A 3D Laplacian-Driven Parametric Deformable Model

Transcription

1 A 3D Laplacan-Drven Parametrc Deformable Model Tan Shen 1,2 Xaole Huang 1 Hongsheng L 1 Edward Km 1 Shaotng Zhang 3 Junzhou Huang 4 1 Department of Computer Scence and Engneerng, Lehgh Unversty, USA 2 Corporate Technology, Semens Ltd., Chna 3 Department of Computer Scence, Rutgers Unversty, Pscataway, USA 4 Department of Computer Scence and Engneerng, Unversty of Texas at Arlngton, USA Abstract 3D parametrc deformable models have been used to extract volumetrc object boundares and they generate s- mooth boundary surfaces as results. However, n some segmentaton cases, such as cerebral cortex wth complex folds and creases, and human lung wth hgh curvature boundary, parametrc deformable models often suffer from over-smoothng or decreased mesh qualty durng model deformaton. To address ths problem, we propose a 3D Laplacan-drven parametrc deformable model wth a new nternal force. Derved from a Mesh Laplacan, the nternal force exerted on each control vertex can be decomposed nto two orthogonal vectors based on the vertex s tangental plane. We then ntroduce a weghtng functon to control the contrbutons of the two vectors based on the model mesh s geometry. Deformng the new model s solvng a lnear system, so the new model can converge very effcently. To valdate the model s performance, we tested our method on varous segmentaton cases and compared our model wth Fnte Element and Level Set deformable models. 1. Introducton Deformable models have been used extensvely n mage segmentaton. Accordng to ther representaton and mplementaton, they can be classfed as ether parametrc (explct) or geometrc (mplct) deformable models. Parametrc deformable models, wth Snakes [5] as a representatve, represent the model boundary parametrcally usng splne curves or parametrc surfaces. They have assocated nternal smoothness forces and are also nfluenced by external mage forces. Usng 3D parametrc models [10] for segmentng objects n medcal mage volumes has the advantages of drectly extractng a smooth boundary surface and guaranteeng coherence between mages slces. The man concern wth 3D parametrc models s the computatonal cost due to a large number of varables. Thus Fnte Element Method (FEM) based parametrc deformable models are wdely used [2], [18], [17]. Usng FEM, the model s represented as a contnuous surface n the form of weghted sums of local polynomal bass functons. By encodng the spatal nformaton and connectvty about the vertces nto the stffness matrx, deformaton of the model s solved through a lnear system, whch s effcent. Because the nternal forces keep the model surface smooth, on one hand, FEM models have been shown to acheve good results n segmentng objects wth smooth boundary such as the left ventrcle [17]; on the other hand, these models often have dffculty delneatng the boundary of objects wth complex folds and creases, or hgh curvature regons. The other class of deformable models s level set based geometrc models [9], [19]. Ths approach represents a curve or a surface mplctly as the zero level set of a hgherdmensonal scalar functon (level set functon). And level set models deform based on the theory of curve evoluton wth ther speed functon specfcally desgned to ncorporate mage gradent nformaton. Geometrc deformable models have been adopted wth consderable success for mage segmentaton snce they are parametrzaton-free and can be easly extended from 2D to 3D. Ther topology flexblty also allows the extracton of multple objects smultaneously. The topology-freedom s not always desrable, however. For nstance, a fxed-topology model may be more sutable when the object of nterest s specfed and ts topology s known [18]. Thus, topology-preservng level set methods have been proposed [4], [16], n whch tests are conducted and steps are taken to process the computed level set functons to prevent undesrable topology changes. In ths paper, we consder the problem of segmentng a specfc object wth known topology n 3D. Both parametrc deformable models and topology-preservng level set models are applcable. Whle parametrc models drectly extract object surfaces represented by trangular meshes, tradtonal FEM models suffer from over-smoothng and can not

