n i P i φ i t i P i-1 i+1 P i+1 P i i-1 i+1 2 r i

Transcription

1 New Algorthms for Controllng Actve Contours Shape and Topology H. Delngette and J. Montagnat Projet Epdaure I.N.R.I.A Sopha-Antpols Cedex, BP 93, France Abstract In recent years, the eld of actve-contour based mage segmentaton have seen the emergence of two competng approaches. The rst and oldest approach represents actve contours n an explct (or parametrc) manner correspondng to the Lagrangan formulaton. The second approach represent actve contours n an mplct manner correspondng to the Euleran framework. After comparng these two approaches, we descrbe several new topologcal and physcal constrants appled on parametrc actve contours n order to combne the advantages of these two contour representatons. We ntroduce three key algorthms for ndependently controllng actve contour parameterzaton, shape and topology. We compare our result to the level-set method and show smlar results wth a sgncant speed-up. 1 Introducton Image segmentaton based on actve contours has acheved consderable success n the past few years [MT96]. Deformable models are often used for brdgng the gap between low-level computer vson (feature extracton) and hgh-level geometrc representaton. In ther semnal paper [KWT88], Kass et al choose to use a parametrc contour representaton wth a sem-mplct ntegraton scheme for dscretzng the law of moton. Several authors have proposed derent representatons [MSMM93] ncludng the use of nte element models [CCA92], subdvson curves [HBS99] and analytcal models [MT91]. Implct actve contour representaton were ntroduced n [MSV95] followng [Set96]. Ths approach has been developed by several other researchers ncludng geodesc snakes ntroduced n [CKS97]. The opposton between parametrc and mplct contour representaton corresponds to the opposton between Lagrangan and Euleran frameworks. Qualfyng the ecency and the mplementaton ssues of these two frameworks s dcult because of the large number of derent algorthms exstng n the lterature. On one hand, mplct representatons are n general regarded as beng less ecent than parametrc contours. Ths s because the update of an mplct contour requres the update of at least a narrow band around each contour. On the other hand, parametrc contours cannot n general acheve any automatc

2 topologcal changes, also several algorthms have been proposed to overcome ths lmtaton [LC91,MT95,LM99]. Ths paper ncludes three dstnct contrbutons correspondng to three dfferent modelng levels of parametrc actve contours: 1. Dscretzaton. We propose two algorthms for controllng the relatve vertex spacng and the total number of vertces. On one hand, the vertex spacng s controlled through the tangental component of the nternal force appled at each vertex. On the other hand, the total number of contour vertces s perodcally updated n order to constran the dstance between vertces. 2. Shape. We ntroduce an ntrnsc nternal force expressons that do not depend on contour parameterzaton. Ths force regularzes the contour curvature prole wthout producng any contour shrnkage. 3. Topology. A new algorthm automatcally creates or merges derent connected components of a contour based on the detecton of edge ntersectons. Our algorthm can handle opened and closed contours. We propose a framework where algorthms for controllng the dscretzaton, shape and topology of actve contours are completely ndependent of each other. Havng algorthmc ndependence s mportant for two reasons. Frst, each modelng component may be optmzed separately leadng to computatonally more ecent algorthms. Second, a large varety of actve contour behavors may be obtaned by combnng derent algorthms for each modelng component. 2 Dscretzaton of actve contours In the remander, we consder the deformaton over tme of a two-dmensonal parametrc contour C(u t) 2 IR 2 where u desgnates the contour parameter and t desgnates the tme. The parameter u belongs to the range [0 1] wth C(0 t)= C(1 t) f the contour s closed. We formulate the contour deformaton wth a Newtonan law of 2 = + f nt + f ext (1) where f nt and f ext correspond respectvely to nternal and external forces. A contour may nclude several connected components, each component beng a closed or opened contour. Temporal and spatal dscretzatons of C(u t) are based on nte derences. Thus, the set of N t vertces fp t g, =0:::Nt ; 1 represents the contour C(u t) at tme t. The dscretzaton of equaton 1 usng centered and rght derences for the acceleraton and speed term leads to: p t+1 = p t +(1; 2)(pt ; pt;1 )+ (f nt ) + (f ext ) : (2) In order to smplfy the notaton, we wllwrte p nstead of p t the vertex poston at tme t. At each vertex p,we dene a local tangent vector t, normal

