Representing Non-Manifold Shapes in Arbitrary Dimensions

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1 Represening Non-Manifold Shapes in Arbirary Dimensions Leila De Floriani,2 and Annie Hui 2 DISI, Universiy of Genova, Via Dodecaneso, Genova (Ialy). 2 Deparmen of Compuer Science, Universiy of Maryland, College Park, MD (USA). deflo@disi.unige.i, huiannie@cs.umd.edu Absrac In his paper, we describe and analyze several represenaions for complex shapes, i.e., muli-dimensional, nonmanifold objecs wih pars of mixed dimensionaliy, based on simplicial complexes. We review our work on compac and scalable represenaions for 2D and 3D simplicial complexes embedded in 3D Euclidean space. We propose a dimension-independen represenaion for simplicial complexes in which all he simplexes are uniquely and explicily encoded. Finally, we describe a decomposiion approach o he represenaion of non-manifold objecs in arbirary dimensions based on heir decomposiion ino nearly manifold componens. Keywords: shape modeling, non-manifold shapes, geomeric daa srucures. Inroducion We consider he problem of represening and manipulaing non-manifold and non-regular objecs in arbirary dimensions described by simplicial complexes. A manifold objec is a subse of he Euclidean space for which he neighborhood of each poin is homeomorphic o an open ball or o an open half-ball. Objecs, ha do no fulfill his propery a one or more poins, are called non-manifold objecs, and if hey also conain pars of differen dimensionaliies are called non-regular. Cell and simplicial complexes are widely used o represen muli-dimensional geomeric objecs in geomeric and solid modeling, in finie elemen analysis and in visualizaion. In paricular, simplicial complexes have received grea aenion, since heir combinaorial properies make hem easier o encode and manipulae. In he lieraure, represenaions have been proposed for 2D simplicial and cell complexes describing he boundary of 3D non-manifold and non-regular objecs [2, 3, 9, 20], bu such daa srucures do no scale well wih he degree of non-manifoldness (i.e., he number of geomeric singulariies) of he complex, hus becoming inefficien when dealing wih objecs wih few non-manifold singulariies. Represenaions developed for cell or simplicial complexes in arbirary dimensions, (e.g., [, 4, 7]), are resriced o a subclass of complexes, namely manifold or pseudo-manifold complexes. Here, we presen firs represenaions specific for 2D and 3D simplicial complexes embedded in he 3D Euclidean space, and hen we describe a new dimension-independen represenaion for simplicial complexes ha we are currenly developing. Anoher approach o represen non-manifold shapes consiss of decomposing he shape ino manifold componens. Some approaches have been proposed in he lieraure for uniformly-dimensional non-manifold shapes [8, 0,, 8]. We review here a sound decomposiion which produces a unique descripion of a non-manifold muli-dimensional objec as a combinaion of nearly manifold componens. This is no only he basis for a compac, dimension-independen, daa srucure for shapes described by simplicial complexes, bu i also provides a basis for non-manifold feaure recogniion and for compuing opological invarians of such shapes. The remainder of his paper is organized as follows. In Secion 2, we summarize some background noions. In Secion 3, we presen he Triangle-Segmen (TS) daa srucure for describing 2D simplicial complexes, while in Secion 4, we describe he Non-Manifold Indexed Daa Srucure wih Adjacencies (NMIA) for describing 3D simplicial complexes. In Secion 5, we presen a new dimensionindependen daa srucure for simplicial complexes. In Secion 6, we describe a decomposiion-based approach for represening non-manifold objecs. Finally, in Secion 7, we draw some concluding remarks. 