2 (c) (d) (e) Fgure 1. Cerebral cortex segmentaton on smulated MRI mages provded by BranWeb ( The external forces of, and (c) are defned n (23). The nternal forces of and (c) are based on the stffness matrx A n (2). The result of our method, the result of a tradtonal FEM based 3D deformable model, (c) the result of the FEM based model wth weaker nternal force, (d) the result of the Actve Contour Wthout Edges [1] after surface reconstructon, and (e) the result of Dstance Regularzed Level Set Evoluton [6] after surface reconstructon. gve satsfactory results when segmentng objects wth complex surface structure or hgh curvature regons. Topologypreservng level set, on the other hand, wll requre extra processng steps to reconstruct the boundary surface from ts bnary map result, and to prevent change of topology. We propose a new method that modfes the nternal forces exerted on a non-fem parametrc deformable model n order to overcome over-smoothng and drectly extract object boundary as a hgh-qualty trangular mesh. The proposed new model s represented as a smplex mesh. Its nternal forces are derved from the mesh Laplacan, whch s wdely used n mesh shape optmzaton and edtng [13], [21]. By decomposng the Laplacan of each vertex nto two orthogonal vectors one on and another perpendcular to the vertex s tangental plane, we observe that () the tangental vector s always desrable n preservng mesh qualty, () the perpendcular vector has double effects on the postve sde, t helps overcome local mnma caused by nose n mages durng segmentaton, and on the negatve sde, t s found to cause the mesh s oversmoothng problem. The new 3D deformable model we ntroduce has nternal forces that keep fully the vector projected onto the tangental plane snce t helps mantan mesh qualty. The contrbuton of the other vector perpendcular to the tangental plane s automatcally adjusted through a weghtng functon defned based on current mesh qualty. We demonstrate that, on segmentng objects wth complex or hgh-curvature boundary surfaces such as the bran cerebral cortex (Fgure 1) or lung (Fgure 6), our method acheves better segmentaton results than tradtonal FEM based 3D deformable model and level set methods [1], [6], n terms of accuracy and qualty of the resultng boundary surface mesh. Furthermore, the new model can stll deform effcently snce model deformaton s derved by solvng a lnear system n whch the matrx s very sparse. 2. Background In ths secton, we brefly revew the mathematcal defnton of tradtonal FEM based deformable models usng a contnuous pecewse-lnear bass functon 1. We analyze the nternal forces on these models to dscuss the cause of the over-smoothng problem D FEM based Deformable Models A 3D FEM deformable model s an elastc surface dscretzed as a smplex mesh (or fnte element trangulaton), whch can deform under the nfluence of nternal smoothness forces and external mage forces derved from the model s energy functon. The mesh Λ s represented as a graph G = (V,E), wth vertces V and edges E. Gven the bass functon φ of the th vertex v and the number of vertces n = V, the deformable model s represented by Λ(x) = n =1 φ (x)v, n whch x R 3. And the bass functon φ (v j ) s defned as φ (v j ) = { 1 = j, 0 j, where φ s a contnuous pecewse lnear functon, and v and v j are theth and jth vertces of the model. Gven the external force vector FV ext = [f1 ext,f2 ext,...,fn ext ] T, mnmzng the energy functon s done by solvng AV = F ext V, (2) where matrxas the stffness matrx [8], whose sze sn n. V = [v 1,v 2,...,v n ] T s the vector of control vertces on the surface. More specfcally, the th row and jth column element ofa,a j, s defned as { a j = Λ ( φ φ j )dλ = j or (,j) E. 0 otherwse (3) Usng fnte dfferences n tme, the model can deform teratvely [2]. (2) s optmzed as: (V (t) V (t 1) )/τ +AV (t) = F ext V (t 1), (4) 1 We use FEM based 3D deformable models to refer to FEM based 3D deformable models usng a contnuous pecewse-lnear bass functon.