3 vector n, metrc parameter and curvature k.we propose to dene the tangent vector at p, as the drecton of the lne jonng ts two neghbors: t =(p +1 ; p ;1 )=(2r ) where r = kp +1 ; p ;1 k=2 s the half dstance between the two neghbors of p. The normal vector n s dened as the vector drectly orthogonal to t : n = t? wth (x y)? =(;y x). The curvature k s naturally dened as the curvature of the crcle crcumscrbed at trangle (p ;1 p p +1 ).Ifwe wrte as, the orented angle between segments [p ;1 p ] and [p p +1 ], then the curvature s gven by k = sn( )=r. Fnally, the metrc parameter measures the relatve spacng of p wth respect to ts two neghborng vertces p ;1 and p +1.If F s the projecton of p on the lne [p ;1 p +1 ], then the metrc parameter s: = kf ; p +1 k=(2r ) = 1 ;kf +1 ; p ;1 k=(2r ). In another words, and 1 ; are the barycentrc coordnates of F wth respect to p ;1 and p +1 : F = p ;1 +(1; )p +1. n P -1 P F 2 r φ P +1 P -1 P C 1 k t P +1 P -1 P F P * * F P +1 Fg. 1. Left: The geometry of a dscrete contour denton of t, n, k,, and F. Rght: The nternal force assocated wth the curvature-conservatveowsproportonal to p? ; p. Other dentons for the tangent and normal vectors could have been chosen. However, our tangent and normal vectors dentons has the advantage of provdng a smple local shape descrpton: p = p ;1 +(1; )p +1 + L(r )n (3) p where L(r )= r ( (1 ; ) tan 2 ) tan wth =1f jj <=2 and = ;1 f jj >=2. Equaton 3 smply decomposes vertex poston p nto a tangental and normal component. The mportance of ths equaton wll be revealed n sectons 3. 3 Parameterzaton control For a contnuous actve contour C(u t), the contour parameterzaton s characterzed by the metrc functon: g(u t) = k. If g(u t) =1 then the parameter of C(u t) concdes wth the contour arc length. For a dscrete the parameterzaton corresponds to the relatve spacng between vertces and s characterzed by g = kp ; p ;1 k. For a contnuous representaton, parameterzaton s clearly ndependent of the contour shape. For a dscrete contour represented by nte derences, shape

4 and parameterzaton are not completely ndependent. The eect of parameterzaton changes s especally mportant at parts of hgh curvature. Therefore, parameterzaton s an mportant ssue for handlng dscrete parametrc contours. In ths secton we propose a smple algorthm to enforce two types of parameterzaton: 1. unform parameterzaton: the spacng between consecutve vertces s unform. 2. curvature-based parameterzaton: vertces are concentrated at parts of hgh curvature. Ths parameterzaton tends to optmze the shape descrpton for a gven number of vertces. To modfy a contour parameterzaton, only the tangental component ofthe nternal force should be consdered. Indeed, Kma et al [KTZ92] have proved that only the normal component of the nternal force appled on a contnuous contour C(u t) has an nuence on the resultng contour shape. Therefore, f t, n are the tangent and normal vector at a pont C(u t), then the contour evoluton may be = f nt = a(u t)t + b(u t)n. Kmaet al [KTZ92] show that only the normal component of the nternal force b(u t) modes the contour shape whereas the metrc functon g(u t) k evoluton s on a(u t) and t) = + b(u The tangental component ofthenternal force a(u t)t constrans the nature of the parameterzaton. We propose to apply ths prncple on dscrete parametrc contours as well by decomposng the nternal force f nt nto ts normal and tangental components: (f nt ) = (f tg ) +(f nr ) wth (f tg ) n = 0 and (f nr ) t =0. More precsely, snce the tangent drecton t at a vertex s the lne drecton jonng ts two neghbors, we use a smple expresson for the tangental component: (f tg ) =(? ; )(p +1 ; p ;1 )=2r (? ; )t where? s the reference metrc parameter whose value depends on the type of parameterzaton to enforce. 3.1 Unform vertex spacng To obtan evenly spaced vertces, we smply choose:? = 1. Ths tangental 2 force moves each vertex n the tangent drecton towards the mddle of ts two neghbors. When the contour reaches ts equlbrum,.e. when (f tg ) = 0, s then equdstant from p ;1 and p +1. It equals to (f tg ) = 2 2 t t t. Because the second dervatve R s the rst varaton of the 2 strng nternal energy ( k2 du), ths force s somewhat related to the classcal snakes approach proposed n [KWT88]. 3.2 Curvature based vertex spacng To obtan an optmal descrpton of shape, t s requred that vertces concentrate at parts of hgh curvature and that at parts are only descrbed wth few vertces.

