Mesh Composition on Models with Arbitrary Boundary Topology

Transcription

1 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, ACCEPTED Mesh Composton on Models wth Arbtrary Boundary Topology Juncong Ln, Xaogang Jn, Charle C.L. Wang*, and Kn-Chuen Hu Abstract Ths paper presents a new approach for the mesh composton on models wth arbtrary boundary topology. After cuttng the needed parts from exstng mesh models and puttng them nto the rght pose, an mplct surface s adopted to smoothly nterpolate the boundares of models under composton. An nterface s developed to control the shape of the mplct transent surface by usng sketches to specfy the expected slhouettes. After that, a localzed Marchng Cubes algorthm s nvestgated to tessellate the mplct transent surface so that the mesh surface of composed model s generated. Dfferent from exstng approaches n whch the models under composton are requred to have parwse mergng boundares, the framework developed based on our technques have the new functon to fuse models wth arbtrary boundary topology. Index Terms Mesh composton; Arbtrary boundary topology; Modelng by slhouette; Localzed Marchng Cubes; Implct surface. INTRODUCTION R ESEARCHES for provdng a user-frendly modelng system to create complex 3D models effectvely have been studed wth a long hstory n computer graphcs. It s common for desgners to start modelng wth a vague mage of the shape n mnd wth reference to exstng models or the features of models. The geometrc modelng nterface s expected to have the ablty to capture desgn features of shapes and form a new shape that nherts the features of exstng shapes. The term desgn feature here does not refer to the features such as holes or drans but to the aesthetc elements wth geometry detals. The wdely nvestgated mesh composton approach s for ths purpose. However, current exstng approaches (ref. [3], [2], [4], [7], [24], [45], [52]) are wth constrants at the topology of boundares on the models under composton. More specfcally, parwse mergng boundares are usually expected. Although the approaches usng mplct surfaces (such as [22], [29], [30] and [44]) can somewhat solve the problem, the models constructed by them do ether need to have smple transent surface shape (as n [22] and [44]) or convert all meshes nto mplct representaton (ref. [29] and [30]) whch easly yelds shape approxmaton errors on the parts under composton. The work presented n ths paper ams at overcomng above drawbacks and provdng a more powerful modelng nterface for creatng complex 3D models from the exstng ones. For example as llustrated n Fg., the composed models by our approach are wth arbtrary boundary topology. After cuttng the needed parts from exstng mesh models and *Correspondng Author. Juncong Ln and Xaogang Jn are currently wth the State Key Lab of CAD&CG, Zhejang Unversty, P.R.Chna; Charle C.L. Wang and Kn-Chuen Hu are wth the Department of Mechancal and Automaton Engneerng, The Chnese Unversty of Hong Kong, P.R.Chna. Part of the work s fnshed when Juncong Ln was a research assstant n The Chnese Unversty of Hong Kong. Manuscrpt receved (nsert date of submsson f desred). Please note that all acknowledgments should be placed at the end of the paper, before the bblography. u xxxx-xxxx/0x/$xx x IEEE puttng them nto the rght pose, the Radal Bass Functon (RBF) based mplct surface [48] wll be employed as the transent surface to nterpolate the boundares of models under composton smoothly. Smply usng the RBF-based mplct surface wll have no shape control on the transent surface. To solve ths problem, a new nterface s developed to control the shape of transent surface by usng sketches to specfy the expected slhouette. The most dffcult part of usng mplct surface to fuse mesh models comes from how to convert the mplct surface nto a mesh consstent wth the boundary of gven models under composton. A localzed Marchng Cubes algorthm has been developed n ths paper for ths purpose. Lastly, a method based on Laplacan mesh processng [45] has been developed as a postprocessng step to reconstruct geometry detals on the tessellated transent surface.. Prevous Work The work proposed n ths paper relates to several prevous researches n the areas of: modelng by examples, model completon and repar, smlarty-based mesh edtng and mplct surface tessellaton. Modelng by example Several approaches n lterature (ref. [3], [2], [4], [7], [22], [24], [29], [30], [43], [45], [52]) explore a desgn by example strategy to construct complex 3D models from exstng ones. The methods presented n [3], [2] and [24] focused on the cut-and-paste operaton on meshes. They could produce seamless composed models. However, the joned models n ther approaches were requred to have boundares topologcally equvalent to a dsk for the necessary mappng between source and target models. The methods n [45] and [52] can transplant and merge meshes also. However, these methods were actually deformaton based; thus, boundary openngs wth smlar topologcal structures were requred. In more detal, t s hard to drectly jon two models where one has two openngs whle the other s only wth one openng. Funkhouser et al. n [4]

2 2 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, MANUSCRIPT ID Fg.. An example to llustrate the functonalty of our mesh composton framework: (a) the faces are scssor from three exstng head models; (b) the faces under composton are placed together around a sphere and the poston of faces can be easly changed on the sphere; (c) after specfyng the slhouette of transent surface so that form a horn on the top of the three-head model, the mesh of transent surface s generated by our localzed Marchng Cubes algorthm n (d); (e) we can then further propagate geometry detals onto the transent mesh surface; (f) we then place the three-head model on to the neck of a dnosaur around a truncated cone together wth some other decoraton agan, the models can be easly moved accordng to the truncated cone; (g) the fnal three-head monster model can be generated effectvely. nvestgated a data-drven approach to construct new 3D models by assemblng parts from exstng ones. Recently, Hassner et al. [7] appled the graph mnmal-cut operaton to the task of 3D model composton. Both [4] and [7] used the technque n [24] to sttch those retaned portons together. In summary, there are varous topologcal lmtatons n all above methods. Recently, Sharf et al. [43] devoted to reduce the burden of users n placng models. Stmulated by the snappng noton whch has been vastly used n graphcs applcatons, they extended the noton to 3D to solve both the postonng and blendng problem of two surfaces by a soft Iteratve Cloesest Pont (ICP) algorthm. Ther tool allowed even lttle kds to construct complex models easly. Another recent work to automatcally adjust the shape of mergng boundary s presented n [9]. However, they are stll strct wth the topology of boundares. The topology constrants embedded n above approaches can be somewhat overcome