Snakes with topology control

Transcription

1 The Viual Computer manucript No. (will be inerted by the editor) Snake with topology control Stephan Bichoff, Leif P. Kobbelt RWTH Aachen, Lehrtuhl für Informatik VIII, Aachen, Germany {bichoff Received: date / Revied verion: date Abtract We preent a novel approach for repreenting and evolving deformable active contour by retricting the movement of the contour vertice to the grid-line of a uniform lattice. Thi retriction implicitly control the (re-) parameterization of the contour and hence make it poible to employ parameterization independent evolution rule. Moreover, the underlying uniform grid make elf-colliion detection very efficient. Our contour model i alo able to perform topology change but more importantly it can detect and handle elf-colliion at ub-pixel preciion. In application where topology change are not appropriate we generate contour that touch themelve without any gap or elf-interection. Key word active contour model topology control implicit parameterization 1 Introduction For the egmentation and hape recontruction from noiy image data, contour extraction cheme baed on deformable model have become a tandard technique. The major reaon for their ucceful ue in many application i the poibility to integrate phyical and topological knowledge into the egmentation proce and thu interpolate the image information where it i detroyed by noie. Variou repreentation have been propoed which adapt to the extreme requirement in an active contour model. Explicit contour repreentation can be proceed very efficiently and their phyical propertie can be controlled in a very intuitive manner. Implicit contour repreentation require more Correpondence to: Stephan Bichoff Lehrtuhl für Informatik VIII Aachen Germany bichoff@informatik.rwth-aachen.de Phone: Fax: ophiticated implementation but they are free of parameterization artifact and they allow the contour to change it topology in a natural manner (ee the next ection for a more detailed decription). Our new approach inherit from both the explicit and the implicit framework: The repreentation of the contour i baically explicit, it evolution however i governed by parameterization independent rule imilar to thoe of the fat marching method in the level et framework. Self-colliion of the contour can be detected eaily and the algorithm can flexibly decide if the contour topology hould change or hould be preerved. The main contribution of our propoed cheme are: Flexible topology control. In contrat to previou work we are able to efficiently detect and reolve elf-colliion without globally reparameterizing the contour. Depending on the uer preference, our algorithm can be tuned to preerve the contour topology a well a to merge the colliding contour. Automatic reampling. The reolution and parameterization of our contour i automatically determined by an underlying uniform lattice. A a conequence, there i no need for a complicated global reampling procedure when the contour i deformed. Simplicity. The baic operation ued in our cheme are conceptually traightforward and can be implemented eaily. All computation during the evolution are local and no handling of pecial cae i neceary. In particular, there i no need to maintain and update elaborate data tructure, like narrow band of voxel, or to approximate and dicretize differential equation. Speed. Due to the robutne of the evolution procedure and the flexible control of the time tep, we can ue an explicit Euler integration cheme to trace the contour through the embedding force field. Overview In Section 2 we give a hort overview of previou and related work. Section 3 briefly introduce active contour model. In Section 4 we decribe our new cheme and give implementation hint. Reult for ynthetic a well a for real

2 2 Stephan Bichoff, Leif P. Kobbelt data are preented in Section 5. In Section 6 we give concluion and propoe future work. 2 Previou and related work t Previou work In recent year, image egmentation baed on active contour model ha become a powerful tool. Epecially in medical imaging application, like the egmentation of organic tructure or the dicrimination of brain tiue, thee model are ubiquitou [1,6,10,16,23]. Depending on the repreentation of the contour hape a the range or the kernel of a function, active contour model can be claified a either explicit or implicit. Image egmentation baed on explicit active contour model ha firt been introduced in 1987 by Ka et. al [13]. In their work, a contour i repreented by a parametric model (a o-called nake) and it evolution i governed by minimizing an energy-functional and applying a emi-implicit integration cheme. Since then numerou refinement and extenion to the original cheme have been propoed [3 5,8,12, 19]. Several author have introduced different explicit repreentation e.g. finite element model [3] and ubdiviion curve [12]. The explicit active contour model ha alo been generalized to higher dimenion, uch a to egment volume data like MRI can of human organ [3, 5, 22]. Early explicit active contour model could not handle topology change, like merging or plitting of contour. In order to overcome thi limitation everal author have propoed topologically adaptive contour model which are e.g. baed on repeated reampling of the contour on an affine grid [7, 8, 15, 18, 20]. Implicit model, on the other hand, repreent the contour a the (zero-) level et of a calar field and were firt introduced by Sethian and Oher in 1988 [24]. Since then, level et method have been applied in numerou