5 To obtan such parameterzaton, we present a method where edge length s nversely proportonal to curvature. If e s the edge jonng p and p +1,then we compute ts edge curvature K +1 as the mean absolute curvature of ts two vertces: K +1 =(jk j + jk +1 j)=2. Then at each vertex p,we can compute the local relatve varaton of absolute curvature K 2 [;1 1] as: K =(K +1 ; K;1 )=(K+1 + K;1 ).To enforce a curvature-based vertex spacng, we compute the reference metrc parameter? as:? = 1 2 ; 0:4 K. When vertex p s surrounded by two edges havng the same curvature then K =0and therefore? s set to 1 whch mples that 2 p becomes equdstant from ts two neghborng vertces. On the contrary, when the absolute curvature of p +1 s greater than the absolute curvature of p ;1 then K becomes close to 1 and therefore? s close to 0.1 whch mples that p moves towards p Results of vertex spacng constrants To llustrate the ablty to decouple parameterzaton and shape propertes, we propose to apply an nternal force that modes the vertex spacng on a contour wthout changng ts shape. We dene a curvature conservatve regularzng force that moves p n the normal drecton n order to keep the same local curvature: f nr =(L(r? ) ; L(r ))n : (5) Ths equaton has a smple geometrc nterpretaton f we note that the total nternal force f nt = f tg + f nr s smply equal to p? ; p where p? s the pont havng the same curvature as p but wth a metrc parameter?.from rght of gure 1, we can see that f tg corresponds to the dsplacement between F? and F whereas f nr corresponds to the derence of elevaton between p? and p. Gven an open or closed contour we teratvely apply derental equaton 2 wth the nternal force expresson descrbed above. Fgure 2 shows an example of vertex spacng constrant enforced on a closed contour consstng of 150 vertces. The ntal vertex spacng s uneven. When applyng the unform vertex spacng tangental force (? =0:5), after 1000 teratons, all contour edge lengths become equal wthn less 5 percent wthout greatly changng the contour shape, as shown n gure 2 (upper row). The dagram dsplays the dstrbuton of edge curvature as a functon of edge length. Smlarly, wth the same number of teratons, the contour evoluton usng the curvature-based vertex spacng force tends to concentrate vertces at parts of hgh curvature. The correspondng dagram clearly shows that edge length s nversely proportonal to edge curvature. 3.4 Contour resoluton control In addton to constranng the relatve spacng between vertces, t s mportant to control the total number of vertces. Indeed, the computatonal complexty of dscrete parametrc contours s typcally lnear n the number of vertces. In order to add or remove vertces, we do not use any global contour reparameterzaton as performed n the level-set method [Set96] because of ts hgh computatonal cost. Instead, we propose to locally add or removea vertex f the edge length does

6 Fg. 2. (up) contour after applyng the unform vertex spacng tangental force (bottom) contour after applyng the curvature-based vertex spacng constrant. not belong to a gven dstance range, smlarly to [IP95,LV95]. Our resoluton constrant algorthm proceeds as follows. Gven two thresholds s mn and s max correspondng to the mnmum and maxmum edge length, we scan all exstng contour edges. If the current edge length s greater than s max and 2 s mn then a vertex s added. Otherwse f current edge length s less than s mn andfthesum of the current and prevous edge length s less than s max, then the current vertex s removed. In general, ths procedure s called every fr resoluton =5deformaton teratons. 4 Shape regularzaton The two nternal forces dened n prevous secton have lttle nuence on the contour shape evoluton because they are only related to the contour parameterzaton. In ths secton, we deal wth the nternal force normal component whch determnes the contour shape regularzaton. The most wdely used nternal forces on actve contours are the mean curvature moton [MSV95], Laplacan smoothng, thn rod smoothng or sprng forces. Laplacan Smoothng and Mean Curvature Moton have the drawback of sgncantly shrnkng the contour. Ths shrnkng eect ntroduces a bas n the contour deformaton snce mage structures located nsde the contour are more lkely to be segmented than structures located outsde the contour. Furthermore, the amount of shrnkng often prevents actve contours from enterng nsde ne structures. To decrease the shrnkng eect, Taubn [Tau95] proposes to apply a lnear lter to curves and surfaces n order to reduce the shrnkng eect of Gaussan