by the mplct surface based approaches (e.g., [22], [29], [30] and [44]). However, the models constructed by [22] need to have planar openngs whch lmt the type of models that can be created. The approach n [44] lmted the transent mplct surface to be polyhedral prmtves, whose shape s too smple. The authors n [29] and [30] convert models nto mplct representaton and merge them. Although mesh surface can be reconstructed from the mplct representaton, ths explctmplct-explct converson ntroduces unexpected shape approxmaton errors. All these lmtatons wll be overcome by our scheme n ths paper. Model completon and repar Model completon usually has two requrements: ) boundary based the completed surface patch should seamlessly match the boundary, and 2) context based the patch should also contan geometry detals smlar to those on the exstng surfaces. Earler researches all focused on the boundary condton. The authors of [7] and [49] conducted the completon by solvng certan partal dfferental equatons, whle others (e.g., [23] and [38]) were volumetrc methods that represented the surface as the boundary between nsde and outsde volumetrc regons. Recently, several context-based methods were proposed n [26], [32], [35] and [42] to repar holes accordng to surface context nformaton. Kraevoy and Sheffer [26] flled the holes usng a mappng between the ncomplete mesh and a template model. Sharf et al. n [42] extended texture synthess technques from 2D to 3D, and represented ntrnsc geometry propertes and performed geometry completon drectly n the 3D doman. In [32] and [35], the 3D geometry synthess problem s transformed nto a 2D doman usng a conformal parameterzaton algorthm and then solved n the target regons usng mage completon technques. These approaches all focus on the completon of holes on one model. Dfferently, our approach wll solve the problem of geometry completon among several models wth topologcal ncompatble mergng boundares, and use sketches lke the ones n [20] and [25] to control the shape of transent surface. The mesh composton method presented n ths paper s also smlar to the model rapr approaches usng volumetrc technques (e.g., [5] [23] [34] [56]), where volumetrc representaton s conducted to fll holes and fx topologcal errors on a gven model. Here, we employ mplct surfaces to nterpolate the gap between composng models wth arbtrary boundary topology. Smlarty-based mesh edtng Recently, sem-local smlarty based shape descrptor has been nvestgated n the applcatons of shape matchng, retreval, modelng and smoothng (e.g., [6], [42], [5], [53] and [54]). Zelnka and Garland [53] proposed a concept of geodesc fans to fathfully dentfy regons on a surface that are geometrcally smlar. Ther descrptor s a vector of n dscrete samples taken at n fxed sample postons accordng to some samplng pattern gven n geodesc polar coordnates. Based on [53], they later presented a curvature map method n [6] to create the unque sgnature for a surface pont. They also adopted a smlar technque to transfer geometry detals onto models n [54]. The post-processng geometry detal reconstructon step n our mesh composton framework borrows some dea from [6], [53] and [54] but usng a dfferent shape descrptor. Recently, a smlarty based mesh smooth-

3 AUTHORS: MESH COMPOSITION ON MODELS WITH ARBITRARY BOUNDARY TOPOLOGY 3 ng method has been presented n [5], whch extends the NL-means mage flter to the pecewse smooth surface represented by trangle soups. Implct surface tessellaton Marchng cubes (MC) algorthm was frst ntroduced by Lorensen and Clne [28] and has become the most commonly used method for tessellatng mplct surfaces. As frst noted by Duerst [0], the orgnal MC algorthm [28] may produce sosurfaces wth holes due to topologcally nconsstent decsons on the reconstructon of ambguous faces, where the borders used by one ncdent cube do not match the borders of the other ncdent cube. Several approaches addressng ths problem have been publshed (see [2] and [33] for a revew). Recently, Lewner et al. [27] presented an effcent and robust mplementaton of Chernyaev s Marchng Cubes 33 algorthm [6] whch ensures a topologcally correct tessellaton. None of above MC algorthm and t varants cover the problem of tessellatng a part of mplct surface. In ths paper, we develop a new algorthm based on the confguraton table n [27] to tessellate the porton of nterest on the transent mplct surface. Although the authors n [36] and [40] also presented method to trm an mplct surface by another mplct, our algorthm presented n ths paper s the frst localzed Marchng Cubes algorthm to generate meshes of a partal mplct surface trmmed by an explct mesh surface. Marchng trangles (MT) s another class of sosurface approxmaton approaches. It s frstly appeared n [8]. The MT algorthm employs the local 3D constrant to reconstruct a Delaunay trangulaton of an arbtrary topology manfold surface. Ths method s further enhanced n [] and [4] by adaptng the sze of trangles to the curvature of surface and closng cracks at the end of mesh growng. However, the drawback nherent to all contnuaton methods stll exsts that t s dffcult to determne seed trangles on the connex part of a surface. Furthermore, the MT-lke algorthms dependng on seed searchng are usually much slower than the MC algorthm and ts varants..2 Contrbutons The approach presented n ths paper has the followng techncal contrbutons. Slhouette-based shape modelng of transent mplct surface: We develop a method to control the shape of RBFbased mplct transent surface, where the shape of transent surface can be easly controlled by usng sketches to specfy the expected slhouettes. Localzed Marchng Cubes Algorthm: In order to tessellate the transent mplct surface nto a mesh connectng the models under composton, a localzed Marchng Cubes algorthm s nvestgated. Although t s not an absolutely new algorthm, ths localzaton s the frst method to contour trmmed mplct surface wth complex boundares. Based on these two techncal contrbutons, a new model composton framework can be constructed to fuse models wth arbtrary boundary topology, whch s a functon that has not been provded n exstng model composton approaches n lterature. Rest of the paper s organzed as follows. Secton 2 presents the RBF mplct surface based shape modelng of transent surface. The localzed Marchng Cubes algorthm s then detaled n secton 3. Secton 4 descrbes other algorthms for the mesh composton framework. Expermental results and necessary dscussons are gven n secton 5. Lastly, our paper ends wth the concluson secton. 