3 Fgure 2. The nternal forcef nt ofv s represented as a summaton of a set of vectors pontng from the vertex to all ts neghborng vertces, and the vertex deforms accordng to the summaton of nternal force vectorf nt and external force vectorf ext. where V (t) s the vector of the model s vertces at the t-th teraton, and τ s the tme step sze. (4) can be wrtten n a fnte dfferences formulaton, as MV (t) = V (t 1) +τf ext V (t 1), M = (I +τa). Based on the bass functon n, snce each vertex n the smplex model s only connected to a small number of vertces, M s a sparse and postve defnte matrx. The soluton of (5) can be obtaned very effcently Analyss of the Conventonal Internal Force (4) can be further reformulated accordng to the nternal and external forces where (5) V (t) = V (t 1) +τ(f nt +F ext V (t 1) ), (6) F nt = AV (t), (7) so the th element f nt n F nt represents the nternal force on the th vertex. The nternal force on the th vertex v s defned based on (7) f nt = (a v + a j v j ). (8) Snce the stffness matrx A also has the property a + a j = 0, for anyv V, (9) then (8) can be wrtten as f nt = a j (v v j ). (10) As shown n Fgure 2., the nternal force on the th vertex, f nt, s a summaton of a set of vectors whch are pontng from the th vertex to ts neghborng vertces. When applyng the deformable model to segment an object, the control vertex moves accordng to the summaton of ts nternal force vector and the external force vector, as shown n Fgure 2.. In the fnal converged result, the summaton of nternal and external force vectors of each vertex s approxmately equal to zero. The nternal force vectors of the FEM model keep the model s surface smooth and also help t overcome the negatve effects of mage nose. However, f the target object has a complex surface (e.g., the cerebral cortex n Fgure 1) or contans hgh curvature boundary (e.g., the bottom of the lung n Fgure 6), the nternal force vectors mght cause the surface to be over-smoothed and thus have vertces devate away from desred object boundares. To allevate ths problem, one may decrease the weght of the nternal force to make the external force have more power n controllng the model s deformaton, by utlzng an automatc parameter adjustment technque [12] or by recevng parameter adjustment nput from an experenced user. Often, to allow the model to ft the detals on object boundary, parameter adjustment nvolves lowerng the weght of the nternal force and decreasng the modeldeformaton step sze. However, such adjustment would decrease the model mesh s qualty and ncrease the number of teratons necessary for the model to converge. A soluton to restore the mesh qualty s to adopt Remeshng wth Detal Preservng methods [7], [13]. Remeshng can be appled to the model mesh after every few teratons. But typcal remeshng technques are tme-consumng. Further, when the nternal force s made weaker, the model may suffer from a sharp degeneraton n mesh qualty due to strong external forces pullng the model n dfferent drectons as well as effects of mage nose. It s lkely that the mesh qualty would become too poor to be recovered, causng dffculty n remeshng and leadng to poor segmentaton results. 3. Methodology We present a new 3D deformable model derved from the mesh Laplacan, wth the objectve of achevng effcent detal-preservng segmentaton whle mantanng hgh model-mesh qualty The Novel Internal Force Mesh Laplacan The new Laplacan-drven parametrc deformable model we propose s stll represented as a smplex mesh, but t has a novel nternal force defnton derved from the mesh Laplacan [13]. For theth vertexv, the unformly weghted Laplacanδ s δ = v j w v = (v j v ), (11)