2 Background In his Secion, we review some basic noions abou Euclidean simplicial complexes in arbirary dimensions, and abou opological relaions. A Euclidean simplex σ of dimension k (or a k-simplex) is he convex hull of k + linearly independen poins in he n-dimensional Euclidean space E n, 0 k n. Any Euclidean p-simplex σ, wih 0 p k, generaed by a subse of he verices of σ, is called a p-face (or, simply, a face) of σ. A finie collecion Σ of Euclidean simplexes forms a (Euclidean) simplicial complex if and only if (i), for each simplex σ Σ, all faces of σ belong o Σ, and (ii), for each pair of simplexes σ and σ, eiher σ σ = or σ σ is a face of boh σ and σ. If d is he maximum of he dimensions of he simplexes in Σ, we call Σ a d-dimensional simplicial complex, or a simplicial d-complex. The carrier of a Euclidean simplicial d-complex Σ embedded in E n, wih 0 d n, is he subse of E n defined by he union,

2 as poin ses, of all he simplexes in Σ. The boundary of a simplex σ is he se of all faces of σ in Σ, differen from σ iself. The sar of a simplex σ is he se of simplexes in Σ ha have σ as a face. The link of a simplex σ is he se of all he faces of he simplexes in he sar of σ which are no inciden in σ. Any simplex σ such ha he sar of σ conains only σ is called a op simplex. Le Σ be a simplicial d-complex and le σ Σ, wih 0 p d. We define he following opological relaions on he simplexes of Σ: For 0 q p, boundary relaion R p,q (σ) consiss of he se of q-simplexes in he se of faces of σ. For p + q d, co-boundary relaion R p,q (σ) consiss of he se of q-simplexes in he sar of σ. For p > 0, adjacency relaion R p,p (σ) is he se of p- simplexes in Σ ha are (p )-adjacen o σ (i.e., hey share a (p )-face wih σ). Adjacency relaion R 0,0 (σ), where σ is a verex, consiss of he se of verices σ such ha {σ, σ } is an -simplex of Σ. 3 A Represenaion for 2D Simplicial Complexes The Triangle-Segmen (TS) daa srucure [6] describes a 2D simplicial complex embedded in he 3D Euclidean space by encoding all verices, and op simplexes. The op simplexes are riangles and wire-edges, i.e., edges no bounding any riangle. For each verex v, he TS daa srucure sores parial relaion R 0,2(v), which encodes one riangle for each conneced componen of he sar of v ha consiss of riangles, and parial relaion R 0,(v), resriced o wire-edges, which encodes he all wire-edges inciden in v. For each riangle, he following wo relaions are sored: boundary relaion R 2,0 (), which encodes he indexes o he verices of, and parial relaion R 2,2() ha, for each edge e of, encodes he riangle(s) ha are immediaely preceding and succeeding in couner-clockwise order around edge e. In Figure, we show an example of he riangles in parial relaion R 2,2(). In he example of Figure, relaion R 2,2( ) a non-manifold edge e consiss of he riangles immediaely preceding and following along edge e in counerclockwise order, namely, 2 and 4. There is no riangle a he oher wo edges of. 4 e Figure : Encoding parial relaion R 2,2 a a non-manifold edge in he TS daa srucure In he TS daa srucure, edges are no encoded explicily, wih he excepion of wire-edges. The TS is a nonmanifold exension of he 2D version of he Indexed daa 2 3 srucure wih Adjacencies. The Indexed daa srucure wih Adjacencies (IA)[7] can represen d-dimensional manifold complexes and i sores, for each d-simplex σ, relaions R d,0, i.e., he indexes of he verices of σ, and R d,d, i.e., he d-simplexes adjacen o σ along a (d )-face, and for each verex v, parial relaion R0,d, which consiss of one d-simplex inciden a v. When applied o a manifold complex, he TS exhibis an overhead of only one bye per verex compared wih encoding i hrough he IA daa srucure. We have shown in [5] ha edge-based daa srucures, like he Loop Edge-use (LE) [5], and he Direced Edge (DE) [2] daa srucures, require a leas wice he sorage cos of he TS. A specializaion of he Parial Eniy daa srucure o 2D simplicial complexes requires four imes he space of he TS [5]. All opological relaions can be rerieved in opimal ime (i.e., in ime linear in he number of simplexes involved in he relaion) from he TS daa srucure as shown in [6]. In [6], algorihms are also presened for performing verexpair conracion (i.e., collapsing wo verices ino a new verex) and verex spli (i.e., is inverse ransformaion) on a simplicial complex encoded in he TS daa srucure. 