7 smoothng. However, these two methods only remove the shrnkng eect for a gven curvature scale. For nstance, when smoothng a crcle, ths crcle would stay nvarant only for one gven crcle radus whch s related to a set of lterng parameters. Therefore, n these methods, the choce of these parameters are mportant but dcult to estmate pror to the segmentaton process. A regularzng force wth hgher degrees of smoothness such asthethn RodSmoothng causes sgncantly less shrnkng snce t s based on fourth dervatves along the contour. However, the normal component of ths force 4 n)n s dependent on the nature of the parameterzaton whch s a serous lmtaton. 4.1 Curvature duson regularzaton We propose to use the second dervatve of curvature wth respect to arc length as the governng regularzng force: f nt = d2 k ds 2 n (6) Ths force tends to duse the curvature along the contour, thus convergng towards crcles, ndependently of ther rad, for closed contours. For the dscretzaton of equaton 6, we do not use straghtforward nte derences, snce t would lead to complex and potentally unstable schemes. Instead, we propose a geometry-based mplementaton that s smlar to equaton 5: f nr =(L(r?? ) ; L(r ))n (7) where? s the angle at a pont p? for whch d2 k =0. The geometrc nterpretaton of equaton 7 s also straghtforward, snce the nternal force f nt = f tg + f nr ds 2 corresponds to the dsplacement p? ; p. The angle? s smply computed by? = arcsn(k? r ) where k? s the curvature at p?. Therefore, k? s smply computed as the local average curvature weghted by arc length: k? = kp p ;1 kk +1 + kp p +1 kk ;1 kp p +1 k + kp p ;1 k Furthermore, we can compute the local average curvature over a greater neghborhood whch results n ncreased smoothness and faster convergence. If > 0 s the scale parameter, and l +j s the dstance between p and p +j then we compute k as : P k? = j=1 l ;jk +j + l +j k P ;j j=1 l wth l +j = ;j + l +j jx k=1 kp +k;1 p +k k: Ths scheme generalzes the ntrnsc polynomal stablzers proposed n [Del94] that requred a unform contour parameterzaton. Because of ths regularzng force s geometrcally ntrnsc, we can combne t wth a curvature-based vertex spacng tangental force, thus leadng to optmzed computatons. Fnally, the stablty analyss of the explct ntegraton scheme s lnked to the choce of. We have found expermentally, wthout havng a formal proof yet, that we obtan a stable teratve scheme f we choose 0:5.

8 5 Topology constrants Automatc topology changes of parametrc contour has been prevously proposed n [LC91,MT95,LM99]. In McInerney et al approach, all topologcal changes occur by computng the contour ntersectons wth a smplcal decomposton of space. The contour s reparameterzed at each teraton, the ntersectons wth the smplcal doman beng used as the new vertces. Recently Lachaud et al [LM99] ntroduced topologcally adaptve deformable surfaces where selfntersectons are detected based on dstance between vertces. Our algorthm s also based on a regular lattce for detectng all contour ntersectons. However, the regular grd s not used for changng the contour parameterzaton and furthermore topology changes result from the applcaton of topologcal operators. Therefore unlke prevous approaches, we propose to completely decouple the physcal behavor of actve contours (contour resoluton and geometrc regularty) wth ther topologcal behavor n order to provde a very exble scheme. Fnally, our framework apples to closed or opened contours. A contour topology s dened by the number of ts connected components and whether each of ts components s closed or opened. Our approach conssts n usng two basc topologcal operators. The rst operator llustrated n gure 3 conssts n mergng two contour edges. Dependng whether the edges belong to the same connected component or not, ths operator creates or remove a connected component. The second topologcal operator conssts n closng or openng a connected component. Fg. 3. Topologcal operator applyng on (left) two edges on the same connected component or(rght) two derent connected components. Our approach for modfyng a contour topology can be decomposed nto three stages. The rst stage creates a data structure where the collson detecton between contour connected components s computatonally ecent. The second determnes the geometrc ntersecton between edges and the last stage actually performs all topologcal modcatons. 5.1 Data structure for the detecton of contour ntersectons Fndng pars of ntersectng edges has an aprorcomplexty ofo(n 2 ) where n s the numberofvertces (or edges). Our algorthm s based on a regular grd of sze d and has a complexty lnear wth the rato L=d where L s the length of the contour. Therefore, unlke the approach proposed n [MT95], our approachsnot regon-based (nsde or outsde regons) but only uses the polygonal descrpton of the contour.

