Internatonal Journal of Computatonal Vson and Bomechancs Internatonal Journal for Computatonal Vson and Bomechancs, Vol. 1, No., July-December 008 Vol.1 No. (July-December, 015 Serals Publcatons 008 ISSN 0973-6778 Dynamc Neghborhoods n Actve Surfaces for 3D Segmentaton Julen Olver, Julen Mlle, Romuald Boné & Jean-Jacques Rousselle Unversté Franços Rabelas de Tours, Laboratore Informatque 64 Avenue Jean Portals, 3700 Tours, France E-mals: julen.olver, julen.mlle, romuald.bone, rousselle@unv-tours.fr Actve contours and surfaces are deformable models used for D and 3D mage segmentaton. In ths paper, we propose two methods developed n order to accelerate 3D mage segmentaton process. They are adaptatons on actve surfaces of two methods developed for D actve contour. We use them on a dscrete 3D surface model (mesh evolvng wth the greedy algorthm. Those methods wll be compared to the classcal greedy algorthm and to a recent fast adaptaton of the level set method. Keywords: Actve surface, mesh, greedy algorthm, fast level set 1. INTRODUCTION Actve contours or snakes, ntally developed by Kass et al. n [1], are powerful segmentaton tools thanks to ther nose robustness and ablty to generate lnked closed boundares. Ther 3D extensons, actve surfaces, were developed accordng to several mplementatons (see [] [3] for surveys on 3D deformable models. Among these mplementatons, meshes are explct dscrete representatons [4], whch represents the surface as a set of nterconnected vertces. The model s deformed by drect modfcatons of vertces coordnates. Several evoluton algorthms have been developed to deform them. One of the most popular s the greedy algorthm [5] because of ts effcency. An adaptaton of the greedy algorthm on 3D surface was proposed by Bulptt and Efford n [6]. Conversely, mplct mplementatons, based on the level set framework [7], handle the surface as the zero level of a hypersurface, defned on the same doman as the mage (for 3D mages, the hypersurface s a 3 applcaton. Level sets are often chosen for ther natural handlng of topologcal changes and adaptveness to an y dmenson. Ther algorthmc complexty s proportonal to the mage resoluton, makng them tme-consumng. Despte the development of acceleratng methods (lke the narrow band technc [7], the fast marchng method [8] and the recent fast level set [9], ther computatonal cost remans hgh, preventng ther use n tme-crtcal applcatons. Moreover, mesh surfaces have several advantages over ther mplct counterparts. Ther representaton s more ntutve, and thus allow easer modelng of a pror knowledge and user nteracton. The man drawback s that meshes do not modfy ther topology naturally (techncs for detecton of topologcal chan ges must be mplemented besde the evoluton algorthm. In many applcatons, the topology of the area of nterest s known n advance. When segmentng mages n whch pror knowledge about the object topology s avalable, we beleve that mesh-based approaches should be prvleged over mplct surfaces. In ths paper, we deal wth a 3D trangular mesh drven by greedy algorthm. The model s able to perform remeshng, thus provdng geometrcal versatlty (n the same manner that D reparameterzaton techncs overcome the lack of geometrcal flexblty of tradtonal snakes. Several methods have been developed n [10] n order to accelerate D actve contours. In ths paper, we study 3D adaptatons of these methods n order to accelerate dscrete actve surfaces, as an extenson of the work n [11]. The outlne of ths paper s as follows: secton presents the 3D model and ts energes, the greedy algorthm for actve surfaces and the remeshng prncple. Secton 3 and 4 descrbe the shfted neghborhood method and the lne search method. Secton 5 descrbes the Fast Level Set mplementaton for actve contours. Secton 6 shows our expermental results on 3D models, comparng the performances of our acceleraton methods wth the basc greedy algorthm and the recent Fast Level Set mplementaton method. Secton 7 concludes wth our work expectng the future developments.. THE 3D ACTIVE SURFACE MODEL In a contnuous doman, a 3D deformable model s represented by a parameterzed surface S mappng a couple of parameters (u, v to a space pont (x, y, z T : S : 3 (u, v (x(u, v, y(u, v, z(u, v T (1 The parameter doman s normalzed : = [0, 1]. The surface s attached to the mage I, whch s a 3 functon, mappng each voxel (x, y, z to a gray level I(x, y, z. Basng ourselves on the work by [5], the surface s endowed wth the energy functonal E(S. S S S E( S 10 01 11 u v uv