4 A Represenaion for 3D Simplicial Complexes The Non-Manifold Indexed daa srucure wih Adjacencies (NMIA) [3] describes a hree-dimensional simplicial complex embedded he 3D Euclidean space, by encoding is verices, and all is op simplexes. We call a op 2-simplex a dangling face, and a op -simplex a wire-edge. The NMIA daa srucure sores boundary relaions R i,0, i =, 2, 3, for each op simplex (erahedron, dangling-face, and wireedge), adjacency relaion R 3,3 (), which encodes, for each erahedron, he four face-adjacen erahedra of, and parial relaions o describe non-manifold siuaions a nonmanifold verices and edges. Similarly o he TS daa srucure, for each verex v, he NMIA daa srucure sores parial relaions R 0,(v), which encodes he wire-edges inciden a v, R 0,2(v), which encodes one dangling-face for each wo-dimensional conneced componen of he sar of v, R 0,3(v), which encodes one erahedron for each hree-dimensional conneced componen of he sar of v. Noe ha in he manifold case, we would encode only R 0,3(v), which consiss of jus one erahedron a v. Similarly o he TS daa srucure, he NMIA daa srucure encodes non-manifold siuaions a edges implicily, since hese laer are no explicily described in he srucure. Thus, for each non-manifold edge e of a dangling face f, i encodes he op simplexes preceding and following f around e (in Figure 2, a edge e of dangling-face df, i encodes 4 and ). Similarly, for each non-manifold edge e of a erahedron, he NMIA srucure encodes he op simplexes preceding and following around e, when ha op simplex is no adjacen o along a face (for example, in Figure 2, a edge e of erahedron, i encodes 4, bu no 2 ). The NMIA daa srucure is a non-manifold exension

3 4 3 e 2 Figure 2: Encoding simplexes a a non-manifold edge in he NMIA daa srucure of he 3D insance of he IA daa srucure. I scales well o he manifold case, since i exhibis only an overhead of 6 byes per verex wih respec o he IA. All opological relaions can be rerieved in opimal ime from he NMIA daa srucure, wih he excepion of R,3 and R 0,3 relaions, since, for rerieving he erahedra inciden a a nonmanifold edge e, he dangling-faces inciden a e need o be raversed as well. Algorihms for performing edge collapse and is inverse, verex-spli, on he NMIA daa srucure are described in [3]. 5 A Dimension-Independen Represenaion for Simplicial Complexes A very general and dimension-independen daa srucure for cell complexes is he Incidence Graph (IG) [9], which can be specialized o simplicial complexes. I encodes all he simplexes of a simplicial d-complex Σ, and he immediae boundary, and immediae co-boundary relaions of each simplex. Namely, i sores, for each p-simplex σ, where 0 < p d, boundary relaions R p,p (σ), and, for each q- simplex σ, where 0 q <d, co-boundary relaions R q,q+ (σ). Alhough simple o manipulae, he incidence graph is a verbose daa srucure. Thus, we have defined a new dimension-independen daa srucure for simplicial complexes, called he Incidence Simplicial (IS) Daa Srucure, which sores all simplexes of he complex plus he same boundary relaions R p,p as he IG, bu parial co-boundary relaions R p,p+ insead of he full co-boundary relaions. For each p-simplex σ, R p,p+(σ) encodes jus one (p+)-simplex for each conneced componen in he sar of σ. Figure 3 shows he difference beween he IG and he IS daa srucures for 2D simplicial complexes. In he IG, he complee co-boundary relaion R 0, (v) is encoded, which consiss of all seven edges inciden a v. In he IS, only he parial co-boundary relaion R 0,(v) is encoded, and i consiss of only hree edges, since here are hree conneced componens in he link of v. We can easily see ha he IS daa srucure is more compac han he IG, especially when here are few non-manifold singulariies. If here are m (p+)-simplexes in he sar of a simplex σ, organized ino q conneced componens, he IG encodes all m (p+)-simplexes, while he IS encodes only one (p+)-simplex for each conneced componen, and hus q (p+)-simplexes. In general, q is