application, among them image egmentation, fluid dynamic and computer viion. We refer to the book [25, 27] for a thorough overview. In order to overcome the computational complexity of level et method, fat marching and narrow band method have been introduced [2, 26]. Implicit repreentation can eaily handle topological change of the contour, o in contrat to explicit repreentation pecial care ha to be taken, if topological change of the level et have to be avoided [11]. Contribution In thi paper, we introduce an active contour model that combine propertie of the implicit a well a the explicit framework. The evolution of the contour i driven by Huygen principle and hence reemble that of the level et method. The topology of the contour i repreented explicitly by a control polygon and can be compared to the traditional nake approach. Note, however, that we only addre the evolution proce and the topology control of the contour we do not propoe any new way of defining image gradient force or otherwie improve the quality of the egmentation. Thi i reflected by the fact that we will conider the force that drive the contour a a black box which i provided by the uer and which incorporate all the external c(,t) Fig. 1 Evolving contour: A contour c(, t) i parameterized by arc length and time t. The movement of each contour point c(, t) can be decompoed into tangential and normal component. Whilt the tangential component affect the parameterization of the contour, only the normal component modifie the contour hape. and internal force that account for the quality of the final egmentation. The combination of implicit and explicit technique provide u with a greatly improved control over the topology of the contour. Colliion can accurately and robutly be detected and reolved without incurring a run-time overhead. Purely implicit model, in contrat, provide no colliion detection at all. For purely explicit model colliion detection i poible, but expenive and often inaccurate. In particular, in our model the uer can chooe whether two colliding contour hould clah, an operation that i not poible in implicit framework or whether they hould merge, an operation that i inefficient in explicit framework. 3 Active contour model In thi ection we give a hort introduction to active contour model. For implicity we will retrict ourelve to the twodimenional cae although mot of the following can readily be generalized to higher dimenion. The idea of active contour model i to track the evolution of a imple, cloed curve, the o-called contour. The contour can be repreented either implicitly a the level et of a function [25, 27] or explicitly by a parametric repreentation [3, 12,13]. Here we will focu on the latter cae. Conider a contour c(, t) R 2 where [0, 1] parameterize the contour arc and t R 0 deignate time, ee Figure 1. The evolution of c can then be decribed by the following equation c t = α t + β n (1) where t i the tangent, n i the outward normal and α and β are arbitrary function decribing the tangential and the normal peed of the contour repectively. Here and in the following we will aume that the contour i cloed, i.e. that c(0, t) = c(1, t) and that it conit of only one component. n

3 Snake with topology control 3 There are numerou way to define the function α and β. In the claical etup they are choen uch that the contour minimize an energy functional E(c) = E internal (c) + E external (c). (2) E internal repreent the internal energy of the contour, and i in general a weighted combination of membrane and thinplate energy which penalize tretching and bending, rep. It i ued to regularize, i.e. to mooth the contour, and hence to avoid artifact like overhooting or ripple. E external repreent the external energy which i in general a potential field derived from the underlying egmentation problem, e.g. attraction to image feature. It can be hown, that for each choice of peed function (α, β) there exit other peed function (0, β) uch that the reulting contour hape are equivalent [9, 14]. Hence, the tangential component α in general only affect the parameterization of the contour while β determine the contour hape. For parameterization-le formulation, like the implicit level et formulation or our r-nake formulation (ee Section 4), Equation 1 can be implified to c t = βn i.e. the contour only evolve in normal direction. In practice, the contour c i repreented either dicretely by the vertice of a polygon or continuouly by e.g. B-pline, ubdiviion curve or other bai function. For our purpoe, we dicretize the contour c a a polygon in pace and in time by a equence of vertice, o-called naxel, C (t i ) = c i 1,..., c i n i, i N where n i deignate the number of vertice of the nake at timetep t i R 0. By approximating derivative through finite difference, the continuou Equation 2 can then be tranformed into a dicrete update rule for the naxel poition. In thi dicrete etup, the tangential peed α can be thought of a regularizing the vertex ditribution, e.g. toward uniform or curvature dependent vertex pacing. However, in general ome local or global vertex inertion/deletion trategie have to be implemented in order to adapt the number of vertice to the contour length. 