2 SHAPE MODELING OF TRANSIENT SURFACE We begn ths secton wth the reasons for selectng RBFbased mplct surface and a bref of ts mathematcal formulas, and then descrbe how t s appled to model the transent surface nterpolatng the mergng boundares on gven models and how the shape control of slhouettes can be gven. 2. RBF-based Implct Surface There are many dfferent mplct surfaces n lterature. By nvestgaton, we fnd that Radal Bass Functons (RBF) based mplct surface s the best canddate for mesh composton on models wth multple boundary openngs snce t holds the followng two mportant strengths. Both the poston and the normal at a surface pont can be specfed on a RBF-based mplct surface so that we can generate a transent surface smoothly nterpolatng the boundary of models under composton specfed normal vectors are conducted to ensure the tangental smoothness. Also, because of the ablty to control both postons and normals, t provdes us the possblty to control the shape of surface slhouettes. A RBF-based mplct surface can descrbe a closed two-manfold surface wth arbtrary topology n a unque and compact mathematcal representaton so that the varaton of topology can be easly mplemented on ths mathematcal representaton, whch leads to a new functon that s not provded on other exstng mesh composton paradgms. Followng [48], the RBF-based mplct surface s based on the thn-plate nterpolaton and can be expressed by a weghted sum of approprate radal bass functons plus an affne term as N v N { v { x, y, z }} = Γ( v) = p( v) + λ φ( v ), () = where λ s are weghts, = are the locaton constrants, and p (v) s a lnear affne functon of poston n the form of p( v) = p0 + px + p2 y + p3z. For the radal bass 3 functons, we employ φ ( r) = r whch s wdely used n 3D problems. For N poston constrants, they can be wrtten as N lnear equatons j j N Γ( v ) = p( v ) + λ φ( v v ) = f (2) = wth j =,..., N and f j s the functon value shown at the locaton v j. The unknown λ s n above lnear equaton system can be solved by addng the followng compatble condton N = N N = = N λ λ x λ y λ z = 0. = = = = j j

4 4 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, MANUSCRIPT ID Fg.2. Illustraton for determnng the 3D shape of slhouette curve through sketchng: (a) the method to determne depth coordnates, where two planes the projecton plane and the depth plane are conducted; (b) the projecton of slhouette curve on the projecton plane (n blue) and the depth plane (n green) can be smoothed wth the help of splne curve fttng and adjusted wth the control ponts; (c) crosssecton vews of Γ by smplfy askng the transent surface to only nterpolate sampled poston constrants (top) versus the transent surface wth normal constrants added (bottom) t s clear that we cannot keep the sampled ponts on surface slhouette wthout normal constrants. 2.2 Shape Modelng Equpped by the RBF-based mplct surface modellng method, the transent surface Γ to jon the scssored models can be elegantly constructed. The shape of transent mplct surface s controlled by two types of constrants: the boundary constrants and the slhouette constrants. To acheve poston nterpolaton on the boundares of models under composton, all vertces on the boundares are substtuted nto Eq.(2) where each yelds a lnear equatons. Snce the transent surface should exactly pass all the vertces on mergng boundares, the functon values of Γ shown on the vertces are zero (.e., f = 0 ). However, smply settng these poston constrants leads to vansh sde condtons so that the computaton of Γ fals. Based on ths reason and also to ensure the smooth nterpolaton, Turk and O'Bren n [48] added the normal constrants to the lnear equaton system. Here, we also defne normal constrants n a smlar manner but wth more ponts. Specfcally, we frst add all vertces of the mergng boundary and ther n-rngs neghbors on the models under composton nto the RBF system as nterpolatng ponts. n = 3 s chosen snce t acheves a balance of the computatonal speed and the smoothness of reconstructed mplct surface. The set of all these vertces are denoted by V *. Then, for each vertex v V * and ts surface normal n, a normal constrant s added by placng a poston constrant at v + τn wth the functon value f = τ where τ s a small value. We use τ = 0.Lmn n all of our examples, where L mn s the shortest (non-zero) edge length on boundares. The surface normal at a vertex s computed by angle-weghted average of the normals [2] on ts adjacent faces. Usng large value of τ may make the normals ntersect each other, therefore a small value s chosen for τ. Above normal constrants are added to ensure the smooth transton n mesh composton. In some cases, f nonsmooth transton s wanted, we can adjust the orentaton of n to satsfy dfferent nterpolaton requrements. In order to control the shape of transent surface Γ, the slhouette constrants are added nto the RBF system to vary the shape of Γ only determned by boundary constrants. A slhouette curve of Γ s determned by two steps through sketchng. After drawng a stroke on the screen plane the stroke s stored as a lst of ponts P lst, we frstly fnd the two mergng boundares A and B by the closest boundary ponts to the startng and endng ponts of P lst n the screen plane. Note that A and B could be on the same model under composton. The two endng ponts P and Q on the slhouette of these two openngs are then located. After that, a plane Ω contanng P and Q, and whose normal vector s closest to the vewng vector, s determned the plane s called as projecton plane. All ponts n P lst are frstly projected onto Ω to get temporary depth coordnates. Secondly, we construct another plane Π, named as depth plane, passng P, Q and the vewng vector. The reason for ntroducng Π s to gve users an nterface to control the shape of slhouette better n 3D. The ponts n P lst on Ω are then further projected onto Π to determne ther 3D coordnates. A sxth-order Bezer curve C wth 6 control ponts are employed to approxmate the projected ponts n P lst. Among the sx control ponts, two control ponts at each end of the curve (red ones n Fg.2(b)) are fxed to ensure the tangental smoothness. The left two control ponts (blue ones n Fg.2(b)) are allowed to move to further modfy the shape of slhouette curve. The coordnate of control ponts can be determned and adjusted by ther projecton on Ω and Π. Ponts are unformly sampled on C f to serve as poston constrants of the slhouette. Smply askng the transent mplct surface Γ to nterpolate the sampled ponts from C f cannot ensure that they are on the slhouette of Γ (see Fg.2(c)). From the defnton of slhouette n computer vson, we know that for a pont on a slhouette, the surface normal at ths pont s perpendcular to the vewng drecton. Therefore, the normal constrants on the sample ponts from C f also need to be added. Wth the tangent vector t p at a sampled pont p C f and the vewng vector n v, the normal constrant at pont p (.e., n p = t p nv ) s added to model the RBF-based mplct surface Γ. It s easy to conduct slhouettes to control the shape of transent mplct surface. Examples on the shape modelng of transent mplct surface wll be shown later n the expermental result secton. Poston and normal constrants on both the surface boundares and the slhouette curves usually result n about several thousand lnear equatons n the RBF system. To effcently solve ths dense lnear equatons system, the publc avalable lbrary LU-GPU [5], whch s accelerated by GPUs, s employed. A dense lnear equaton system wth dmenson 3,000 can be solved n less than 5 seconds (.e., n an nteractve speed). f

