erm should be dominan, seing µ>1. Oherwise, if deficiencies are assumed, he righ penaly erm should be domi-

6 (a) (b) (c) (d) (e) (f) Figure 2. (a, d) Images of symmerical objecs up o a projecive ransformaion. The objecs are disored eiher by deficiencies (a) or by excess pars (d). (b, e) Segmenaion (red) of he images in (a) and (d) respecively, using unbiased dissimilariy measure beween φ and is ransformed reflecion as in Eq. (25). Objec segmenaion is furher spoiled due o he imperfecion in is reflecion. (c, f) Successful segmenaion (red) using he biased dissimilariy measure as in Eq. (27). nan, seing µ<1. Fig. 2c,f show segmenaion of symmerical objecs wih eiher deficiencies or excess pars, incorporaing he shape erm (27) wihin he segmenaion funcional. We used µ =0.5 and µ =2for he segmenaion of Fig. 2c and Fig. 2f, respecively Recovery of he ransformaion We now look for he opimal alignmen funcion T p ha minimizes E S defined in eq. (27). The operaion of T p on H ɛ ( ˆφ) is equivalen o he ransformaion of he coordinae sysem of ˆφ(x) by a projecive ransformaion H. T p H ɛ ( ˆφ(x)) = H ɛ ( ˆφ(Hx)) (29) The marix H is defined in Eq (17). The eigh unknown raios of is enries h ˆ ij = h ij /h 33 are recovered hrough he segmenaion process, alernaely wih he evoluion of he level se funcion φ. The parameers ĥij are obained by minimizing (27) wih respec o each. h ˆ ij = δ ɛ (T p ( ˆφ)) [(1 H ɛ (φ)) µh ɛ (φ)] T p( ˆφ) dx Derivaion of Tp( ˆφ) ĥij is done similarly o [25] Unified segmenaion funcional ĥij (30) Symmery-based, edge-based, region-based and smoohness consrains can be inegraed o esablish a comprehensive cos funcional for segmenaion: E(φ) =E PS + E LEN + E GAC + E RA + E S (31) wih he equaions ( 6, 11, 9, 13, 27). The evoluion of φ in each ime sep, φ( +1)=φ()+φ is deermined by φ (φ) =φ RB + φ LEN + φ GAC + φ RA + φ S (32) using a weighed sum (w i ) of equaions (7, 12, 10, 14, 28). Refinemen of he segmenaion can be obained for images wih muliple channels, I : R n, e.g. color images. The region-based erm φ RB and he alignmen erm φ RA sum of he conribuions of each channel I i. Figure 5 demonsraes segmenaion of a color image. Furher exploraion could address he use of Belrami flow [29] Algorihm The algorihm for segmenaion of a symmerical objecs in he presence of projeciviies is summarized as follows: 1. Choose an iniial level-se funcion φ( =0)ha deermines he iniial conour wihin he image. 2. Se iniial values for he ransformaion parameers h ˆ ij. For example se H = I where I is he ideniy marix. 3. Compue u + and u according o (1), based on he curren conour inerior and exerior, defined by φ(). 4. Generae ˆφ, he symmerical counerpar of φ. 5. Updae he alignmen erm T p by recovering he ransformaion parameers h ij. according o (30) 6. Updae φ using he gradien descen equaion (32). 7. Repea seps 3-6 unil convergence. 5. Experimens We exemplify he proposed algorihm for he segmenaion of skew-symmerical objecs. The images are displayed wih he iniial and final segmening conours. Segmenaion resuls are compared o hose obained using he funcional in Eq. (31) wihou he symmery erm. The conribuion of each erm in he gradien descen equaion (32)

7 (a) (b) (c) Figure 3. (a) Inpu image of a roughly symmerical objec wih he iniial segmenaion conour (red). (b) Segmenaion (red) wihou he symmery consrain. (c) Successful segmenaion (red) wih he proposed algorihm. Figure 4. (a) Inpu image of a roughly symmerical objec wih he iniial segmenaion conour (red). (b) Segmenaion (red) wihou he symmery consrain. (c) Successful segmenaion (red) wih he proposed algorihm. Original image couresy of George Payne. URL: hp://cajunimages.com (a) (b) (c) Figure 5. (a) Inpu image of a roughly symmerical objec wih he iniial segmenaion conour (red). (b) Segmenaion (red) wihou he symmery consrain. (c) Successful segmenaion (red) wih he proposed algorihm. Original image couresy of Richard Lindley. URL: hp:// is normalized o [ 1, 1] avoiding he need o guess heir relaive weighs. In Fig. 3 he upper par of he guiar is used o exrac is lower par correcly. In he buerfly image, Fig. 4, a lef-righ reflecion of he evolving level se funcion is used o suppor accurae segmenaion of is lef wing. In Fig. 5 we used he image colors in addiion o he symmery consrain o faciliae he exracion of he swan and is reflecion. 6. Summary This paper has wo major conribuions. Firs, i presens a level-se framework for he segmenaion of symmerical objecs disored by projecive ransformaions. Second, i shows he explici form of he homography ha relaes he image of a skew-symmerical objec o is symmerical counerpar. This homography capures he projecive disorion in he objec symmery. I is recovered hrough he segmenaion process, hus revealing an imporan geomeric informaion on he objec of ineres. Acknowledgmen This research was suppored by MUSCLE: Mulimedia Undersanding hrough Semanics, Compuaion and Learning, a European Nework of Excellence funded by he EC 6h Framework IST Programme.