4 Parameterization free active contour model In thi ection we preent a implified type of nake that we call retricted nake (r-nake). Although r-nake lack ome of the original nake flexibility, they can be ued in a wide range of etting and allow for topology preerving, interection-free evolution of a contour. An r-nake i a pecial type of nake. Intead of letting the naxel move freely, we impoe certain retriction on their movement. Mot importantly the naxel may only move along the line of a given, fixed grid. Whenever a naxel run into a gridpoint, it i automatically plit. Finally we aume that the nake move only normal to itelf and that it may not elf-interect. The above retriction allow u on the one hand to efficiently detect and avoid colliion. On the other hand, they automatically reample the nake according to the reolution of the underlying grid. 4.1 Definition In the following we aume that the Euclidean plane i ubdivided by a Z Z integer grid into unit quare that we call pixel. The ide of the pixel are called grid egment in contrat to nake egment that join two conecutive naxel. Conider an interection-free, cloed nake S = 1,..., n, uch that S divide the Euclidean plane into an interior and an exterior part. We call S a retricted nake, r-nake for hort, if the following three propertie hold: 1. (Supporting egment) Each naxel S lie on a grid egment which we call the upporting egment of. To be more precie, for each naxel there are two gridpoint f Z Z ( from ) and t Z Z ( to ) uch that f t = 1 and an affine parameter 0 d < 1 ( ditance ) uch that poition p on the Euclidean plane i given a (ee Figure 2). f interior p = (1 d ) f + d t d exterior Fig. 2 Snaxel: In thi and the other figure naxel are repreented a arrowhead uch a to indicate their from and to vertice. 2. (Orientation) All naxel are conitently oriented. By convention, each naxel point from the interior of S to the exterior of S (ee Figure 2). 3. (Uniquene) No two conecutive nake egment of an r-nake S may lie in the ame pixel. Note that thi condition follow readily from condition 2, and i merely tated for convenience. Hence naxel configuration a hown in Figure 3 are forbidden and notche that are thinner than one pixel cannot be repreented. t

4 4 Stephan Bichoff, Leif P. Kobbelt a) b) Fig. 3 Forbidden naxel configuration: The configuration hown above are forbidden, a the darkened naxel point from the interior to the interior of the contour. 4.2 Implementation For repreenting an r-nake, we ue a imple data tructure. Each naxel object ha 8 member, namely truct Snaxel { float d; /* affine parameter */ float v; /* peed */ int fx, fy; /* "from" vertex */ int tx, ty; /* "to" vertex */ Snaxel *next, *prev; /* connectivity */ } The next and prev pointer are ued to arrange all naxel of an r-nake in counter-clockwie direction in a doublylinked lit. An r-nake can be initialized by reampling an arbitrary cloed, elf-interection free curve on the Z Z grid. If the curve i e.g. given by a igned ditance function, the reampling can eaily be performed by a Marching Cube like algorithm [17]. A typical r-nake i hown in Figure Evolving an r-nake In thi ection we decribe how an r-nake evolve according to the impact of external and internal force. In the following we will alway aume 1. that the r-nake move in normal direction and 2. that the r-nake move outward only. In general the tangential component of a force affect only the contour parameterization but not it geometry [14]. Retriction 1 i hence very natural in the parameterizationle level et framework. A the parameterization of an r- nake i automatically adapted according to the underlying grid and hence doe not need to be adjuted by tangential force, we alo apply retriction 1 for our etup. Retriction 2 i baically for convenience only. The following expoition could alo be formulated without thi retriction but then it would be more elaborate. Note that retriction 2 can alo be circumvented by alternately revering the orientation of the r-nake after each update tep, hence exchanging the inide and the outide, ee alo [21]. In general, the evolution of a nake i determined by variou factor and parameter, like external and internal force. For the ake of generality and implicity, however, we aume the exitence of a black box v, which, given an arbitrary naxel compute the (calar) peed v of in direction normal to the r-nake. Thi black box peed function i aumed to take the application dependent internal and external energie into account. Suppoe that for each naxel a normal n i given (thi will be explained in more detail below). In general the normal n of the naxel will not coincide with the direction of the upporting egment of. A the naxel can only move along it upporting egment, we have to project the normal onto the egment and compute the projected peed ṽ of the naxel. A one can ee from Figure 5 the projected naxel peed can be eaily computed a v ṽ = n d where d = t f i the unit vector pointing in direction of the upporting egment of. Thi formula reult in the following update rule for the naxel poition d d + t ṽ Fig. 4 A typical r-nake: Note that the two darkened naxel hare the ame upporting egment. Such a ub-pixel configuration could not be modeled with nake in level et formulation [11]. where t i the timetep. (The computation of t i decribed in Section 4.4.) Note that the projected peed ṽ i in general larger in magnitude than the original peed v, a d i in general not parallel to n. There are numerou way to approximate a normal n in a naxel. Although often ufficient, thee cheme tend to exhibit ome artifact a i demontrated in the following example. Conider an r-nake which evolve with contant (unit) peed v 1 and let t = 0.5. After updating all vertice we expect that the r-nake ha moved outward by 0.5 unit. Actually, however, becaue we ue only approximated normal, cup and creae may appear a i hown in Figure 6.