4 wherew s the number of vertces connected wthv. Gven the number of vertcesn, the elementl j n then n Laplacan matrx L s represented as l j = w = j, 1 (,j) E, 0 otherwse. (12) In Fgure 3., the Laplacan of the th vertex s shown as the blue arrow. Servng as the nternal force, the Laplacan moves the th vertex to the centrod of ts neghborng vertces. Ths procedure can be decomposed nto two substeps. Frstly, v s moved on ts tangental plane n favor of havng trangles wth equal areas. Secondly,v s further moved along ts normal drecton to reach the centrod. Accordng to these two steps, δ can be decomposed nto two vectors assocated wth the vertex s tangental plane, whch are the red vectorδ tang on the tangental plane and the green vectorδ perp perpendcular to the tangental plane, as shown n Fgure 3.. δ tang has the desred effect of balancng the areas of neghborng trangles (Fgure 3.(c)), and δ perp moves the vertex along the normal drecton whch shrnks the model and makes t a smooth surface. When segmentng objects wth complex surfaces, the above mentoned oversmoothng problem s caused by δ perp. If smply decreasng the nfluence of the nternal force, the weght of δ tang s also reduced, whch may severely affect the mesh qualty (Fgure 1.(c)). In our method, by decomposng the nternal force nto two vectors, the over-smoothng problem can be solved as keepng the contrbuton ofδ tang and lowerng the nfluence ofδ perp (Fgure 1.). The decomposton of nternal forces has been used on 2D parametrc deformable contours n [3]. Based on a control vertex s tangental vector, the method [3] decomposed the nternal force on the 2D parametrc contour nto two parts so as to automatcally control the dstances between neghborng vertces and the total number of control vertces. Ths method can not be extended to 3D, however, snce 3D models have much more complex geometry and shape representaton than 2D models. In 3D, the nternal force vector of a control vertex n the tradtonal FEM based deformable model could also be decomposed nto two vectors assocated wth the tangental plane of the vertex. We dd not follow ths approach because the stffness matrx of the FEM model has the property of reducng the moblty of vertces [8], thus the nternal force vector projected on the tangental plane vanshes, unable to effectvely preserve mesh qualty. Furthermore, the stffness matrx n an FEM model needs to be updated after each teraton snce t depends on the spatal postons of the connected vertces. In contrast, the Laplacan matrx used by our model has the property that ts weghts do not depend on the vertex poston, thus the weghts only need to be calculated once and can be precomputed after the model s ntalzed. Therefore, we prefer the mesh Laplacan over FEM Internal Forces Based on Mesh Laplacan δ prep δ tang n Fgure 3. can be calculated based on: δ perp = ( n δ ) n. (13) where n s the normal of v. In ths paper, we defne t as the average of the normals of the vertex s adjacent surfaces. Thenδ tang n the tangental plane s = δ δ perp = v j w v δ perp. (14) Usng (14) as the new nternal force and applyng fnte dfferences, the deformaton equaton for theth vertex s v (t) +τ(w v (t) v (t) j +(δ perp ) (t) ) = v (t 1) +τf ext, (15) whereτ s the step sze,v (t) s theth vertex s locaton at the t-th teraton. In (15), the unknown parameters arev (t),v (t) j and (δ perp ) (t). Snce we use fnte dfferences, we assume the shape of the model does not change sgnfcantly n each teraton, so we approxmate (δ perp ) (t) wth (δ perp ) (t 1),.e., (δ perp ) (t) = (δ perp ) (t 1), (16) where(δ perp ) (t 1) can be obtaned from the current mesh s geometrc shape drectly. We then have v (t) +τ(w v (t) v (t) ) (t 1) ) = v (t 1) +τf ext. j +(δ perp (17) When the th vertex moves accordng to (17), the new nternal force δ tang moves the vertex only on ts tangental plane wthout shrnkng the model, whch overcomes the over-smoothng problem and keeps the mesh qualty effectvely. However, testng the model on 3D medcal mages, we fnd the model becomes more senstve to nose. The reason s that the vector perpendcular to the tangental plane δ perp has the property of helpng the vertex pass some local mnma caused by nose and stop at strong boundares. And δ perp also has postve effects on mprovng mesh qualty, snce movng the vertex along ts normal drecton also reduces the areas of ts neghborng trangles and make them more smlar to an equlateral trangle. To keep the benefts of δ perp, we adopt a weghtng functon ω(v) to control ts contrbuton. The nternal force s thus redefned as: f nt = v j w v ω(v )δ perp. (18) To defne ω(v), we use the vertex s neghborng trangle s radus ratoǫ[14], whch s mapped to[0,1] as ǫ = 2 r ns R cr, (19)