much smaller han m (in he manifold case, for insance q =, while m may be arbirarily large). The rerieval algorihms for boundary relaions are he same in IG and IS. They are based on he ransiive closure df e 2 e 3 e v e 5 e 6 e 7 (a) e 4 e 2 e 4 Figure 3: Comparison beween IG and IS daa srucure for 2D simplicial complexes: (a) in he IG, relaion R 0 (v) = {e, e 2, e 3, e 4, e 5, e 6, e 7 } is encoded. (b) in he IS daa srucure, parial relaion R 0(v) = {e 2, e 4, e 7 } is encoded. of he R p,p encoded in boh daa srucure. The rerieval of co-boundary relaion R p,q, wih q > p is again he ransiive closure of co-boundary relaions expressible as a series of R p,p+. In he IG hese laer are sored, while in he IS hey have o be rerieved. Noe ha R d,d is encoded in he IS as well, since R d,d is he same as R d,d. The IS daa srucure scales very well o manifold cases. I exhibis an overhead of one bye and one bi per verex for manifold 2-complexes, and of four byes and four bis per verex for manifold 3-complexes. When applied o a 2D simplicial complex, he IS has an overhead of approximaely 2n compared wih he TS, where n is he number of edges in he complex. We have seen ha he sorage cos of he IS daa srucure is abou.4 he cos of he TS daa srucure in boh he manifold and non-manifold cases. 6 A Decomposiion Approach o Non-manifold Modeling A decomposiion of a non-manifold shape ino simpler pars can be obained by spliing he shape a hose elemens (verices, edges, faces, ec.) where singulariies occur. In order o be effecive, he decomposiion process should remove as many singulariies as possible, wihou inroducing arificial, or arbirary, cus hrough manifold pars. Under hese assumpions, a decomposiion ino manifold componens is possible, in general, only for wo-dimensional complexes. In hree or higher dimensions, a decomposiion ino manifold componens may need o inroduce arificial cus hrough he objec. In six or higher dimensions, a decomposiion ino manifold componens is no feasible in general, since he class of d-manifolds has been proven o be no decidable for d 6 [6]. In [7], we have defined a decomposiion which is unique, since i does no make any arbirary choice in deciding where he objec has o be decomposed, and naural, since i removes singulariies by spliing he complex a non-manifold simplexes only. We have called such a decomposiion he sandard decomposiion of he original complex, and shown ha he componens of such decomposiion, ha we called Iniial Quasi-Manifolds (IQMs), admi a local characerizaion in erms of combinaorial properies around each verex. A d-dimensional IQM is a simplicial d-complex Σ in which all op simplexes have dimension d and such ha he sar of each verex of Σ is (d )-conneced, i.e., can be raversed by moving beween adjacen d-simplexes hrough e 7 v (b)

4 heir common (d )-face. Up o dimension wo, he class of iniial quasi-manifolds coincides wih ha of manifolds. In general, in hree or higher dimensions, an IQM is no always a manifold (and no even a pseudo-manifold). However, in dimension d 3, if an IQM is embeddable in R d, i mus be a pseudo-manifold complex (i.e., a (d )-conneced complex in which every (d )-simplex is on he boundary of one, or wo d-simplexes). Based on he above heory, we have developed an algorihm for compuing a sandard decomposiion of a simplicial complex ino IQM componens. Figure 4(b) shows an example of a decomposiion of he complex depiced in Figure 4(a) ino hree iniial quasi-manifold componens. The connecion among componens is described hrough he verices bounding he k-simplexes, which are shared by more han one componen. A verex v of Σ, which is shared by several IQM componens, is called a spli verex. The copy of spli verex v in a componen C i, o which verex v belongs, is denoed as v i and i is called a verex copy. In he example shown in Figure 4, verex v is spli ino verices v, v 2 and v 3 in he decomposiion. A sandard decomposiion can be encoded also wih oher daa srucures, for insance, by using an IS represenaion for each IQM componen. Being unique, a sandard decomposiion