9 The two dmensonal Eucldean space wth a reference frame (o x y) s decomposed nto a regular square grd whch sze d s user-dened. The nuence of the grd sze d s dscussed n secton 6.1. In ths regular lattce, we dene a pont of row and column ndces r and c as the pont of Cartesan coordnates o grd + rx + cy where o grd s the grd orgn pont. Ths pont srandomly determned each tme topology constrants are actvated n order to make the algorthm ndependent of the orgn choce. Furthermore, we dene a square cell of ndex (r c) as the square determned by the four ponts of ndces (r c), (r +1 c), (r +1 c+1)and (r c+1). In order to buld the sampled contour, we scan all edges of each connected components. For each edge, we test f t ntersects any row or columns of the regular lattce. Snce the row and column drectons correspond to the drectons x and y of the coordnate frame, these ntersecton tests are ecently computed. Each tme an ntersecton wth the row or column drecton s found, a grd vertex s created and the ntersectng contour edge s stored n the grd vertex. Furthermore, a grd vertex s stored n a grd edge structure. A grd edge s ether a par of grd vertces or a grd vertex assocated wth an end vertex (when the connected component s an opened lne). Fnally, the grd edge s appended to the lst of grd edges nsde the correspondng grd cell. Grd Vertex Assocated Contour Edge Contour Edge Grd Edge Grd Cell End Vertex Grd Edge Fg. 4. (left) The orgnal contour wth the regular grd the contour decomposed on the regular grd (rght) Denton of grd vertex, grd edge, grd cell and contour edge assocated wth a grd edge. 5.2 Fndng ntersectng grd edges In order to optmze memory space, we store all non-empty grd cells nsde a hash table, hashed by ts row and column ndces. The number of grd cells s proportonal to the length L of the contour. In order to detect possble contour ntersectons, each entry to the hash table s scanned. For each cell contanng n grd edges wth n>1, we test the ntersecton between all pars of grd edges (see gure 4, left). Snce each grd edge s geometrcally represented by a lne segment, ths ntersecton test only requres the evaluaton of two dot products. Once a par of grd edges has been found to ntersect, a par of contour edges must be assocated for the applcaton of topologcal operators (see secton 5.3). Because a contour edge s stored n each grdvertex, one contour edge can be assocated wth each grd edge. Thus, we assocate wth each grd edge, the mddle of these two contour edges (n terms of topologcal dstance) as shown n gure 4, rght.