5 AUTHORS: MESH COMPOSITION ON MODELS WITH ARBITRARY BOUNDARY TOPOLOGY 5 Γ s a closed mplct surface parttoned by the boundares on models under composton, only part of the mplct surface belongs to the transent regon. Therefore, a tessellaton method s needed to provde the followng functons: Dstngush the nterested and non-nterested regons on the transent mplct surface automatcally; Construct mesh compatble to the mesh connectvty on boundares of models under composton; Generate well-shaped trangles. Exstng mplct tessellaton methods rarely satsfy all these requrements. We propose a new algorthm the Localzed Marchng Cubes algorthm plus remeshng for solvng the above ssues. Our algorthm conssts of four steps: ) Cubes constructon and classfcaton, 2) Topology guaranteed tessellaton (an example of the step result has been llustrated n Fg.3(b) and 3(d)), 3) Qualty optmzed gap trangulaton (an example step result s shown n Fg.3(f)), and 4) Transent surface remeshng (resultng n Fg.3(g)). Fg.3. Tessellatng the transent mplct surface usng our new Localzed Marchng Cubes algorthm: (a) the models wth nonparwse boundares under composton, (b) partally fnshed transent mesh surface gap strps are left to be further trangulated, (c) the fnshed model, (d) the zoom and mesh vew of (b) the blue curve s the back of the boundary on the other sde, (e) the zgzag boundary n (d) are smoothed, (f) progressve result wth gap strps trangulated, and (g) the fnal remeshed transent surface. Fg.4. Illustraton for the cubes constructon and classfcaton n our Localzed Marchng Cubes algorthm: (a) the transent mplct surface n the workng space are separated nto portons by the boundares, (b) the boundary surface cubes BS-Cubes, (c) some seed IS-Cubes (n blue) are found near BS-Cubes, and (d) the VS- Cubes (n green) and the IS-Cubes (n blue) are separated through a floodng algorthm wth the seed IS-Cubes. 3 LOCALIZED MARCHING CUBES ALGORITHM The transent surface constructed so far s represented mplctly n a scalar functon. We need to tessellate the transent surface nto meshes. However, as the transent surface 3. Cubes Constructon and Classfcaton The algorthm proposed here s a varaton of the famous Marchng Cubes (MC) algorthm, whose sprt s to subdvde the nterested space Ψ of an mplct surface nto cubc sub-spaces (named as cubes) and then tessellate the surface based on the nsde/outsde flags of the eght nodes on each cube. Smlarly, the frst step of localzed MC s also to construct cubes. However, dfferent from orgnal MC, we need further classfy the constructed cubes nto dfferent categores so that can trm the transent mplct surface Γ nto nterested and non-nterested portons. In detal, four types of cubes are defned n our algorthm: E-Cube: For a cube, f ts eght nodes are all nsde Γ (.e., Γ( p ) > 0 ) or all outsde Γ (.e., Γ( p ) < 0 ), the cube wll be empty durng the tessellaton denoted by E- Cube as no trangle wll be generated n these cubes. All the other cubes are surface cubes, S-Cube, whch can be further classfed nto three types below. BS-Cube: For an S-Cube, f t ntersects any boundary edge on the models under composton, t s defned as a boundary ntersectng cube. For any S-Cube not ntersectng the boundares, f one of ts 26 neghborng cubes s a boundary ntersectng cube, ths S-Cube s defned as a boundary neghborng cube. To be robust n the later regon separaton, both the boundary ntersectng cubes and the boundary neghborng cubes are classfed nto boundary surface cubes, BS-Cube. The red cubes n Fg.4 are BS-Cubes. VS-Cube and IS-Cube: Wth the BS-Cubes, the left S- Cubes are then separated nto the vald surface cubes (named as VS-Cube) that le on the nterested regon of the transent mplct surface Γ and the nvald surface cubes (called IS-Cube) on the non-nterested regon of Γ. VS-Cubes are shown n green color n Fg.4 whle IS-Cubes are dsplayed n blue. Note that n the mesh composton framework, the workng space Ψ of our localzed MC s a regon slghtly larger than the boundng box of all mergng boundares on the models, and the sze of cubes are selected as 0.8 tmes of the average length of edges on the mergng boundares. The cubes n Fg.4 are shown wth a larger sze for llustraton.

6 6 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, MANUSCRIPT ID We conduct a floodng algorthm to separate the IS-Cubes from the VS-Cubes. Frstly, the BS-Cubes are clustered nto groups. For example there are three groups for the BS-Cubes shown n Fg.4(b). On the models under composton, 3- rngs of trangles near to the mergng boundares are named as boundary trangles. Any type-undetermned S- Cube ntersectng a boundary trangle can serve as a seed IS- Cube. Startng from the seed IS-Cube, a floodng algorthm can be used to fnd all other IS-Cubes lnked to ths seed IS- Cube. The steps of fndng seed IS-Cubes and floodng are repeated untl no more seed IS-Cubes can be found. That means we have determned all IS-Cubes n the workng envelope e.g., the blue cubes n Fg.4(d). The blue cubes n Fg.4(c) llustrate the seed IS-Cubes adopted for floodng. Then, all the left type-undetermned S-Cubes are classfed nto VS-Cubes (e.g., the green ones n Fg.4(d)). The pseudocode of the seed IS-Cube search algorthm Functon Seed- CubeSearch (c) and the IS-Cube floodng algorthm Functon CubeFloodng (s) s lsted below. Both are mplemented as recursve functons. Functon SeedCubeSearch (c) Input: a BS-Cube c Output: a seed IS-Cube. for any of c s 26 neghbors c n 2. f c n s E-Cube OR BS-Cube OR IS-Cube, then contnue; 3. f c n ntersect a boundary trangle, then return c n ; 4. for any of c s 26 neghbors c n 5. s = null; 6. f c n s a BS-Cube, then s = SeedCubeSearch (c n ); 7. f s s NOT null, then return s; 8. return null; Functon CubeFloodng (d) Input: a seed IS-Cube d Output: the floodng result. Assgn d to an IS-Cube; 2. for any of d s 26 neghbors d n 3. f d n s E-Cube OR BS-Cube OR IS-Cube, then contnue; 4. CubeFloodng (d n ); For each group of BS-Cubes, we just randomly select one BS-Cube c as nput to call Functon SeedCubeSearch (c). A seed IS-Cube s wll be determned by ths functon. We then call Functon CubeFloodng (s) to propagate the regon of IS- Cubes. If the return of Functon SeedCubeSearch (c) s null, we need to select a BS-Cube from another group to fnd the seed IS-Cube by callng ths functon agan. The seed cube search and the cube floodng wll be teratvely conducted untl no more seed IS-Cube can be found. In both of these two functons, the searches are conducted locally. Thus, the classfcaton procedure can be completed quckly. 