5 Snake with topology control 5 n The normal lie on different ide of the upporting egment of (Figure 8b). In thi cae the final projected peed hould be et to f t ṽ = v 1/( n d ) Fig. 5 Projecting the naxel peed: A naxel can only move along their upporting egment, the normal peed ha to be projected onto the egment. n b n a a In thi cae, the peed ṽ reulting from the projection of n i too high and reult in an overhooting effect. Analogou effect can be oberved in cae of a concave corner. b Fig. 7 Normal approximation: In each naxel we approximate normal from the left and from the right by taking the normal of the two nake egment adjacent to. 0.5 >0.5 Fig. 6 A contour evolving with unit peed v 0.5 may neverthele develop cup becaue of the arbitrary approximation of the normal n of the naxel. To avoid the overhooting and to avoid the neceity to approximate vertex normal altogether, we employ a contruction following Huygen principle [27]. For thi we imagine for a moment, that the r-nake i not dicrete but continuou and that it locally evolve with contant peed. The interection of the reulting continuou contour with the grid will then determine the new naxel poition of the dicrete contour. Conider the cae of a naxel on a convex corner. We approximate normal both from the left and from the right by taking the normal of the two nake egment adjacent to. To be more precie, let be the naxel under conideration, a it predeceor and b it ucceor (ee Figure 7). Then we et n a = (p p a ) n b = (p b p ) Thee two normal give rie to two projected naxel peed ṽ a and ṽ b. Two cae have to be ditinguihed, ee Figure 8. Both normal lie on the ame ide of the upporting egment of (Figure 8a). The final projected peed hould be taken a ṽ = min{ṽ a, ṽ b } n Let u now conider the cae of a naxel on a concave corner. Here again there are two poibilitie for the relative orientation of the normal n a, n b and the upporting egment, ee Figure 8c and 8d. Both cae lead to the ame formula, namely ṽ = max{ṽ a, ṽ b } 4.4 Determining the timetep To compute the optimal timetep t, we proceed a follow. Firt we note that the number of vertice of an r-nake C doe not change, a long a the r-nake doe not cro a grid vertex. Hence a natural upper bound for the timetep t can be determined a follow: We firt compute for each naxel it peed v, and then et t = min C {(1 d )/ṽ } to update all naxel imultaneouly with thi timetep. In thi way we can be ure that the r-nake doe not cro a gridpoint during the naxel update, i.e. it i alway guaranteed, that d + t ṽ 1. Note that a thi i a global bound on the timetep, the timetep i expected to decreae when the number of naxel increae. Hence, for a larger number of naxel, the algorithm ha to perform more update cycle. 4.5 Splitting naxel Whenever a naxel run into a gridpoint x = t, i.e. whenever d + t ṽ = 1

6 6 Stephan Bichoff, Leif P. Kobbelt a) b) a) b) x n a n a n b n b c) d) c b a c b a c) d) n a n b Fig. 9 Splitting naxel: a) Original r-nake. b) Snaxel ha run into gridpoint x from the outh. c) Snaxel i plit into three new naxel a, b, c that run to the eat, north and wet repectively. d) Some timetep later. n a n b ten remain double and triple vertice a depicted in Figure 11. Thee vertice are conceptually ditinct, a they have different upporting egment. However, they hare the ame patial poition uch that the left and/or right normal are not well-defined. In uch a cae, we only ue the well-defined normal to compute the projected peed. If there i none, a in the cae of a center triple vertex, we et ṽ = v. Fig. 8 Snaxel peed are computed by applying Huygen principle on continuou contour: The original contour (dark) i locally propagated with contant peed. The interection point of the reulting continuou offet contour (light) with the grid are then ued to determine the new naxel poition of the original contour. we plit it into three new naxel a, b, c. The three new naxel emanate from x in the other direction but have the ame poition a. To be more precie, we et a = b = c = 0 f a = f b = f c = x and t a, t b, t c accordingly. Figure 9 depict thi operation. After a naxel plit, condition 3 of Section 