174 Internatonal Journal of Computatonal Vson and Bomechancs S S 0 0 P( S dudv ( u v Segmentaton of a regon of nterest n mage I s performed by determnng the optmal surface S * mnmzng E. The energy functonal depends on two knds of energes. The frst one quantfes the geometrcal regularty of the surface, whereas the second one s the external term, dependng on the dstance between the surface and the salent edges of the mage. Frst and second order partal dervatves of S wth respect to u and v are smoothng terms. Coeffcents j weght the sgnfcance of regularzng components wth respect to the external term. 10 and 01 are the elastcty coeffcents, 0 and 0 are the rgdty coeffcents and 11 s the resstance to twst. Snce our segmentaton s boundarybased, P s the external term attractng surface ponts towards salent boundares on the mage. It must decrease as the edge magntude ncreases, hence we choose P = I. It should be noted that other attractng functons could be chosen, e.g. a dstance map wth respect to the thresholded edge mage. To mplement the actve surface, we use the dscrete representaton descrbed n [1], whch s a trangular mesh made up of n vertces, denoted p = (x, y, z T 3, and edges connectng the vertces (makng a set of adjacent trangles. In order to represent the connectvty noton, each vertex p has a set of adjacent vertces, denoted A. The mesh s bult from successve subdvsons of an cosahedron [13] [14], th us leadng to a sphere-lke surface wth a homogeneous vertex dstrbuton (see fgure 1. Fgure 1: The basc cosahedron and ts frst two tesselatons Intally developped for D actve contours by Wllams and Shah n [5], the greedy algorthm s an energy mnmzng method frst proposed as an alternatve to the varatonnal method [1] and the dynamc programmng [15]. It has been recently used for D segmentaton n [16] and [17]. In [6] one may fnd an extenson of ths algorthm for 3D meshes. Global energy mnmzaton s performed va successve local optmzatons. Each vertex s endowed wth ts own energy. Hence, unlke n eq., the energy functonal s the sum of vertex energes. n E( S E( p (3 1 At each teraton, a cubc neghborhood of sde length w around each vertex s consdered (see fg.. The energy s computed at each voxel belongng to the neghborhood and the vertex s moved to the locaton leadng to the lowest energy. Ths approach avods to perform gradent descent of a partal dervatves equaton, derved from the energy functonal usng classcal Euler-Lagrange scheme [13]. Hence, t s not necessary to derve analytcally the energes, wth respect to vertex coordnates. Fgure : Cubc centered neghborhood Unlke n the classcal greedy algorthm, our methods wll deal wth neghborhoods whch are not necessarly (t centered around the vertces. We defne, the neghborhood of the th vertex at teraton t: ( t ( t ( t p r s r[0, 1] 3 (4 s ( s, s, s ( t T x y z w s the shft vector, representng the coordnates of the th vertex at teraton t relatvely to an orgnal voxel chosen on the corner of the neghborhood. At the begnnng, all vertces are centered n ther neghborhood, (0 T hence we have s ( w /, w /, w /. The ntal poston beng p, we denote p a tested locaton n the neghborhood. Once all energes have been computed the new locaton of vertex p s chosen: p ( t 1 arg mn E( pk p (5 k The energy of a vertex at locaton p s a weghted sum of dscrete nternal and external energes, normalzed on the whole neghborhood. E(p = E cont (p + E curv (p + E grad (p + E bal (p (6 The coeffcents,, and are the weghts defnng the relatve nfluence of the energes. The contnuty E cont and the curvature E curv are dscrete mplementatons of frst and second order surface dervatves of eq., respectvely. Parameters controls the surface elastcty whereas s the rgdty. They have a smlar role than coeffcents j n the contnuous model. Let us descrbe the adaptaton of the dfferent energes to our 3D model. The energes are ntutve extensons of the D actve contour ones, sutable to our mesh representaton. Whle the dscretzaton of nternal energes s smple for a parametrc snake, t s not obvous how to mplement the surface dervatves of eq. n a trangular mesh. For a dscrete planar contour, the contnuty s the dstance between successve neghbors. However, lke n [5], ( t