10 Our contour edges ntersecton algorthm has the followng propertes: () f one par of grd edges ntersects then there s at least one par of contour edges that ntersects nsde ths grd cell and () f a par of contour edges ntersects and f the correspondng ntersectng area s greater than d d then there s a correspondng par of ntersectng grd edges. In another words, our method does not detect all ntersectons but s guaranteed to detect all ntersecton havng an area greater than d d. In practce, snce the grd orgn o grd s randomly determned each tme the topology constrant s enforced, we found that our algorthm detected all ntersectons that are relevant for performng topology changes. 5.3 Applyng topologcal operators All pars of ntersectng contour edges are stored nsde another hash table for an ecent retreval. Snce n general two connected components ntersect each other at two edges, gven a par of ntersectng contour edges, we search for the closest par of ntersectng contour edges based on topologcal dstance. If such a par s found, we perform the followng tasks. If both edges belong to the same connected component, then the they are merged f ther topologcal dstance s greater than a threshold (usually equal to 8). Ths s to avod creatng too small connected components. In all other cases, the two edges are merged wth the topologcal operator presented n gure 3. Fnally, we update the lst of ntersectng edge pars by removng from the hash table all edge pars nvolvng any of the two contour edges that have been merged. 5.4 Other applcatons of the collson detecton algorthm The algorthm presented n the prevous sectons merges ntersectng edges regardless of the nature of the ntersecton. If t corresponds to a self-ntersecton, then a new connected component s created, otherwse two connected components are merged. As n [MT95] our framework can prevent the mergng of two dstnct connected components whle allowng the removal of self-ntersectons. To do so, when a par of ntersectng contour edges belongng to dstnct connected components s found, nstead of mergng ths edges, we algn all vertces located between ntersectng edges belongng to the same connected component (see gure 5). Thus, each component pushes back all neghborng components. In gure 5, rght, we show an example of mage segmentaton where ths repulsve behavor between components s very useful n segmentng the two heart atrums. 6 Results 6.1 Topology algorthm cost We evaluate the performance of our automatc topology adaptaton algorthm on the example of gure 6. The contour consstng of 50 vertces, s deformed from a crcular shape towards a vertebra n a CT mage. The computaton tme for buldng the data structure descrbed n secton 5.1 s dsplayed n gure 6,

11 Fg. 5. (from left to rght) Two ntersectng connected components after mergng the two pars of ntersectng edges after algnng vertces along the ntersectng edges. Example of 2 actve contours reconstructng the rght and left ventrcles n a MR mage wth a repulsve behavor between each components. rght, as a functon of the grd sze d. It vares from 175 ms to 1 ms when the grd sze ncreases from 0.17 to 10 mage pxels on a Dgtal PWS 500 Mhz. The computaton tme for applyng the topologcal operators can be neglected n general. When the grd sze s equal to the mean edge dstance (around 2 pxels), the computaton tme needed to detect edge ntersectons becomes almost equal to the computaton tme needed to deform the contour durng one teraton (4.8 ms) Computaton Tme Grd Sze Fg. 6. Segmentaton of a vertebra n a CT mage Topology algorthm computaton tme. When the grd sze ncreases, the contour samplng on the regular grd becomes sparse and therefore some contour ntersectons may not be detected. However, we have vered that topologcal changes stll occur f we choose a grd sze correspondng to 20 mage pxels wth contour ntersectons checked every 20 teratons. In practce, we choose a conservatve opton wth a grd sze equal to the average edge length and wth a frequency for topology changes of 5 teratons whch mples an approxmate addtonal computaton tme of 20 percent. 6.2 Segmentaton example Ths example llustrates the segmentaton of an aortc arch angography. Fgure 7 shows the ntal contour (up left) and ts evoluton towards the aorta and the

12 man vessels. External forces are computed as a functon of vertex dstance to a gradent pont to avod oscllatons around mage edges and are projected on the vertex normal drecton. The contour s regularzed by a curvature dusve constrant. The contour resoluton constrant s appled every 10 teratons whch makes the resamplng overhead very low. Topology constrants are computed every 5 teratons on a 4 pxel grd sze to fuse the self-ntersectng contour parts. Intersectons wth mage borders are computed every 10 teratons and the contour s opened as t reaches the mage border. Fg. 7. Evoluton of a closed curve towards the aortc arch and the branchng vessels. 7 Comparson wth the level-set method The man advantage of the level-set method s obvously ts ablty to automatcally change the contour topology durng the deformaton. Ths property makes t well-suted for reconstructng contours of complex geometres for nstance treelke structures. Also, by mergng derent ntersectng contours, t s possble to ntalze a deformable contour wth a set of growng seeds. However, the major drawbacks of level-sets methods are related to ther dcult user nteracton and ther computatonal cost, although some speed-up algorthms based on constranng the contour evoluton through the Fast-Marchng method [Set96] or by usng an asynchronous update of the narrow-band [PD98] have been proposed. The formal comparson between both parametrc and level-set approaches have been recently establshed n the case of geodesc snakes [ABF99]. In ths secton, we propose a practcal comparson between both approaches ncludng mplementaton ssues. 7.1 Level-set mplementaton The level-set functon s dscretzed on a rectangular grd whose resoluton corresponds to the mage pxel sze. The evoluton equaton s dscretzed n space usng nte derences n tme usng an explct scheme, leadng to : t+t j = t j + t jkr t j j k [MSV95] where j denotes the propagaton speed term and t s the dscrete tme step. The propagaton speed term s desgned to attract C towards object boundares extracted from the mage usng a gradent operator wth an addtonal regularzng term: (p) = (p)((p) + c). (p) denotes the contour curvature at pont p whle c s a constant resultng n a balloon force [Coh91] on the contour. Fnally, (p) 2 [0 1] s a multplcatve coecent dependent on the mage