4.1 might be violated, a i depicted in Figure 10. To reetablih the r- nake property we perform a cleaning conquet: All naxel that violate condition 3 imply are removed. The cleaning conquet ha to be applied recurively to the neighborhood of the plit naxel and to the neighborhood of each removed naxel. Performing a naxel plit and the following cleaning conquet reetablihe the r-nake property. Nonethele there of- 4.6 Colliion detection and avoidance In our etup it i eay to detect and avoid (elf-) interection of r-nake. For each grid egment, we tore the naxel that are upported by thi egment (at mot two). If memory requirement are an iue, thi can be accomplihed by a hahtable, which i indexed by the naxel from and to coordinate, ee e.g. [8]. Hence it i eay to detect potential colliion partner: They are upported by the ame grid egment and hence have the ame hah-key. Whenever a potential colliion i detected, we adapt the timetep uch that the two correponding naxel will not cro, but jut touch each other. Depending on the application, we may chooe whether the two colliding naxel will clah and come to a halt (topology preervation) or whether they will merge (topology change), ee Figure 12. Clahing Becaue the contour propagate only outward and may not elf-interect, the two colliding naxel will tay in their poition forever. Hence, we flag them a frozen and exclude them from the remaining update tep (Figure 13). We can further decide, whether frozen naxel are affected by the

7 Snake with topology control 7 a) b) x c) d) Fig. 12 Colliion detection and handling. The above equence how 5 nake evolving and colliding. In the top row, the colliding nake change topology and merge. In the bottom row, the colliding nake keep their topology and come to a halt. a) b) a b e) f) a b c) d) Fig. 10 Cleaning conquet: a) Original r-nake. b) Snaxel ha run into gridpoint x. c) Snaxel i plit into three new naxel. d) Without a cleaning conquet both naxel a and naxel b would violate condition 3. e) Snaxel a and b are removed by the cleaning conquet. f) Some timetep later. a) b) Fig. 11 Multiple vertice: Triple (a) and double (b) vertice may appear after a naxel plit. Although thee vertice hare the ame patial poition, they are conceptually ditinct, a they are upported by different grid egment. Due to the parameter independent evolution rule thi degenerate nake parameterization doe not affect the numerical robutne of the algorithm. cleaning conquet or not. Thi will reult in different behavior a i demontrated in Figure 14. Merging In cae topology change of the nake are permitted, the two colliding naxel a and b can be merged by imply relinking their neighboring naxel. For thi we jut et Fig. 13 Colliion detection and handling: a) Original r-nake, a potential colliion i detected a naxel a and b are upported by the ame grid egment. b) The timetep i adapted uch that naxel a and b do not cro but jut touch each other. c) If the topology of the nake mut not change, the naxel that have collided are frozen and excluded from further updating. d) If topology change are wanted, naxel a and b are removed and their previou/following naxel are connected uch that the two part of the nake merge. a->next->prev = b->prev b->prev->next = a->next b->next->prev = a->prev a->prev->next = b->next

8 8 Stephan Bichoff, Leif P. Kobbelt Fig. 14 Depending on whether frozen naxel are removed by the cleaning conquet, recee will diappear a oon a they completely touch each other (top) or are conerved (bottom). and remove a and b (Figure 13). After that we perform a cleaning conquet to remove puriou bad naxel. Hence, in contrat to [8] and [18] thi operation doe not require any reampling. Notice that the topology change are very imple operation even in thi explicit repreentation etting. Thi i due to the fact that the retriction for the naxel movement guarantee that colliion alway happen at the contour vertice (and not at the egment). 