Dynamc Neghbourhoods n Actve Surfaces for 3D Segmentaton 175 t s preferable to use the absolute dfference between ths dstance and a default length d, whch may be the mean dstance computed over all vertces. Extendng ths prncple to the mesh, E cont mantans the vertces evenly spaced along the surface. Mnmzng t reduces the gap between the mean squared dstance d and the dstance between the consdered vertex and ts adjacent vertces. E ( p d p p cont j ja d 1 1 n j n 1 A ja p p (7 The second nternal energy s the curvature E curv, whch mnmzaton results n a local smoothng effect, by makng the vertex get closer to the centrod of ts adjacent vertces. In a D snake, curvature s equvalent to the squared dstance between the vertex and the mddle of ts two neghbors. By extenson to 3D, the curvature of the tested pont p s the squared dstance between p and the centrod of the neghbors of vertex p. E 1 ( p p p (8 A j A curv j Note that for a gven mesh vertex p, E curv (p = 0 f p and all ts adjacent vertces le on the same plane. To attract vertces towards salent edges, the external energy E grad s a functon of normalzed gradent magntude g of mage I. In presence of nosy data, the mage s smoothed wth a gaussan flter pror to gradent operaton. In the followng equatons, G s a gaussan kernel wth standard devaton, s the convoluton operator and g max s the maxmal edge ntensty n the mage. g(p = I(p G /g max E grad (p = g(p (9 As regards gradent magntude, real 3D edge detecton s obtaned by convolvng the mage wth the Zucker- Hummel operator [18], whch usually yelds better edge localzaton than usng drect fnte dfferences. It s made up of three 3 3 3 masks ZH x, ZH y and ZH z, each mask flterng the mage n one dmenson. k1 0 k1 k 0 k k1 0 k1 ZH k 0 k k 0 k k 0 k x 3 3 k1 0 k 1 k 0 k k1 0 k 1 k1 k k1 k k3 k k1 k k1 ZH y 0 0 0 0 0 0 0 0 0 k k k k k k k k k 1 1 3 1 1 k1 k k1 0 0 0 k1 k k1 ZH k k k 0 0 0 k k k z 3 3 k1 k k 1 0 0 0 k1 k k 1 (10 (11 (1 3 k1 ; k ; k3 1 (13 3 To ncrease the capture range, we ntroduce a balloon energy E bal derved from the nflaton force proposed n [19]. It allows the mesh to be ntalzed far from the object boundares. E bal ( p p ( p kn (14 where n s the unt nward normal, defned at vertex p. The normal of vertex p s the normalzed sum of the normals of the neghborng trangles [14]. Rgorously, the normal of a trangle t s the unt vector orthogonal to the plane defned by t. In the followng expressons, T s the set of neghborng trangles of p. The normal n t of a gven trangle s the normalzed cross product between two vectors belongng to the correspondng plane. nt tt ( p ( t p t p 1 t p 3 t1 nt ; nt st ( pt p ( t p 1 t p 3 t1 n (15 tt t where p tj, j = 1...3 are vertces of trangle t (p must be one of them. s t = ±1 s the sgn changng the orentaton of n t nsurng that t ponts towards the nteror of the surface. Such a calculaton of the normal vector s necessary to a correct balloon mplementaton. The moton resultng from the balloon energy mnmzaton s ether an expanson or a retracton of the surface, dependng on the sgn of coffecent. Ths one must be chosen regardng the ntal poston of the surface wth respect to the target object. In order to adapt local topology, remeshng s performed after each teraton of the greedy algorthm. The mesh s allowed to add or delete vertces to keep the dstance between adjacent vertces homogeneous, resultng n a stable vertex dstrbuton [0] [14] [1]. It nsures that every couple of adjacent vertces (p, p j satsfes the constrant: d mn p p j d max (16 where d mn and d max are two user-defned thresholds, such that d max d mn. We choose ther values near the neghborhood wdth w, so that the surface samplng s consstent wth the moton range of the vertces. Addng or deletng vertces modfes local topology, thus topologcal constrants should be verfed. To perform vertex addng or deletng, p and p j should share exactly two common adjacent vertces: A A j =. When p p j > d max, a new vertex s created at the mddle of lne segment p p j and connected to p a and p b (see mddle part of fgure 3. When p p j < d mn, p j s deleted and p s translated to the mddle locaton (see rght part of fgure 3. 3. THE SHIFTED NEIGHBORHOOD METHOD In order to mprove actve surfaces completon tme, we used the shfted neghborhood method developed for D snakes