3.2 Topology Guaranteed Tessellaton of VS-Cubes In ths step, trangles are generated n the VS-Cubes based on the effcent and robust mplementaton [27] of the Marchng Cubes 33 algorthm n [6]. The algorthm runs through 4 steps for each VS-Cube: Step ) The case number and confguraton s determned based on the nsde/outsde flags on eght nodes of the VS-Cube; Step 2) Lookng up faces that need to be further tested n the above determned confguraton; Step 3) For each face needs to be further tested, testng and determnng ts correspondng subcase; Step 4) Lookng up the tlng for a determned subcase and generatng trangles n ths VS-Cube. Ths mplementaton n fact depends on three tables: A case table maps each of the 256 possble confguratons of a cube to one of the 5 cases and to a specfc number desgnatng ths confguraton; A test table stores the further tests to be performed to resolve topologcal ambguty for each confguraton; A tlng table encodes the method to trangulate a cube for each confguraton and subcase. The tessellaton results are guaranteed to be a two-manfold mesh surface. As only VS-Cubes are tessellated, there s a gap between the gven model and the resultant mesh (e.g., see Fg.3(d)), where s occuped by BS-Cubes and wll be further trangulated n the followng step. 3.3 Qualty Optmzed Gap Trangulaton The resultant mesh surface from prevous step s usually wth zgzag boundares (e.g., the mesh n Fg.3(d)), whch wll affect the results of gap trangulaton. Therefore, before trangulatng the gap, the boundares are smoothened (e.g., see Fg.3(e)). The optmal trangulaton problem has been well defned n [50]. Gven two pecewse lnear curves C P and C Q wth m and n vertces, a boundary brdge trangulaton (BBT) s defned as an ordered collecton of trangles M = {T, T 2,, T N }. For two pecewse lnear curves C P and C Q wth m and n vertces, respectvely, there are a total of m + n 2 m + n 2 ( m + n 2)! = = m n ( m )!( n )! dstnct boundary brdge trangulatons. Fnd one partcular BBT from ths huge pool that wll optmze a certan gven objectve by exhaustve search s nether plausble nor practcal. To solve ths problem, authors n [50] converted ths combnatoral search problem nto a shortest path problem, whch can be computed by the Djkstra's algorthm. When applyng ths technque to compute an optmzed gap trangulaton, two problems needs to be solved: ) what s the property measured on the graph lnk (.e., on the formed trangle), and 2) how to determne the startng vertces. We have the second problem because that the pecewse lnear curves here are closed whle the ones n [50] are open. The frst problem s solved by a heurstc that we wsh to have resultant trangles as regular as possble. Thus, the followng metrc s employed to measure the qualty of a trangle T J = 2 3 r / l.0, (3) where r s the radus of T s nscrbed crcle, and l s the maxmum length of three edges on T. The more regular the trangle T s, the smaller value s gven on the metrc J. We solve the second problem by another heurstc to select the closest two vertces on C P and C Q as the startng vertces. Fgure 3(f) shows an example of trangulated gaps flled wth qualty optmzed trangles.

7 AUTHORS: MESH COMPOSITION ON MODELS WITH ARBITRARY BOUNDARY TOPOLOGY 7 Fg.5. Transent prmtves are used to help place component models under composton: (a) the sphere and (b) the truncated cone n blue. 3.4 Transent Surface Remeshng So far we have obtaned the fully tessellated transent mesh surface whch seamlessly sttches the compostng components together as shown n Fg.3(f). An Area Equalzng Remeshng procedure akn to [4] s conducted to further mprove the qualty of resultant trangles. Gven a target edge length L, the followng steps were performed repeatedly for about 5 runs: Splt edges longer then 4L / 3 at ther mdpont; Collapse edges shorter than 4L / 5 nto ther mdpont; Flp edges to mnmze the devaton from valence 6; Iteratvely move each vertex p to ts area-weghted T centrod g for 0 tmes by: p p + λ I n n )( g p ) ( wth n be the normal vector of p and λ = 0. be a dampng factor used to avod oscllatons. Only the edges and vertces not on the boundary of transent surface are processed here. Fgure 3(g) shows an example transent mesh surface remeshed from the one gven n Fg.3(f). 4 OTHER ALGORITHMS FOR MESH COMPOSITION Other algorthms to support the mplementaton of mesh compostons are presented n ths secton. Although they are wth mnor techncal contrbutons, they are essental components to support our mesh composton framework. 4. Prmtve Based Components Placement As mentoned n other mesh composton papers (ref. [3] and [43]), the placement of model components under mergng s a very tedous job whch needs a lot of nteractvtes. To reduce ths burden, we developed a prmtve based components placement scheme. After scssorng the models usng the method n [4], the boundary on one model component s frstly selected. A selected prmtve s then placed on the selected boundary the prmtve could be a cube, a cylnder, a sphere, or a truncated cone. After that, other components can be placed, rotated and translated on the surface of the prmtve. Meanwhle, the dmensons of the prmtve can also be adjusted so that the poston of placed components are changed automatcally by keepng ts relatve local coordnate wth the transent prmtves. Wth these prmtves, t s easer to place the models under composton. Fgure 5 shows examples usng two dfferent prmtves a sphere and a truncated cone. Note that ntroducng these prmtves s only to reduce the work-loadng for placng the components under composton. Usng other technques (e.g., [43]) to place the components wll not affect the composton technque presented above. 4.2 Geometry Detal Propagaton The transent mesh surface produced by tessellatng the trmmed mplct surface lacks of geometry detals. Two methods have been mplemented n our framework so that can add geometry detals n post-processng steps, whch can be consdered as a 3D extenson of the mage completon n [46] akn to [54]. Sgnals for geometry detals The dfference between a surface S wth geometry detal and ts shape after low-pass flterng (.e., S ) can be consdered as sgnals. Drectly encodng/decodng ths dfference n Eucldean space s named as the dstance-map method, whch always leads to unexpected dstortons. Thus, the authors n [45] adopted the dfferences of Laplacan coordnates on S and S as the sgnal for geometry detals. More specfcally, for a vertex v S, the sgnal s defned as ξ = δ δ where δ and δ are the Laplacan coordnates of v on S and S. To overcome the dstorton of Laplacan operator on rregular meshes, we adopt the cotangent weghted Laplacan operator but not the unform Laplacan operator n [45]. The cotangent weghted Laplacan coordnate s consstent wth the curvature flow surface farng method n [9], whch s employed here to generate the smooth surface S by the curvature flow operator. After obtanng S, the geometry detal sgnal s recorded by encodng ξ = δ δ. Reversely, once the geometry detal sgnal ξ s gven on every vertex of a smooth surface S, we frst compute ts correspondng Laplacan coordnate δ usng the current poston of vertces. δ can then be calculated by δ = δ + ξ. Lastly, we re-compute the poston of every vertex by ther curvature flow Laplacan coordnates (.e., δ ) through a Least-Square