can be he basis for searching in daa bases of 3D and higher-dimensional shapes. Moreover, i can be successfully used for exracing form feaures of differen dimensionaliies from nonmanifold objecs as well as for compuing opological invarians. In [7], a daa srucure for a simplicial complex based on is sandard decomposiion has been proposed. Any h- dimensional IQM componen can be described by an IA daa srucure, since he sar of each verex in an IQM can be raversed by using relaions R h,h plus R0,h, which encodes one h-simplex for each verex. The relaions among he componens in he decomposiion are represened as a hypergraph H. The nodes in H correspond o IQM componens and each hyperarc corresponds o a spli verex v, and i connecs all componens C i sharing v. In he hypergraph shown in Figure 4(c), a hyperarc associaes v wih he hree componens C, C 2 and C 3 hrough he hree verex copies of v shown in Figure 4(b). For every verex copy v i corresponding o a spli verex v, he daa srucure encodes references o he componen conaining v i, and o he hyperarc corresponding o v. The hypergraph suppors a verexbased raversal among componens conneced hrough he same hyperarc. Given a verex copy v i from any componen C i, we can follow he reference o is hyperarc and find all oher verex copies v j conneced wih v, as well as all oher componens sharing v. The IQM daa srucure encodes only he op simplexes and he verices. Boh he IQM and he IS daa srucures can encode d-dimensional simplicial complexes. They boh can suppor he rerieval of all boundary relaions, and of co-boundary relaion R d,d and R 0,q in opimal ime, i.e. in ime linear in he number of simplexes involved. The oher co-boundary relaions R p,q (σ), 0 < p < q can be no rerieved from he IQM daa srucure in opimal ime since he rerieval algorihms on he laer involves rerieving R 0,q for each verex of σ. The sar of σ is a subse of he union of he sars of all he verices of σ. We have shown in [5] ha he IQM daa srucure is slighly larger han he TS daa and he NMIA daa srucures for 2D and 3D complexes, respecively, bu i has he same sorage cos for manifold complexes. we w df v (a) df 2 w C C 2 w we w w 2 v w 2 (b) C 2 C 3 v 2 v v 3 C (c) v v 2 v 3 df df 2 Figure 4: A 3D complex (a); is IQM decomposiion, and associaed decomposiion graph 7 Concluding Remarks We have presened a review and an analysis of dimensionindependen and dimension-specific represenaions for nonmanifold shapes, ha we have developed or are currenly developing. Now we are working on an implemenaion of he IS daa srucure in arbirary dimensions and on opological simplificaion operaors, for applicaion o he idealizaion of finie elemen meshes generaed from CAD models. Such dimension-independen daa srucure can also find applicaions in modeling of shapes in four-dimensional space described by ime-varying erahedral meshes. A common issue in represening and manipulaing nonmanifold objecs is he availabiliy of large-size simplicial represenaions for describing such objecs, whose complexiy can easily exceed he capabiliy of compuaional ools for analyzing hem (for insance, in finie elemen simulaions). In hese cases, adapively simplified meshes, i.e., simplicial complexes in which he level of he deail varies in differen par of he objec hey describe, are ofen required. On he oher hand, accurae mesh simplificaion algorihms are oo ime consuming o be performed on-line. Thus, a muli-resoluion model, which encodes he modificaions performed by a simplificaion algorihm in a compac represenaion, is an effecive soluion. Dimensionindependen non-manifold represenaions can be inegraed wih a muli-resoluion framework, giving rise o a powerful ool for modeling non-manifold objecs a variable resoluions [4]. We have also discussed a general approach o modeling non-manifold muli-dimensional shapes based on uniquely decomposing hem ino nearly manifold pars. Such decomposiion serves boh as he basis for a compac, dimension- C 3