176 Internatonal Journal of Computatonal Vson and Bomechancs Fgure 3: Remeshng operatons: vertex addng and deletng n [10]. We adapted ths greedy-based method on actve surfaces. For each vertex and at each teraton, we modfy the neghborhood n order to drect the searchng space of each vertex to the drectons that seems the most nterestng. To defne where these drectons are, we use the nformaton of the drecton followed by each vertex durng the last teraton. So each neghborhood wll be shfted from one voxel n the drecton followed prevously. At each teraton, we compute the next shft appled to the vertex wth: ( t 1 ( t 1 ( t d ( 1, 1, p p (17 ( t The vector quantty d represents the dsplacement appled on the neghborhood of the th vertex at teraton t. s a shft lmtng functon, boundng the vector coordnates between two scalars: max( b1, mn( b, ux ( b1, b, u max( b1, mn( b, uy (18 max( b1, mn( b, uz ( t 1 The dsplacement d gven by equaton (17 allows ( t 1 us to defne the new shft vector s for each vertex of the actve surface. Thus we have: ( t 1 ( ( 1 (1, w, t t s s d (19 Algorthm 1 Shfted Neghborhood Method: 3D Model 1: for t 1 to T do : for 1 to n do (t+1 3: p = arg mn E( p ( t k pk ( t 1 ( t 1 ( t 4: d ( 1,1, p p ( 1 ( ( 1 5. (1,, s w s d t N t p t r s r[0, w 1] ( 1 ( 1 ( 3 6: 7: end for 8: end for The next teraton of the greedy algorthm wll be helded wth these new neghborhoods. At ths stage, we can defne the algorthm for the shfted neghborhood method. Ths last conssts n computng the new neghborhood (t+1 wth equatons (4, (17 and (19 at the end of each teraton, once all the vertces of the actve surface have been modfed. When ncluded n the greedy algorthm for an actve surface of n vertces and T teratons, the shfted neghborhood method s descrbed n algorthm 1. 4. THE LINE SEARCH METHOD The lne search method [10] orgnally appled for twodmensonal actve contours allows to reduce completon tme effcently. We adapted ths method on 3D actve surfaces. The prncple of ths approach s to antcpate on the next teraton of the greedy algorthm usng the nformaton taken from the prevous one. Ths method s launched at the end of each teraton of the greedy algorthm, once all the vertces have been translated. The drecton followed by each vertex p s memorzed and we look toward t for a fxed number of voxels, whch creates a lnear neghborhood. These lasts are compared by computng ther global energes n a smlar way as t s done wth the cubc neghborhood (see equaton (6. The voxel gvng the lowest energy s then chosen for the new locaton of the current vertex. As a result, for each vertex, two neghborhoods are scanned consecutvely: the cubc centered neghborhood and the lnear neghborhood. The second algorthm descrbes the lne search method ntegrated n the greedy algorthm for actve surfaces. Let T be the number of teratons to be done by the greedy algorthm and l the number of voxels to be explored (length of the lnear neghborhood. 5. LEVEL SET AND FAST LEVEL SET IMPLEMENTATION In order to compare our methods wth recent breaktroughs n the mage segmentaton doman we adapted the Fast Level Set mplementaton method to our 3D segmentaton problem. Dynamc neghborhoods n actve surfaces for 3D segmentaton 9 Algorthm : Lne Search Method: 3D Model 1: for t 1 to T do : for 1 to n do p arg mn E( p ( t 1 3: ( t pk p 4: Determne the drecton v p k p ( t 1 ( t ( t 1 ( t p 5: Lne Search: m = ( t arg mn E( p kv k[0, l] ( t 1 ( t1 6: Update: p p v 7: end for 8: end for m Basng ourselves on the work of Osher et al. [] and [7], we mplemented our actve surface model wth level sets. We consder the parameterzed surface S defned n secton, on whch we add a tme dependency. S : + 3 (u, v, t (x(u, v, t, y(u, v, t, z(u, v, t T (0 The surface evolves accordng to the followng partal dfferental equaton:
Dynamc Neghbourhoods n Actve Surfaces for 3D Segmentaton 177 S( u, v, t Fn (1 t where n s the nward normal vector and F s the speed functon appled on the surface ponts. The surface boundary s mplctly represented as the zero-level of a functon : 3 +. It comes: (S(u, v, t, t = 0 (u, v, t 0 ( Functon evolves on ts zero-level accordng to the equaton: ( S( u, v, t, t F( u, v, t ( S( u, v, t, t (3 t If the speed functon F s defned over the entre doman 3 + then eq. 3 can be extended such as: ( x, t 3 F( x, t ( x, t x, t 0 (4 t The man advantage of the level-set mplementaton s ts ablty to automatcally handle topologcal changes. Indeed, the curve can naturally splt or merge wth others wthout any addtonnal mplementaton. Unfortunately the level-set method requres sgnfcant computatonnal tme. Several methods have been developed n order to accelerate the level-set mplemen taton. Th e n arrow ban d mplementaton [8] allows to decrease the complexty of the algorthm from O(n to O(n, n beng the sze of the mage grd. The author consders a band around the zero-level of the level-set functon. The partal derental equaton (PDE s solved only nsde ths regon and not n the entre defnton doman of. Although ths technque allows to sgnfcally reduce computatonal tme of the level-set mplementaton, t stll lmts the use of the level-set n real tme applcatons such as object trackng. Sh et al have recently developed n [9] an acceleraton method based on the narrow band mplementaton called the Fast Level Set method. As t was orgnally desgned for D segmentaton, we extended t to 3D. The Fast Level Set method uses two lsts to represent the surface: the lst of outsde boundary ponts L out and the lst of nsde boundary ponts L n. L out = {x (x > 0, y N(x, (y < 0} L n = {x (x < 0, y N(x, (y > 0} (5 where N(x s the dscrete neghborhood of x. The authors assume (x to take only four nteger values, accordng to the poston of x: 1 f xlout 1 f x Ln (x 3 f x soutsdes and xlout (6 3 f x snsde S and xln The authors defne the dcrete optmalty condton for the surface as: The surface S wth boundary ponts L n and L out s optmal f the speed functon F satsfes: F(x < 0 x L out and F(x > 0 x L n (7 Whle ths optmalty condton s not reached, the speed functon F s calculated for every pont n L n and L out. If F(x > 0 at a pont n L out the surface s moved outward. If F(x < 0 at a pont n L n the surface s moved nward. Once the surface has moved, the two lsts are updated. The man dea of ths method s thus to make the surface evolve wthout solvng the PDE of 4 whch requres sgnfcant computatonnal tme but only by computng the speed functon F. Agan, segmentaton s boundary-based, hence the surface should stop on strong edges. As a result, F s a functon of curvature and gradent magntude. F(x = I(x (x (8 The curvature regularzes and s weghted by coecent. It s expressed as follows: dv = xx ( y z yy ( x z zz ( x y xy x y xz x z yz y z wth notatons x ( 3/ x y z, xx,... x x For more detals and testngs on the method one can refer to [9]. The next secton descrbes and compares the results obtaned on tested mages wth our greedy acceleraton technques. 6. EXPERIMENTAL RESULTS In ths secton, we present our experments on 3D mages. We compare the shfted neghborhood method and the lne search method to the classcal greedy algorthm. Ths comparatve study also ncludes the results obtaned wth mplct surface modelng mplemented wth the fast level set method. Each tested mage s made of several slces of gray objects embedded n whte backgrounds, hghly corrupted wth gaussan nose. Derent values of neghborhood wdth w are tested (obvously, the shfted neghborhood s not expermented wth w = 3. For each mage, the surface s ntalzed as a sphere wth dentcal center and radus for all evoluton methods, ndependently from ts mplementaton (a mesh or a level set. In order to evaluate segmentaton, we use a functon takng nto account the overall dstance between the estmated boundary and ground truth. Let be the set of voxels belongng to the real boundary, and the set of voxels belongng to the estmated boundary. For each voxel on the estmated boundary, we consder the dstance to the nearest voxel on the real boundary, and conversely. The modfed Hausdor dstance mean ntroduced n [3] measures the average fttng of the surface to the real boundary. mean (, = max(h mean (,, h mean (, h mean (, = 1 mn p q q p (30