5 (c) Fgure 3. Illustraton of the novel nternal force. The nternal force vector of the vertex s shown n the neghborng trangles, gven the vertex s normal drecton, the nternal force vector can be decomposed nto two vectors accordng to the vertex s tangental plane, and (c) v moves on the tangental plane n favor of havng trangles wth equal areas. where r ns and R cr are the rad of the nscrbed and crcumscrbed crcles of the trangle. ǫ = 1 ndcates a well shaped equlateral trangle and ǫ = 0 means a degenerated trangle. The radus rato of the th vertex r(v ) s defned as the average radus rato of all ts neghborng trangles. A hghr(v) means the vertex s neghborng trangles are n good shape and ω(v) s set as close as possble to 1. Then the nternal force vector can be mapped onto the tangental plane to make sure the vertex deforms wthout shrnkng. Once r(v) decreases because of nose or some other factors, we assgn ω(v) a relatvely small value. Then δ perp s partally preserved to reduce the effect of nose and to restore the neghborng trangles of the vertex to a good shape. In ths paper, we ntroduce two ways to set ω(v) based on the radus rato. We frst defneω(v) as a Ferm functon (sgmodal), whch s ω(v) = 1+e 1 s(r(v) µ), (20) where s s the steepness, and µ controls the overall mesh qualty level. s andµare emprcally set as 20 and 0.7. Secondly, takng the relatve frequency of radus rato nto account, we adopt ts cumulatve densty functon (CD- F) to map from r(v) to weght [0,1]. In ths way, the weghtng functon s defned as a normalzed dscrete summaton overr(v) for all the vertces: ω(v ) = c k = v k V c k, { 1 V r(v k ) r(v ) 0 r(v k ) > r(v ). (21) Based on our experence, (20) s more sutable to segment objects wth hgh curvature boundares and (21) s more sutable to segment objects wth complex surfaces. Compared wth (14), (18) has a weghtng functon rangng from 0 to 1, whch s used to control the contrbuton of δ perp n Fgure 3. Consderng all the control vertces, the lnear system for the new deformable model s defned based on (14) as M [ V (t) D (t) M = ] [ V (t 1) = D (t 1) ] [ F ext +τ V (t 1) 0 [ ] [ M00 M 01 L00 L = (I +τ 01 0 M ], ] ), (22) where L 00 s the n n Laplacan matrx n (12), L 01 s a n n dagonal matrx derved from the weghtng functon ω(v) to control the weghts of δ perp. D (t) and D (t 1) are the n 1 vectors encodng δ perp of all the vertces at tme t and t 1 respectvely. M 11 s the dentty matrx of sze n. The sze ofm s2n 2n, and t s very sparse. Solvng (22) s stll very effcent. In 3D, (22) s solved three tmes, for thex,y and z components separately. 4. Experments In ths secton, we show experments that test our new model. We also compare ts performance wth several other deformable models: an FEM model (Secton 2.1) wth normal or weak nternal forces, and two level set based algorthms Actve Contour Wthout Edges (ACWE) [1] and Dstance Regularzed Level Set Evoluton (DRLSE) [6]. Durng comparson, all models use the same ntalzaton. The test datasets nclude a 3D volume of a synthetc object whose boundary has hgh curvature regons (Fgure 4), a 3D smulated MRI bran mage from BranWeb (Fgure 1), and 10 human lung volumetrc datasets from the NCI Lung Image Database Consortum (LIDC) (Fgure 6). The parameter settngs for the varous models and the models performance measures are summarzed n Table Expermental Setup On the 3D synthetc dataset and the lung datasets, we compared our model wth the FEM model. Both models are ntalzed n the same way,.e., spheres or ellpsods near the object wth user-specfed rad. To segment the cerebral cortex from the MRI bran mage, we compared our model