5 Reult Synthetic Data Figure 15 how the evolution of two r-- nake. They are initialized at the outide and at the inide of a polygon, then they are propagated with unit peed v 1. The figure how naphot of the contour at equiditant time interval. A can be een, the contour obey Huygen principle and nicely handle concave a well a convex corner. Real Data In Figure 16 we applied our algorithm to the problem of recontructing the brain cortex from an MRI image. In thi cae the grid reolution ha been et to the reolution of the image ( ), but higher reolution would alo be poible. Firt, the MRI image ha been pre-proceed by a 3 3 Gauian filter. Then we initialized two circular r- nake for each of the two hemiphere. The naxel peed are et proportional to the underlying image intenitie. The egmentation proce took le than three econd. In practice, the reult could be enhanced by additionally applying well-known tandard egmentation technique, like uing internal force on the r-nake or applying cale-pace technique on the image [13]. Drawback Becaue the naxel of an r-nake are retricted to move along grid egment, the algorithm ometime exhibit preference for certain direction. In particular, when two r-nake collide in diagonal direction, ripple in the order of one pixel magnitude may appear, a i depicted in Figure 17. Thee ripple in particular occur in ynthetic Fig. 15 Contour evolution: The image above how the evolution of two contour that were initialized a the inner and outer boundary of the polygon. Both contour propagate with unit peed v 1 and are hown at equiditant time interval. The reolution of the underlying grid i dataet where there i no underlying external force that provide a meaningful gradient which guide the naxel to their final detination. Since the ripple repreent feature at ubpixel preciion which can be conidered a ampling artifact of the dicretized underlying calar field, we mooth the frozen naxel of the r-nake in a pot-proceing tep to remove thee artifact. 6 Concluion and future work We have preented a novel approach for the repreentation and evolution of active contour. It major advantage are Eae of implementation Automatic adaption of the contour parameterization Efficient colliion detection and avoidance Topology control We have demontrated it applicability on ynthetic a well a on real image data. Our approach prove to be epecially well uited for the recontruction of convoluted organic tructure, like the cortex of the brain. The major drawback in our current implementation i that we et the time tep by taking a global minimum. Thi could eaily be improved by adjuting the time tep locally and integrating the cleaning conquet into the evolution procedure to avoid inconitent nake configuration. The ripple - problem i of minor relevance in real application ince here the peed function i uually dominated by the external energy force. In the future, we plan to generalize our algorithm

9 Snake with topology control 9 Fig. 16 Contour evolution: The image above how the evolution of two r-nake in order to egment the brain cortex in an MRI image. The peed of a naxel i proportional to the image intenity at the poition of the naxel. Note in particular the gap-le eam that recontruct the intenity valley a hown in the cloeup. a) b) Fig. 17 Ripple: When two r-nake clah together, there may appear ripple in the order of one pixel magnitude (a). Thee ripple can be removed in a pot-proceing tep by moothing the r-nake (b). to higher dimenion and to develop a method uch that poitive a well a negative peed can be applied in one update tep. Reference 1. M. O. Berger. Snake growing. In Proc. Firt European Conf. on Computer Viion, page Springer LNCS, D. L. Chopp. Computing minimal urface via level et curvature flow. Jour. of Comp. Phy., 106:77 91, I. Cohen, L. Cohen, and N. Ayache. Uing deformable urface to egment 3-d image and infer differential tructure. Computer Viion, Graphic and Image Proceing: Image Undertanding, 56(2): , L. Cohen. On active contour model and balloon. Computer Viion, Graphic and Image Proceing: Image Undertanding, 53(2): , L. D. Cohen and I. Cohen. Finite element method for active contour model and balloon for 2d and 3d image. IEEE Tran. on Pattern Analyi and Machine Intelligence, 15(11): , C. A. Davatziko and J. L. Prince. An active contour model for mapping the