fttng system (ref. [45]). Based on ths geometry detal encodng/decodng method, two schemes for propagatng detals on the smooth transent mesh surface are developed. Structured geometry detal propagaton The structural geometry detals are propagated along a user specfed curve patch by patch. Frstly, patches are constructed along the propagaton governng curve. After drawng a governng curve, t s resampled nto m anchor ponts and projected onto the surface S (.e., the smoothed one). The ntersecton pont between the curve and the boundary of the transent surface needs to be determned (e.g., the green pont n Fg.6(c)), whch s named as the bound anchor ponts. By the bound anchor ponts, the resampled ponts are grouped nto two types: ) the anchor ponts fall n the regon wth geometry detal (e.g., the red ponts n Fg.6(c)), and 2) the anchor ponts n the regon wthout geometry detal (e.g., the blue ones n Fg.6(c)). We create rectangular domans centered at every anchor pont. On the tangent plane of an anchor pont, two rectangles are defned. The wdth of the nner rectangle (the red rectangle n Fg.6(c)) along the governng curve s 2 tmes the dstance between the two neghborng anchor ponts, and the wdth perpendcular to the curve s related to the bandwdth of the structured geometry detals and s defned by users. The wdth of the outer rectangle (the green one n Fg.6(c)) s.5 tmes of the nner rectangle wdth. By [], the trangles

8 8 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, MANUSCRIPT ID Fg.6. Structured geometry detal propagaton : (a) user specfed propagaton governng curve, (b) the surface s sampled and projected onto the smoothed model the ntersecton anchor pont between the curve and the boundary of transent surface s computed, (c) two rectangles are constructed on the tangent plane of every anchor pont, (d) the trangles fall n dfferent rectangles are grouped, (e) the parameterzaton result by [], and (f) the reconstructed shape by the structured geometry detal. Fg.7. Unstructured geometry detal propagaton: (a) the model before detal reconstructon the source regon for geometry detal exemplars are selected n blue, and (b) the reconstructed model wth geometry detals. whose projectons to the tangent plane fall n the outer square can be parameterzed nto a rectangular doman (see Fg.6(e)). Only the trangles fall n the nner rectangle show small dstortons on the parameterzaton (.e., the red trangles n Fg.6(d) and 6(e)). Other trangles (e.g., the blue ones n Fg.6(d)) act as buffers to absorb dstortons, whch s smlar to [55]. Also, only the vertces n the nner rectangle are consdered n our structure propagaton algorthm. Startng from the patch assocated wth the bound anchor pont, we fll geometry detals to the vertces n the nner rectangle of each patch. More specfcally, for a patch ΨQ, we search among all patches assocated wth the anchor pont n the regon wth geometry detal (.e., the red ones n Fg.6(c)), and select the patch ΨP that gves the mnmal value on the followng dfference error E= Q ξ Tξ P ( ) / Q. (4) In Eq.(4), Q denotes the set of vertces n ΨQ wth geometry detal assgned, ξ P ( ) represents the geometry detal at the correspondng pont P( ) of vertex v ΨQ n patch ΨP, and T s the transformaton matrx between two local frames. In general, P( ) shall not be a vertex so that ts geometry detal s computed by the barycentrc nterpolaton of detals on vertces of the trangle holdng P( ). After fndng the best match patch ΨP of ΨQ, all vertces n ΨQ wthout geometry detal are assgned by the detal on ts correspondng pont on ΨP. In ths way, we can progressvely assgn geometry detals to the vertces n the nner Fg.8. Comparson between the transent mplct surfaces modeled wthout slhouette constrants (the 2nd column) versus wth slhouette constrants (the 3rd and 4th columns) under dfferent orentatons: from top to bottom, the angle between two components vares from 60, 90, 20, 50 to 80 degrees respectvely. square of all patches along the propagaton curve and then reconstruct ther geometry. Fgure 6 gves such an example. Unstructured geometry detal propagaton To defne a mesh-free geometry smlarty metrc, smlar to [53], we buld a geodesc fan wth eght branches of the length.5 Lmax ( Lmax s the maxmal edge length) on a vertex. Let p, j and q, j denote the samplng ponts on the geodesc fans at two vertces p and q respectvely, where (=0,..,7) s the branch ndex and j (j=0,,2) s the ndex of sample ponts on a branch. The smlarty of geometry detals on p and q s defned as Π ( p, q) = mn{ k τ τ, j 2, j L (ξ ( p, j ), ξ ( q(( + k )%8), j ))} (5) j j where τ, j s the weght of samplng pont on the geodesc fan. τ, j = when p, j s n a trangle whose vertces are all 2 wth ξ assgned; otherwse, τ, j = 0. L2 (ξ p, ξ q ) s the L norm of the dfference between two geometry detal vectors after rotatng them to have the same surface normal drecton. For a surface pont p nsde a trangle, ts geometry detal ξ ( p ) agan s calculated by the barycentrc nterpola-

9 AUTHORS: MESH COMPOSITION ON MODELS WITH ARBITRARY BOUNDARY TOPOLOGY 9 Fg.9. Comparson between the transent mplct surfaces modeled wthout (the 2 nd column) versus wth slhouette constrants (the 3 rd and 4 th columns) under dfferent dstances. Fg.0. A cartoon cat s created by composng two components (a) where the boundary openngs are non-planar. To better control the shape of transent surface, two strokes n green have been added to specfy the slhouettes (b). The constructed transent surface (c) and the fnal result (d) satsfy the user specfed slhouettes. ton. The smlarty defnton here s dfferent from the one used n the structured propagaton, where no rotaton s allowed. The value of Σ Σ jτ, j at a vertex gves the confdence factor of geometry detal the larger the value, the more confdent s the vertex. Our unstructured geometry detal propagaton employs ths confdence factor to determne the fllng order of vertex geometry detal. Specfcally, among all vertces wthout geometry detal, the vertex v p wth the hghest confdence s chosen. The vertex v q showng the hghest smlarty to v p n the user specfed source regon (e.g., the blue ones n Fg.7(a)) s selected, and the geometry detal ξ ( v q ) s then assgned to v p. After that, the confdence factors and the geodesc fans of vertces adjacent to v p are updated, and the search and fllng steps are repeated untl the geometry detal of all vertces n the transent regon are assgned. The strategy employed here s smlar to the mage completon algorthm n [8]. Note that dfferent regons could choose dfferent source regons as templates to fll n geometry detals. Lastly, the surface mesh wth detals s reconstructed by the flled geometry detals (e.g., Fg.7(b)). 