6 wth the FEM model wth normal nternal forces, the FEM model wth weakened nternal forces, and the two level set based algorthms ACWE and DRLSE; all models use the same ntalzaton whch s the pre-segmented bran Whte Matter obtaned by [20]. For our proposed parametrc deformable model wth Laplacan-drven nternal forces, we used (20) to control the weght of δ perp when segmentng the synthetc object and human lungs, and used (21) for segmentaton of the bran cerebral cortex. To focus on evaluatng only the performance of the new Laplacan-drven nternal force, n the experments, we used the same external force to deform our model and the FEM based model. Smlar to T-Snakes [11], the external force s set accordng to the ntensty values and the vertces normals. For theth vertex, t s f ext = { γ n I(v ) T, γ n otherwse, (23) where n s the normal of the vertex, I(v ) denotes the ntensty value of the th vertex, and T s the mean ntensty value of the mage volume. The settngs of step sze τ and the weght of external force γ for our model and the FEM models are shown n Table 1. Note that our model dd not need parameter tunng and used the same parameter settng for all experments. In contrast, the FEM model needs dfferent parameter settngs for dfferent segmentaton tasks, and n the experments, we manually set the FEM model parameters to enable t to grow as much as possble under the nfluence of strong external forces toward object boundary. The convergence crteron for our model and the FEM model s defned based on the model vertces movements: we assume the model reaches the fnal converged result f all the vertces move less than a maxmum of two voxels n a certan teraton. For the two level set methods used n Fgure 1.(d) (e), we emprcally set ther parameters to make the models acheve ther best segmentaton results. To llustrate the models performance, we calculated the Dce Smlarty Coeffcent (DSC) [15] values to measure the segmentaton accuracy and used the radus rato of mesh trangles to measure the mesh qualty, see Table Expermental Results Fgure 4 shows segmentaton results on the 3D synthetc dataset. From the fnal results vewed n 3D and 2D crosssectonal planes, one can clearly see that our model precsely found the object boundary but the conventonal FEM based model generated a smooth surface that dd not ft well at corners on the boundary. The mean radus rato values of the resultng boundary surface meshes by our model and the FEM model are 98.87% and 97.91%, respectvely. Fgure 1 compares the cerebral cortex segmentaton results usng our method, the FEM model wth normal nternal forces, the FEM model wth weakened nternal forces, (2) (c) Fgure 4. Segmentaton results on a 3D synthetc dataset. Volume renderng of the dataset, the segmentaton results vewed n 3D, (c) the segmentaton results vewed n 2D cross-sectonal planes, the result of our method, and (2) the result of the FEM based method. (c) (d) Fgure 5. Vsualzaton of mesh qualty evaluaton by colormappng the result meshes radus rato values usng the cumulatve densty functon of the result n Fgure 1.(c). Result of our method n Fgure 1., result of the FEM model wth normal smoothness constrants n Fgure 1., (c) result of the FEM model wth weakened smoothness constrants n Fgure 1.(c), and (d) the color bar. the ACWE and DRLSE. For our model and the FEM models, the ntalzaton s the so-surface mesh reconstructed from the pre-segmented Whte Matter; for the two level set methods, the ntalzaton s the sgned dstance transform of the Whte Matter. Our result s shown n Fgure 1., and has the hghest mean radus rato value 92.3%. In Fgure 1. (c), FEM based models ether became too smooth or had low mesh qualty. The mean radus rato values of Fgure 1. (c) are 89.1% and 83.4%. Wth a relatvely weak nternal force, the FEM model had to adopt a small step sze to deform, whch ncreased the runnng tme as shown n Table 1. In Fgure 1.(d) (e), a post-processng surface reconstructon step was necessary to obtan the boundary surface from level set s volumetrc segmentaton result, and the reconstructed surface has low mesh qualty, affectng accuracy and vsualzaton. The mean radus rato values of Fgure 1.(d) (f) are77.4% and77.9%. In Fgure 5, we vsualze the mesh qualty comparson results of Fgure 1. (c) by color-mappng the mesh trangles radus ratos usng the cumulatve densty functon (CDF) defned n (21). Here we adopt the CDF of the mesh 1 0