5 independen represenaion of simplicial complexes, and as signaures of non-manifold properies, which is very useful for reasoning on non-manifold shapes. Our curren work is on developing efficien algorihms for updaing a sandard decomposiion when a opological local operaor, such as verex pair conracion, is applied o he underlying complex. Moreover, we are using he decomposiion as he basis for non-manifold feaure recogniion and for compuing opological invarians of non-manifold shapes. 8 Acknowledgemen This work has been parially suppored by he European Nework of Excellence AIM@SHAPE under conrac number REFERENCES [] E. Brisson. Represening geomeric srucures in D dimensions: opology and order. In Proceedings 5h ACM Symposium on Compuaional Geomery, pages ACM Press, 989. [2] S. Campagna, L. Kobbel, and H.-P. Seidel. Direced edges - a scalable represenaion for riangle meshes. Journal of Graphics Tools, 3(4): 2, 998. [3] L. De Floriani and A. Hui. Updae operaions on 3D simplicial decomposiions of non-manifold objecs. In D. Fellner, edior, 9h ACM Symposium on Solid Modeling and Applicaions, pages 69 80, Genova (Ialy), 9 June ACM Press. [4] L. De Floriani and A. Hui. A dimension-independen represenaion for muli-resoluion simplicial complexes submied for publicaion. [5] L. De Floriani and A. Hui. Daa srucures for simplicial complexes: an analysis and a comparison. In M. Desbrun and H. Pomann, ediors, Third Eurographics Symposium on Geomery Processing, pages 9 28, Vienna, Ausria, July 4-6, [6] L. De Floriani, P. Magillo, E. Puppo, and D. Sobrero. A muli-resoluion opological represenaion for nonmanifold meshes. Compuer-Aided Design Journal, 36(2):4 59, February [7] L. De Floriani, M. M. Mesmoudi, F. Morando, and E. Puppo. Non-manifold decomposiions in arbirary dimensions. CVGIP: Graphical Models, 65(/3):2 22, [8] H. Desaulnier and N. Sewar. An exension of manifold boundary represenaion o r-ses. ACM Transacions on Graphics, ():40 60, 992. [9] H. Edelsbrunner. Algorihms in Combinaorial Geomery. Springer Verlag, Berlin, 987. [0] B. Falcidieno and O. Rao. Two-manifold celldecomposiion of R-ses. In A. Kilgour and L. Kjelldahl, ediors, Proceedings Compuer Graphics Forum, volume, pages , Sepember 992. [] A. Gueziec, G. Taubin, F. Lazarus, and W. Horn. Convering ses of polygons o manifold surfaces by cuing and siching. In Conference absracs and applicaions: SIGGRAPH 98, Compuer Graphics, pages ACM Press, 998. [2] E. L. Gursoz, Y. Choi, and F. B. Prinz. Verex-based represenaion of non-manifold boundaries. In M. J. Wozny, J. U. Turner, and K. Preiss, ediors, Geomeric Modeling for Produc Engineering, pages Elsevier Science Publishers B. V., Norh Holland, 990. [3] S. H. Lee and K. Lee. Parial-eniy srucure: a fas and compac non-manifold boundary represenaion based on parial opological eniies. In Proceedings Sixh ACM Symposium on Solid Modeling and Applicaions, pages 59 70, Ann Arbor, Michigan, June 200. ACM Press. [4] P. Lienhard. Topological models for boundary represenaion: a comparison wih n-dimensional generalized maps. Compuer Aided Design, 23():59 82, 99. [5] S. McMains. Geomeric Algorihms and Daa Represenaion for Solid Freeform Fabricaion. PhD hesis, Universiy of California a Berkeley, [6] A. Nabuovsky. Geomery of he space of riangulaions of a compac manifold. Commun. Mah. Phys., 8: , 996. [7] A. Paoluzzi, F. Bernardini, C. Caani, and V. Ferrucci. Dimension-independen modeling wih simplicial complexes. ACM Transacions on Graphics, 2():56 02, January 993. [8] J. Rossignac and D. Cardoze. Machmaker: manifold BReps for non-manifold R-ses. In W. F. Bronsvoor and D. C. Anderson, ediors, Proceedings Fifh Symposium on Solid Modeling and Applicaions, pages 3 4. ACM Press, 9 June 999. [9] K. Weiler. The radial-edge daa srucure: a opological represenaion for non-manifold geomeric boundary modeling. In H. W. McLaughlin J. L. Encarnacao, M. J. Wozny, edior, Geomeric Modeling for CAD Applicaions, pages Elsevier Science Publishers B. V. (Norh Holland), Amserdam, 988. [20] Y. Yamaguchi and F. Kimura. Non-manifold opology based on coupling eniies. IEEE Compuer Graphics and Applicaions, 5():42 50, January 995.