178 Internatonal Journal of Computatonal Vson and Bomechancs In what follows, we compare computatonal tmes obtaned on fnal surfaces havng equvalent qualtes, wth respect to the modfef Hausdo dstance. Typcally, H mean have values around 1, whch corresponds to good fttng of the real boundares. The frst mage s a 400 400 400 data set representng a spral and was chosen n order to test the methods when vertex addng s enabled. The actve surface s ntalzed nsde the 3D model wth only 1 vertces. The fnal meshes for the three methods contan about one thousand vertces. We choose = 0, = 0.5, = and [ 0.6, 1.1]. The second mage s a 00 00 00 model of a vase and s nterestng to test the nfltraton of the model nto the concavtes. For ths partcular 3D dataset, remeshng of the model s dsabled. We use = 0, = 0.5 for the greedy algorthm, 0.4 for the shfted neghborhood method and 0.3 for the lne search, = and = 0.8. We ntalze the meshes wth 56 vertces. The thrd mage represents three ellpsods and allows to have both salent and smooth angles on the same model. The mage sze s 00 00 00. We prevent remeshng of the model and ntalze t wth 56 vertces. The parameters are = 0.5, = 0.3 for the greedy algorthm and 0.4 for the shfted neghborhood method and the lne search, = and [0.3, 0.7]. In order to compare boundary fttng ablty of the methods on real data, we tested them on computed tomography (CT data sets of the abdomen, n whch the actve surface was used to detect boundares of the aorta. Such segmentaton s done n the framework of abdomnal aortc aneurysm dagnoss. The mesh was ntalazed as a small sphere nsde the aorta and nflated thanks to the balloon energy. Fgure 4 shows one slce of each mage and the vsual 3D results obtaned wth the lne search method. Tables 1 and lst computatonal tmes obtaned on synthetc and CT mages, respectvely. Each method and each wndow wdth s systematcally tested on the mages (tests were made on a Pentum IV 1.7 GHz wth 51 Mb RAM. Mesh-based methods (ntal greedy algorthm, LS and SN need fewer teratons than the level set surface to reach the boundares, whch s manly due to the level set front. A voxel neghborng the front (the L out lst needs more than one teraton to change ts status, from outer to nner voxel. An teratons of the level set method s also more computatonally ntensve than one of the explct methods. The dscretzaton of the evolvng front s the same as the mage grd, so that each voxel located on the front needs to be consdered. On the mesh, the moton of a vertex does not aect ts coordnates but also all neghborng trangles, whch cover many voxels (the quantty of voxels depends on the samplng resoluton w. As regards dstance measures, t s nterestng to note that explct methods lead to more accurate results than the level set approach, except on the Vase data set. We may assume that level sets are globally more senstve to nose and prone to boundary leakage ssues, because of the mplct formulaton of the regularzng curvature energy, whch has a lmted range. On the mesh, the nternal energy of a vertex aects a larger porton of the surface. The better performance of the mplct surface on the Vase data set s explaned by the presence of sharp angles at the shape borders. Curvature prevents the mesh from fttng angular parts accurately, whereas the level set surface s not lmted thanks to ts dscretzaton. Indeed, t can easly grow voxels nto small concave parts of the boundary. Fgure 4: D slces of 3D nosy mages (top and surface results obtaned wth the lne search method (bottom