7 Table 1. Parameter settngs and quanttatve evaluaton of our method and other methods. Synthetc Data Human Lung Human Bran Ours FEM Ours FEM Ours Fg. 1. Fg. 1.(c) Fg. 1.(d) Fg. 1.(e) τ (step sze) n/a n/a γ (weght of external force) n/a n/a DSC 99.2% 95.3% 96.3% 93.2% 95.5% 88.7% 93.6% 94.6% 93.8% Iteratons Runnng Tme 79.4s 54.4s 70.1s 61.88s 43.4s 31.3s 71.6s 52.2s 39.6s DSC Our Method FEM Based Method Slce Index (2) Fgure 7. DSC and mesh qualty (radus rato) evaluaton of the segmentaton results n Fgure 6.. DSC results of dfferent axal slces, the vertces radus rato dstrbuton, radus rato dstrbuton of our method, and (2) radus rato dstrbuton of the FEM based method. The left and rght lungs extend from the 53rd slce to the 314th slce. n Fgure 1.(c). Our result contans the most red regons, ndcatng t has the best mesh qualty. We also evaluated our method on10 human lung datasets and show 4 of them n Fgure 6. From the DSC values shown n Table 1, our model produced a notceable mprovement n accuracy compared to the FEM model. In Fgure 7, we plotted the DSC value n each axal slce (0 means there s no segmentaton result n that slce) and showed the dstrbuton of the model vertces radus rato from the dataset n Fgure 6.. Snce the objects extend from the 53rd slce to the 314th slce, from the fgure we can see that our method correctly recovered the objects wth hgh DSC value n each slce, whle the FEM model mssed parts of the objects n the top and bottom slces. Both our model and the FEM model produced hgh-qualty 3D surface meshes, wth mean radus rato values above 97%. To further demonstrate the results, we show 4 sets of comparson results from dfferent cross-sectonal planes n Fgure Concluson and Future Work In ths paper, we proposed a new 3D Laplacan-drven deformable model based on mesh Laplacan amng to segment wth better accuracy objects that have complex surfaces or hgh curvature boundares. The man contrbutons nclude: decomposng the nternal force of each vertex nto two vectors accordng to the vertex s tangental plane, the 3D parametrc deformable model s nternal force can be consdered as consstng of two separated parts; (2) mantanng the vector projected onto the tangental plane and usng a mesh-geometry based weghtng functon to control the contrbuton of the vector perpendcular to the tangental plane, the model can thus be used to segment objects wth complex surface or hgh curvature regons, whle preservng the mesh qualty; (3) deformng the new model s equvalent to gettng the soluton of a lnear system, whch s very effcent. Testng the new model on 3D synthetc, human lung and bran datasets and comparng wth other parametrc and geometrc deformable models, our model acheved better performance wth lttle senstvty to parameter settngs. Acknowledgements. Ths research was supported n part by the NSF grant IIS and NIH grant R21GM References [1] T. Chan and L. Vese. Actve contours wthout edges. IEEE Trans. Image Processng, 10: , [2] L. Cohen and I. Cohen. Fnte-element methods for actve contour models and balloons for 2-D and 3-D mages. IEEE Trans. Pattern Analyss and Machne Intellgence, 15: , [3] H. Delngette and J. Montagnat. Shape and topology constrants on parametrc actve contours. Computer Vson and Image Understandng, 83: , [4] X. Han, C. Xu, and J. Prnce. A topology preservng level set method for geometrc deformable models. IEEE Trans. Pattern Analyss and Machne Intellgence, 25: , [5] M. Kass, A. Wtkn, and D. Terzopoulos. Snakes: Actve contour models. Internatonal Journal of Computer Vson, 1: , [6] C. L, C. Xu, C. Gu, and M. Fox. Dstance regularzed level set evoluton and ts applcaton to mage segmentaton. IEEE Trans. Image Processng, 19: , [7] Y. Lu, W. Wang, B. Lévy, F. Sun, D. Yan, L. Lu, and C. Yang. On centrodal vorono tessellaton energy smoothness and fast computaton. ACM Trans. Graphcs, 28:1 17, [8] D. Logan. A Frst Course n the Fnte Element Method. CL Engneerng, [9] R. Mallad, J. Sethan, and B. Vemur. Shape modelng wth front propagaton: A level set approach. IEEE Trans. Pattern Analyss and Machne Intellgence, 17: , [10] T. McInerney and D. Terzopoulos. Deformable models n medcal mage analyss: A survey. Medcal Image Analyss, 1:91 108, 1996.

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