5 EXPERIMENTAL RESULTS AND DISCUSSION We have mplemented the proposed approach on a PC wth Intel Pentum IV 2.4GHz CPU + 52M RAM. The followng two tests are chosen to study the mplct surface based mesh composton. The frst test s to nvestgate the shape change of RBFbased mplct surface wth the orentaton varaton of mergng boundares on the models under composton. For the sea-mads constructed n Fg.8, we progressvely ncrease the orentaton of the tal starng from 60 degree to 90, 20, 50, untl 80 degree. The transent surfaces modelled wthout slhouette constrants are wth very poor shape (as shown n the second column n Fg.8). However, after addng slhouette constrants, we can easly control ts shape well (as shown n the fourth column n Fg.8). The second test s smlar to above but wth the dstance varaton (see Fg.9). Wth the ncrease of dstance whle keepng the same boundary condtons (both poston and normal constrants on openngs), the transent RBF surface wthout slhouette constrants becomes more and more narrow n the sense of pullng an elastc object (e.g., see the second column n Fg.9). The result can be mproved by addng slhouette constrants (see the fourth column n Fg.9). Besdes the three-head sprt model shown n Fg., some more examples can be found n Fgs.0-2. Fgure 0 shows a mesh composton example wth non-planar boundary openngs. Slhouettes are specfed n Fg.0(b) to desgn the shape of the transent surface. Dfferent from those approaches that fully convert models nto mplct representaton lke [29] and [30], here only the transent part s an mplct surface therefore, the detals on orgnal models are retaned. In Fg., two legs of an elephant are replaced by the legs of amadlo. Detals are reconstructed on the smooth transent surface. The last example shown n Fg.2 s employed to show another beneft of our mplct transent surface based mesh composton approach the ablty of topology varaton. As shown n Fg.2(a), the default mergng result from a bottle and a torus s a mesh model wth genus- topology. However, after addng a stroke to specfy the slhouette of a hole n the mddle of the transent surface, our mesh composton framework can create a more complex transent surface to merge two components nto a model wth genus-2 topology. Ths s a functon that has not been provded n other mesh composton approaches. Although some sketchng approaches provde such functons (lke [3]), ther method s based on a 2D trangulaton step and a mesh nflatng step whch cannot be appled to the mesh composton drectly. When usng the LU-GPU lbrary to compute the RBF surface fttng and our localzed MC algorthm to tessellate the transent surface, the smooth mesh composton step can always be fnshed n less than 0 seconds (.e., at nteractve speed) on our mplementaton. For the postprocessng of geometry detal propagaton (whch s an optonal step), structure propagaton of detal sgnals can also be fnshed n an nteractve speed; however, the unstructured propagaton takes much longer tme usually n few mnutes as t nvolves many tmes of global search. The detal mesh surface reconstructed from assgned geometry detals can be fnshed n less than 0 seconds by usng the TAUCS solver [47].

10 0 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, MANUSCRIPT ID Fg.. A mesh composton example wth unstructured geometry detal propagaton an elephant has ts legs replaced by the amadlo s: (a) the model before detal reconstructon and (b) the reconstructed model wth geometry detals. Fg.2. The topology of transent surface can be easly changed by addng more slhouette curves: (a) the default composton result from a bottle and a torus s a mesh model wth genus- topology, and (b) after specfyng more slhouettes, our approach can create a more complex transent surface to merge two components nto a model wth genus-2 topology. Too close to each other Fg.3. If two mergng boundares are too close to each other, a gap wth more complex topology than strp wll be formed. 5. Lmtatons The current mplementaton of our mesh composton framework shows several lmtatons. The frst lmtaton comes from the localzed Marchng Cubes algorthm. The thrd step of our localzed MC algorthm assumes that the left gap to be flled by an optmal trangulaton s n the topology of a strp. However, ths s not true n the followng extreme case. If two of the boundary openngs are very close to each other so that some parts of them fall n the same cube, we then need to trangulate a gap wth one rng on one sde whle two rngs on the other sde (as shown n Fg. 3). Ths s a more complex problem than computng an optmal strp trangulaton. To avod ths, n our current mplement, we requre the placement of models should let the dstance between any two mergng boundares be greater than 3 w. Here, w s the wdth of cubes adopted n the localzed MC algorthm. One of our future works s planned to develop a constraned Delaunay trangulaton method n 3D to trangulate the gap regon wth more complex topology. Secondly, our mplementaton assumes that every stroke generates a slhouette curve on a plane lnkng to the endponts of two openngs. Ths s relatvely smple. Although we can further adjust the shape of slhouettes on the projecton plane and the depth plane later (e.g., the slhouettes to specfy holes n Fg.2 are generated by ths smple extenson), a more complex nterface s wanted by whch one slhouette can be specfed by several strokes as what s used n charcoal drawngs. We dd not nclude ths work n ths paper because we consder ths as mnor techncal contrbuton. Ths s consdered as another future work. Thrdly, the geometry detal encodng/decodng method presented n ths paper works well for the rlevo-lke detals, whch can be complex n shapes but wth smple topology; however, the Laplacan coordnate based detal encodng/decodng fals for detals wth complex topology such as bowknot. The method for constructng detals wth complex topology on the smooth transent surface wll be nvestgated n our future research, where the methods of volume texture [37] and [39] wll be consdered. Geodesc fans wth dfferent lengths lead to dfferent smlarty comparson results greater support sze s more robust but can hardly dstngush sharp features, whle smaller support sze works well on sharp features but are volated by local normal dsturbances. There s no smple means to choose an approprate support sze for the geodesc fans. We choose the sze through experments. As shown n the tests of Fg.4, dfferent smlartes are shown on the bunny model wth the geodesc fan s lengths as 0.005, 0.0, 0.05 and 0.02 of the dagonal length of the bunny s boundng box. The smlartes obtaned by the geodesc fans wth 0.0 dagonal length are the best. Therefore, we employ ths length for geodesc fans n all our examples. Lastly, the current mplementaton does not consder about the ntersecton between the transent mplct surface and the exstng meshes under composton. We rely on users to fnd the ntersecton and change the shape of transent surface by slhouette curves. Although such case rarely occurs n our tests, t cannot be fully prevented. Ths wll be nvestgated n our future work. Furthermore, keepng the orgnal meshes be rgd may cause problems when the models are not well algned and the mergng boundares are very close to each other. For ths scenaro, the automatc shape adjustment approach as [9] wll be helpful.