Dynamc Neghbourhoods n Actve Surfaces for 3D Segmentaton 179 Table 1 Comparson of Completon Tmes between Classcal Greedy Algorthm, Lne Search Method (LS, Shfted Neghborhood Method (SN and Fast Level Set on Synthetc Data Sets Image Neghborhood Method # teratons Tme (s Hmean w=3 Greedy 400 0.30 0.497 Spral LS 65 0.16 0.499 (400 400 400 w=5 Greedy 195 0.5 0.73 voxels LS 63 0.30 0.734 SN 138 0.31 0.78 w=7 Greedy 145 0.63 0.985 LS 60 0.57 0.997 SN 97 1.11 0.991 Fast level set 1060 44.17 0.995 3 ellpsods w=3 Greedy 47 0.63 0.534 (00 00 00 LS 19 0.41 0.545 voxels Greedy 5 1.05 0.716 w=5 LS 1 0.69 0.734 SN 16 0.98 0.71 w=7 Greedy 18.4 0.961 LS 1 1.86 0.985 SN 14.07 0.97 Fast level set 10 30.8 1.030 Vase w=3 Greedy 155 0.98 0.665 (00 00 00 LS 68 0.7 0.687 voxels w=5 Greedy 10 1.7 0.875 LS 50 0.85 0.896 SN 9 1.08 0.89 w=7 Greedy 109 1.98 1.337 LS 43 1.13 1.554 SN 60.0 1.43 Fast level set 10 47.04 0.815 Table Comparson of Completon Tmes between Classcal Greedy Algorthm, lne Search Method (LS, Shfted Neghborhood Method (SN and Fast Level Set on CT (Computed Tomography Data Sets Image Neghborhood Method # teratons Tme (s Hmean CT 1 w=3 Greedy 580 5.30 1.34 (51 51 810 LS 98 17.7 1.458 voxels w=5 Greedy 330 9.65 1.694 LS 117 7.3 1.734 SN 56 8.03 1.710 w=7 Greedy 40 8.69 1.936 LS 10 6.68.034 SN 168 9.73 1.967 Fast level set 17 30.41 3.345 CT w=3 Greedy 45 18.57 1.15 (51 51 400 LS 96 13.99 1.41 voxels w=5 Greedy 4 7.08 1.439 LS 85 5.30 1.476 SN 187 5.89 1.454 w=7 Greedy 176 6.37 1.646 LS 86 4.95 1.756 SN 136 7.16 1.67 Fast level set 750 185.07.381 CT 3 w=3 Greedy 313 14.34 0.754 (51 51 400 LS 75 10. 0.83 voxels w=5 Greedy 178 5.3 0.965 LS 65 4.47 0.988 SN 137 4.86 0.974 w=7 Greedy 13 4.8 1.104 LS 6 3.81 1.159 SN 96 5.49 1.11 Fast level set 735 18.446.016 7. CONCLUSION AND FUTURE WORK In ths artcle we have descrbed two acceleraton methods for actve surfaces evolvng wth the greedy algorthm. The frst one s based on a smart orentaton of the neghbourhood grd of each vertex regardng the drectons followed n the precedng teratons. The second one uses the same drecton nformaton to make each vertex search for a better poston along an exploraton lne. The man applcaton doman of our methods s tme-dependent segmentaton wth a pror knowledge about the topology of the object, such as 3D vdeos. We compared our two acceleraton methods wth a recent fast mplct mplementaton of actve contour based on the level-set approach. As shown n table 1 and, our methods allowed us to accelerate the greedy algorthm for actve surfaces. Meshbased methods (the greedy algorthm and our acceleraton methods leads to performances turnng out to be far beyond the level set-based method ones. Acceleratons methods tend to make completon tme fall below 1s (for the best confguraton whereas the fast level set approach exceeds 30s. The best acceleraton method for D actve contours was the shfted neghborhood but we detected the lne search as the best to be appled on 3D actve surfaces. The explcaton s that n three dmensons an exploraton lne stays a lne whereas a square neghborhood becomes cubc, rsng the completon tmes added by the shfted neghborhood method. We also tested the Deformed Neghborhood method dscrbed n [10] but t was not ecent on actve surfaces for the same reasons. We can also notce that the contrbuton of the shfted neghborhood method s better wth a large neghborhood. Indeed, the shftngs are dependent of ts sze. We are developng an hybrd model based on the shfted neghborhood method and the physc-based approach of parametrc actve contours [4]. The man dea s to use the nformaton of the force vector to drect the shft of the neghborhood grd. We also plan to upgrade the performances of our methods by usng a mult-resoluton approach. REFERENCES [1] Kass, M., Wtkn, A., Terzopoulos, D.: Snakes: actve contour models. Internatonal Journal of Computer Vson 1 (1988, 31 331. [] Montagnat, J., Delngette, H., Ayache, N.: A revew of deformable surfaces: topology, geometry, and deformaton. Image and Vson Computng 19 (001, 103 1040. [3] Mlle, J., Boné, R., Makrs, P., Cardot, H.: 3D segmentaton usng actve surface: a survey and a new model. In: 5th Int. Conf. on Vsualzaton, Imagng & Image Processng (VIIP, Bendorm, Span (005, 610 615. [4] Zhang, Z., Braun, F.: Fully 3D actve surface models wth selfnflaton and self-deflaton forces. In: IEEE Computer Vson and Pattern Recognton (CVPR, San Juan, Puerto Rco (1997, 85 90. [5] Wllams, D., Shah, M.: A fast algorthm for actve contours and curvature estmaton. Computer Vson, Graphcs, and Image Processng: Image Understandng 55 (199, 14 6.
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