cortex. IEEE Tran. on Medical Imaging, 14(1): , D. DeCarlo and D. Metaxa. Blended deformable model. In Proceeding CVPR 94, page , H. Delingette and J. Montagnat. Shape and topology contraint on parametric active contour. Computer Viion and Image Undertanding, 83: , M. Gage. On an area-preerving evolution equation for plane curve. Contemp. Math., 51:51 62, A. Gupta, T. O Donnell, and A. Singh. Segmentation and tracking of cine cardiac mr and ct image uing a 3d deformable model. In IEEE Conf. on Computer and Cardiology, page , X. Han, C. Xu, and J. L. Prince. A topology preerving deformable model uing level et. In Computer Viion and Pattern Recognition Proceeding, page , J. Hug, C. Brechbühler, and G. Szekely. Tamed nake: A particle ytem for robut emi-automatic egmentation. In Medical Image Computing and Computer-Aited Intervention, number 1679 in LNCS, page , M. Ka, A. Witkin, and D. Terzopoulo. Snake: Active contour model. Internation Journal of Computer Viion, 1: , B. Kimia, A. Tannenbaum, and S. Zucker. On the evolution of curve via a function of curvature i. the claical cae. Journal of Mathematical Analyi and Application, 163: , J.-O. Lachaud and A. Montanvert. Deformable mehe with automatic topology change for coare-to-fine three-dimenional urface extraction. Medical Image Analyi, 3(2): , S. Lobregt and M. Viergever. A dicrete dynamic contour model. IEEE Tran. on Medical Imaging, 14(1):12 23, W. E. Lorenen and H. E. Cline. Marching cube: a high reolution 3d urface recontruction algorithm. In SIGGRAPH 87 proceeding, page , T. McInerney and D. Terzopoulo. Topologically adaptable nake. In International Conference on Computer Viion, page , T. McInerney and D. Terzopoulo. Deformable model in medical image analyi: A urvey. Medical Image Analyi, 1(2):91 108, T. McInerney and D. Terzopoulo. Topology adaptive deformable urface for medical image volume egmentation. IEEE Tranaction on Medical Imaging, 18(10): , T. McInerney and D. Terzopoulo. T-nake: Topology adaptive nake. Medical Image Analyi, 4(2):73 91, J. V. Miller, D. E. Breen, W. E. Lorenen, R. M. O Bara, and M. J. Wozny. Geometrically deformed model: A method for extracting cloed geometric model from volume data. In SIG- GRAPH 91 Proceeding, page , F. Prêteux N. Rougon. Directional adaptive deformable model for egmentation. Journal of Electronic Imaging, 7(1): , S. Oher and J. A. Sethian. Front propagating with curvaturedependent peed: Algorithm baed on hamilton-jacobi formulation. J. Comput. Phy., 79(1):12 49, 1988.

10 10 Stephan Bichoff, Leif P. Kobbelt 25. S. J. Oher and R. P. Fedkiw. Level Set Method and Dynamic Implicit Surface. Springer, J. A. Sethian. A fat marching level et method for monotonically advancing front. Proc. Nat. Acad. Sci., 93(4): , J. A. Sethian. Level Set Method and Fat Marching Method. Cambridge Univerity Pre, Leif P. Kobbelt i a full profeor and the head of the Computer Graphic Group at the Aachen Univerity of Technology, Germany. Hi reearch interet include all area of Computer Graphic and Geometry Proceing with a focu on multireolution and free-form modeling a well a the efficient handling of polygonal meh data. He wa a enior reearcher at the Max-Planck-Intitute for Computer Science in Saarbr ücken, Germany from 1999 to 2000 and received hi Habilitation degree from the Univerity of Erlangen, Germany where he worked from 1996 to In 1995/96 he pent a pot-doc year at the Univerity of Wiconin, Madion. He received hi Mater (1992) and Ph.D. (1994) degree from the Univerity of Karlruhe, Germany. Over the lat year he ha authored many reearch paper in top journal and conference and erved on everal program committee. Stephan Bichoff graduated in 1999 with a mater in computer cience from the Univerity of Karlruhe, Germany. He then worked at the graphic group of the Max-Planck-Intitute for Computer Science in Saarbr ücken, Germany. In 2001 he joined the Computer Graphic Group at the Aachen Univerity of Technology, Germany, where he i currently puruing hi PhD. Hi reearch interet focu on freeform hape repreentation for efficient geometry proceing and on topology control technique for level-et urface.