11 AUTHORS: MESH COMPOSITION ON MODELS WITH ARBITRARY BOUNDARY TOPOLOGY ACKNOWLEDGMENT The authors would lke to thank the Shape Repostory for sharng some of the models. Juncong Ln, Charle C.L. Wang and Kn-Chuen Hu were partally supported by Hong Kong RGC/CERG grant CUHK/42405 and DAG grant CUHK/ Xaogang Jn s supported by the Chna 863 program (Grant: 2006AA0Z34), the Natural Scence Foundaton of Zhejang Provnce (Grant: R0543), the Natonal Natural Scence Foundaton of Chna (Grant: ) and NCET REFERENCES Fg.4. Smlarty maps wth dfferent geodesc fan length: (a) the smlarty of all vertces to the vertces gven n blue color s computed, (b)- (e) the geodesc fan s lengths are gven as 0.005, 0.0, 0.05 and 0.02 of the dagonal length of the boundng box. 6 CONCLUSION In ths paper, we developed a novel mesh composton framework for creatng 3D models from models wth arbtrary boundary topology. The novel mesh composton framework s based on two technques developed here. After placng the components of mergng n ther rght pose, a RBF-based mplct surface s adopted to smoothly nterpolate the boundares of models under composton. To acheve better shape control for the transent part, a new nterface s developed to control the shape of the mplct transent surface by usng sketches to specfy the expected slhouettes. A localzed Marchng Cubes algorthm s nvestgated to tessellate the mplct transent surface so that the mesh surface of composed model s generated. Based on these two techncal contrbutons, our mesh composton framework can fuse models wth arbtrary boundary topology but all exstng mesh composton approaches need to have parwse mergng boundares. Such an exctng new functon provdes a method to buld more complex models by mesh composton easly and effcently. Also, some assstant technques have been presented n ths paper to help ) pose the models under composton more easly and 2) propagate structured and unstructured geometry detals on the smooth transent surface. The examples presented n ths paper show the success of these functons. [] S. Akkouche, E. Galn, Adaptve mplct surface polygonzaton usng marchng trangles, Computer Graphc Forum, vol.20, pp.67-80, 200. [2] C. Andujar, P. Brunet, A. Chca, I. Navazo, J. Rossgnac, and A. Vnacua, Optmzng the topologcal and combnatoral complexty of sosurfaces, Computer-Aded Desgn, vol.37, pp , [3] H. Bermann, I. Martn, F. Bernardn, and D. Zorn, Cut-andpaste edtng of multresoluton subdvson surfaces, ACM Tran. Graphcs, vol.2, no.3, pp , [4] M. Botsch, and L. Kobbelt, A remeshng approach to multresoluton modelng, Proc. of the EUROGRAPHICS/ACM SIGGRAPH Symposum on Geometry Processng, pp , [5] E.V. Chernyaev, Marchng cubes 33: constructon of topologcally correct sosurfaces, Techncal Report CERN CN-95-7, 995. [6] J.C. Carr, R.K. Beatson, J.B. Cherre, T.J. Mtchell, W.R. Frght, B.C. McCallum, and T.R. Evans, Reconstructon and representaton of 3D objects wth radal bass functons, Proc. of SIGGRAPH 200, pp.67-76, 200. [7] U. Clarenz, U. Dewald, G. Dzuk. M. Rumpf, R. Rusu, A fnte element method for surface restoraton wth smooth boundary condtons, Computer Aded Geometry Desgn, vol.5, pp , [8] A. Crmns, P. Perez, and K. Toyama, Regon fllng and object removal by exemplar-based mage npantng, IEEE Trans. on Image Processng, vol.3, no.9, pp , [9] M. Desbrun, M. Meyer, P. Schröder, and A.H. Barr, Implct farng of rregular meshes usng dffuson and curvature flow, Proc. of ACM SIGGRAPH 99, pp , 999. [0] M.J. Duerst, Letters: Addtonal reference to marchng cubes, Computer Graphcs, vol.22, pp.72-73, 988. [] M. Floater, Parameterzaton and smooth approxmaton of surface trangulatons, Computer Aded Geometrc Desgn, vol.4, no.3, pp , 997. [2] H. Fu, C.-L. Ta, and H. Zhang, Topology-free cut-and-paste edtng over meshes, Proc. of Geometrc Modelng and Processng, pp.73-84, [3] H. Fu, O.K.C. Au, C.-L. Ta, Effectve dervaton of smlarty transformatons for mplct Laplacan mesh edtng, Computer Graphcs Forum, vol.26, no., pp.34-45, [4] T. Funkhouser, M. Kazhdan, P. Shlane, P. Mn, W. Kefer, A. Tal, S. Rusnkewcz, and D. Dobkn, Modelng by example, ACM Trans. on Graphcs, vol.23, no.3, pp , [5] N. Galoppo, N.K. Govndaraju, M. Henson, and D. Manocha, LU-GPU: effcent algorthms for solvng dense lnear systems on graphcs hardware, Proc. of the 2005 ACM/IEEE conference on Supercomputng, pp.3, 2005.

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13 AUTHORS: MESH COMPOSITION ON MODELS WITH ARBITRARY BOUNDARY TOPOLOGY 3 [56] Q.-Y. Zhou, T. Ju, and S.-M. Hu, Topology repar of sold models usng skeletons, IEEE Trans. on Vsualzaton and Computer Graphcs, accepted, Juncong Ln s a PhD canddate of the State Key Lab of CAD&CG, Zhejang Unversty. He receved hs BSc degree n Envronmental Engneerng n 2003 from Zhejang Unversty. Hs research nterests nclude mesh edtng and modellng. Kn-Chuen Hu receved hs B.Sc and Ph.D n Mechancal Engneerng n 979 and 990 respectvely from the Unversty of Hong Kong. Before jonng the Chnese Unversty of Hong Kong n 992, he was a consultant n the CAD Servces Centre of the Hong Kong Productvty Councl. He s currently a Professor of the Mechancal and Automaton Engneerng Department at the Chnese Unversty of Hong Kong, and s the drector of the Computer-Aded Desgn Laboratory. He s a member of the edtoral board of the Journal of Computer-Aded Desgn. Hs research nterests nclude computer graphcs, geometrc and sold modelng, vrtual realty, and ther applcatons. Xaogang Jn s a professor of the State Key Lab of CAD&CG, Zhejang Unversty. He receved hs BSc degree n computer scence n 989, MSc and PhD degrees n appled mathematcs n 992 and 995, all from Zhejang Unversty. Hs current research nterests nclude mplct surface computng, specal effects smulaton, mesh fuson, texture synthess, crowd anmaton, cloth anmaton and facal anmaton. Charle C. L. Wang s currently an Assstant Professor at the Department of Mechancal and Automaton Engneerng, the Chnese Unversty of Hong Kong. He ganed hs B.Eng. (998) n Mechatroncs Engneerng from Huazhong Unversty of Scence and Technology, M.Phl. (2000) and Ph.D. (2002) n Mechancal Engneerng from the Hong Kong Unversty of Scence and Technology. He s a member of IEEE and ASME. Hs current research nterests nclude geometrc modelng n computer-aded desgn and manufacturng, bomedcal engneerng, and computer graphcs, as well as computatonal physcs n vrtual realty.