13 Fg. 8. (Fve left gures) Dscrete contour deformaton (Fve rght gures) level-set deformaton. gradent norm at pont p. When C moves across pxels of hgh gradents, ths term slows down the level-set propagaton. A threshold parameter determnes the mnmal mage boundary strength requred to stop a level-set evoluton. We speed-up the level-set by usng a narrow band method [MSV95] whch requres to perodcally rentalze the level-set contour. In order to compare actve contours to level-sets, we compute external forces smlar to level-set propagaton at dscrete contour vertces. Frst, we use mean curvature moton as the governng nternal force and we have mplemented for the external force, a balloon force weghted by the coecent proposed above. Fnally, we were able to use the same gradent threshold n both approaches. 7.2 Torus example We rst propose to compare both approaches on the synthetc mage shown n gure 8. Ths mage has two dstnct connected components. A dscrete contour s ntalzed around the two components. A medum grd sze (8 pxels resoluton) s used and topology constrants are computed every 10 teratons. Throughout the deformaton process, vertces are added and removed to have smlar edge length along the contour. A correspondng level-set s ntalzed at the same place that the dscrete contour. A 7 pxel wde narrow band appeared to optmze the convergence tme. A 0:3 tme step s used. It s the maxmal value below whch the evolvng curve s stable. Fgure 8 shows the convergence of the dscrete contour (left) and the level-set (rght). The dscrete contour converges n 0:42 seconds opposed to 3:30 seconds for the level-set, that s a 7:85 acceleraton factor n favor of the dscrete contour. The derence of computatonal tme s due to the small vertex numberusedfor the dscrete contour (varyng between 36 and 48 vertces) compared to the much greater number of stes (from 1709 up to 3710) updated n the level-set narrow band. 7.3 Synthetc data Ths experments shows the ablty of the dscrete contour topology algorthm to follow dcult topology changes. We use a synthetc fractal mage showng a number of small connected components. Fgure 9, upper row, shows the dscrete contour convergence n the mage whle the bottom row shows the level set convergence. In both cases, the ntal contour s a square located at the mage border. It evolves under a deaton force that stops on strong mage boundares. For the

14 Fg. 9. (upper row) Dscrete contour convergence n a fractal mage (bottom row) Level-set convergence n the same mage. dscrete contour, a small grd sze s used due to the small mage structure sze (4 pxels grd sze and 5 teratons algorthm frequency). A weak regularzng constrant allows the contour to segment the square corners. The contour s checked every 10 teratons to add the necessary vertces. A 0:3 tme step s used for the level-set. Ths hgh value leads to a rather unstable behavor as can be seen n gure 9. As the level-set contours gradually lls-n the whole mage, we have vered that the convergence tme s not mnmzed by usng any narrow bands. Agan, the speed-up s 3.84 n favor of the dscrete contour. 8 Concluson We have ntroduced three algorthms that greatly mprove the generalty of parametrc actve contours whle preservng ther computatonal ecency. Furthermore, these algorthms are controlled by smple parameters that are easy to understand. For the nternal force, a sngle parameter between 0 and 1 s used to set the amount of smoothng. For resoluton and topology constrant algorthms, dstance parameters must be provded as well as the frequency at whch they apply. Gven an mage, all these parameters can be set automatcally to meanngful values provdng good results n most cases. Fnally, we have compared the ecency of ths approach wth the level set method by mplementng parametrc geodesc snakes. These experments seem to conclude that our approach s at least three tmes as fast as the mplct mplementaton. Above all, we beleve that the most mportant advantage of parametrc actve contours s ther user nteractvty. References [ABF99] G. Aubert and L. Blanc-F raud. Some Remarks on the Equvalence between 2D and 3D Classcal Snakes and Geodesc Actve Contours. Internatonal Journal of Computer Vson, 34(1):517, September [CCA92] I. Cohen, L.D. Cohen, and N. Ayache. Usng Deformable Surfaces to Segment 3-D Images and Infer Derental Structures. Computer Vson, Graphcs, and Image Processng: Image Understandng, 56(2):242263, September 1992.