8 References [1] A. Brucksein and D. Shaked. Skew-symmery deecion via invarian signaures. Paern Recogniion, 31(2): , [2] V. Caselles, R. Kimmel, and G. Sapiro. Geodesic acive conours. IJCV, 22(1):61 79, Feb [3] T. Chan and L. Vese. Acive conours wihou edges. IEEE Transacions on Image Processing, 10(2): , Feb [4] Y. Chen, H. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. Gopinah, R. Briggs, and E. Geiser. Using prior shapes in geomeric acive conours in a variaional framework. IJCV, 50(3): , Dec [5] D. Cremers, T. Kohlberger, and C. Schnorr. Shape saisics in kernel space for variaional image segmenaion. Paern Recogniion, 36(9): , [6] D. Cremers and S. Soao. A pseudo-disance for shape priors in level se segmenaion. In VLSM, pages , [7] D. Cremers, N. Sochen, and C. Schnorr. Muliphase dynamic labeling for variaional recogniion-driven image segmenaion. IJCV, 66(1):67 81, [8] L. Davis. Undersanding shape, ii: Symmery. Trans. on Sysems, Man and Cyberneics, 7: , [9] A. Gupa, V. Prasad, and L. Davis. Exracing regions of symmery. In Proceedings of he Inernaional Conference on Image Processing, pages III: , [10] R. I. Harley and A. Zisserman. Muliple View Geomery in Compuer Vision. Cambridge Universiy Press, 2nd ediion, [11] W. Hong, Y. Yang, and Y. Ma. On symmery and muliple view geomery: Srucure, pose and calibraion from single image. IJCV, 60: , [12] K. Kanaani. Symmery as a coninuous feaure: Commen. PAMI, 19(3): , [13] R. Kimmel. Fas edge inegraion. In S. Osher and N. Paragios, ediors, Geomeric Level Se Mehods in Imaging Vision and Graphics. Springer-Verlag, [14] R. Kimmel and A. Brucksein. Regularized laplacian zero crossings as opimal edge inegraors. IJCV, 53(3): , [15] N. Kiryai and Y. Gofman. Deecing symmery in grey level images: The global opimizaion approach. IJCV, 29(1):29 45, [16] A. Laird and J. Miller. Hierarchical symmery segmenaion. In Proc. SPIE, Inelligen Robos and Compuer Vision IX: Algorihms and Techniques, volume 1381, pages , [17] M. Levenon, W. Grimson, and O. Faugeras. Saisical shape influence in geodesic acive conours. In CVPR, volume I, pages , [18] T. Liu, D. Geiger, and A. Yuille. Segmening by seeking he symmery axis. In Proceedings of he Inernaional Conference on Paern Recogniion, pages , [19] L. Lorigo, O. Faugeras, G. W.E.L., R. Keriven, R. Kikinis, A. Nabavi, and C. Wesin. Codimension wo-geodesic acive conours for he segmenaion of abular srucures. In CVPR, pages , [20] G. Marola. On he deecion of he axes of symmery of symmeric and almos symmeric planar images. PAMI, 11(1): , [21] D. Mukherjee, A. Zisserman, and J. Brady. Shape from symmery deecing and exploiing symmery in affine images. Phil. Trans. of he Royal Sociey of London, 351:77 106, Series A. [22] D. Mumford and J. Shah. Opimal approximaions by piecewise smooh funcions and associaed variaional problems. Communicaions on Pure and Applied Mahemaics, 42: , [23] S. Osher and J. Sehian. Frons propagaing wih curvauredependen speed: Algorihms based on Hamilon-Jacobi formulaions. J. of Comp. Physics, 79:12 49, [24] T. Riklin-Raviv, N. Kiryai, and N. Sochen. Unlevel-ses: Geomery and prior-based segmenaion. In ECCV, volume 4, pages 50 61, [25] T. Riklin-Raviv, N. Kiryai, and N. Sochen. Prior-based segmenaion by projecive regisraion and level ses. In ICCV, volume I, pages , [26] T. Riklin-Raviv, N. Kiryai, and N. Sochen. Shape symmery for segmenaion: a level-se approach. Technical repor, Tel- Aviv Universiy, [27] M. Rousson and R. Deriche. Adapaive segmenaion of vecor valued images. In S. Osher and N. Paragios, ediors, Geomeric Level Se Mehods in Imaging Vision and Graphics. Springer-Verlag, [28] M. Rousson and N. Paragios. Shape priors for level se represenaion. In ECCV, pages 78 92, [29] N. Sochen, R. Kimmel, and R. Malladi. A general framework for low level vision. IEEE Transacions on Image Processing, 7: , Special Issue on Geomeric Acive Diffusion. [30] A. Tsai, A. Yezzi, Jr., W. Wells, III, C. Tempany, D. Tucker, A. Fan, W. Grimson, and A. Willsky. A shape-based approach o he segmenaion of medical imagery using level ses. Trans. on Medical Imaging, 22(2): , [31] L. Van Gool, T. Moons, D. Ungureanu, and A. Ooserlinck. The characerizaion and deecion of skewed symmery. CVIU, 61(1): , [32] L. Vese and T. Chan. A muliphase level se framework for image segmenaion using mumford and shah model. IJCV, 50(3): , [33] Y. Yang, K. Huang, S. Rao, W. Hong, and Y. Ma. Symmerybased 3-d reconsrucion from perspecive images. CVIU, 99: , [34] Y. Yang, S. Rao, K. Huang, W. Hong, and Y. Ma. Geomeric segmenaion of perspecive images based on symmery groups. In ICCV, volume 2, pages , [35] H. Zabrodsky, S. Peleg, and D. Avnir. Symmery as a coninuous feaure. PAMI, 17(12): , [36] H. Zabrodsky and D. Weinshall. Using bilaeral symmery o improve 3d reconsrucion from image sequences. CVIU, 67:48 57, 1997.

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