15 [CKS97] V. Caselles, R. Kmmel, and G. Sapro. Geodesc Actve Contours. Internatonal Journal of Computer Vson, 22(1):6179, [Coh91] L.D. Cohen. On Actve Contour Models and Balloons. Computer Vson, Graphcs, and Image Processng: Image Understandng, 53(2): , March [Del94] H. Delngette. Intrnsc stablzers of planar curves. In thrd European Conference on Computer Vson (ECCV'94), Stockholm, Sweden, June [HBS99] J. Hug, C. Brechb ler, and G. Sz kely. Tamed Snake: A Partcle System for Robust Sem-automatc Segmentaton. In Medcal Image Computng and Computer-Asssted Interventon (MICCAI'99), pages , Sprnger, Cambrdge, UK, September [IP95] J. Ivns and J. Porrll. Actve Regon models for segmentng textures and colours. Image and Vson Computng, 13(5):431438, June [KTZ92] B. Kma, A. Tannenbaum, and S. Zucker. On the evoluton of curves va a functon of curvature. the classcal case. Journal of Mathematcal Analyss and Applcatons, 163:438458, [KWT88] M. Kass, A. Wtkn, and D. Terzopoulos. Snakes: Actve Contour Models. Internatonal Journal of Computer Vson, 1:321331, [LC91] F. Letner and P. Cnqun. Complex topology 3d objects segmentaton. In SPIE Conf. on Advances n Intellgent Robotcs Systems, Boston, November [LM99] J.-O. Lachaud and A. Montanvert. Deformable meshes wth automated topology changes for coarse-to-ne three-dmensonal surface extracton. Medcal Image Analyss, 3(2):187207, [LV95] S. Lobregt and M. Vergever. A Dscrete Dynamc Contour Model. IEEE Transactons on Medcal Imagng, 14(1):1223, [MSMM93] S. Menet, P. Sant-Marc, and G. Medon. Actve Contour Models: Overvew, Implementaton and Applcatons. IEEE Trans. on Systems, Man and Cybernetcs, , [MSV95] R. Mallad, J.A. Sethan, and B.C. Vemur. Shape Modelng wth Front Propagaton : A Level Set Approach. IEEE Transactons on Pattern Analyss and Machne Intellgence, 17(2):158174, [MT91] D. Metaxas and D. Terzopoulos. Constraned Deformable Superquadrcs and nonrgd Moton Trackng. In Internatonal Conference on Computer Vson and Pattern Recognton (CVPR'91), pages , Mau, Hawa, June [MT95] T. McInerney and D. Terzopoulos. Topologcally adaptable snakes. In Internatonal Conference on Computer Vson (ICCV'95), pages , Cambrdge, USA, June [MT96] T. McInerney and D. Terzopoulos. Deformable models n medcal mage analyss: a survey. Medcal Image Analyss, 1(2):91108, [PD98] N. Paragos and R. Derche. A PDE-based Level-Set Approach for Detecton and Trackng of Movng Objects. In Internatonal Conference on Computer Vson (ICCV'98), pages , Bombay, Inda, [Set96] [Tau95] J.A. Sethan. Level Set Methods : Evolvng Interfaces n Geometry, Flud Mechancs, Computer Vson and Materals Scence. Cambrdge Unversty Press, G. Taubn. Curve and Surface Smoothng Wthout Shrnkage. In Internatonal Conference on Computer Vson (ICCV'95), pages , 1995.