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Computing handles and tunnels in 3D models Tamal K. Dey Joint work: Kuiyu Li, Jian Sun, D. Cohen-Steiner Department of Computer Science and Engineering Projects in Jyamiti group • Surface reconstrucion (Cocone) • Delaunay meshing (DelPSC) • Shape feature processing (Segmatch, Cskel) Department of Computer Science and Engineering 2/52 Handle loops/Tunnel loops Department of Computer Science and Engineering 3/52 Handle loops/Tunnel loops Department of Computer Science and Engineering 4/52 Topological Simplification Department of Computer Science and Engineering 5/52 Use of Handles and Tunnels • Topological simplification / feature recognition / parameterization • El-Sana and Varshney[1997] use α-hulls to search for tunnels. • Guskov and Wood[2001] propose a surface growing strategy to remove small handles. • Nooruddin and Turk[2003] use morphological operations to remove small handles. • Wood, Hoppe, Desbrun, and Schroder[2004] use Reeb graph to remove handles. • Shattuck and Leahy[2001] use Reeb graph to compute handles. • From WUSTL Model/surface editing (Zhou, Ju, and Hu[2007] use medial axis to detect loops) and work by Grimm et al. Department of Computer Science and Engineering 6/52 Nontrivial loops as first attempt? • Nontrivial loops can neither be handle nor tunnel. • Handle and tunnel loops have to be non- separating. Department of Computer Science and Engineering 7/52 Related work for loops on surface • Loops on a surface • Compute polygon schemas [Vegter-Yap90][Dey-Schipper95] • Linear time algorithm • Compute optimal systems of loops [VL05][EW06] • Verdiere and Lazarus gave an algorithm for computing a system of loops which is shortest among the homotopy class of a given one. • Erickson and Whittlesey gave a greedy algorithm to compute the shortest system of loops among all systems of loops. • Compute optimal cut graph [EH04] • Computing a shortest cut graph is NP-hard. Department of Computer Science and Engineering 8/52 A better idea • Intuitively, a loop on a connected, closed, surface M in R3 is: • handle if it spans a disk (surface) in the bounded space bordered by M. • tunnel if it spans a disk (surface) in the unbounded space bordered by M. Department of Computer Science and Engineering 9/52 Cycle, Boundary and Homology • p-chain: c=aii, i is p-simplex. (e2+e3+e4) is a 1-chain. • If c'=ai'i, then c+c'=(ai+ai')i. • c + c = 0 for Z2 coefficients. • Boundary operator • p = [v0,v1,…,vp] = i=0p[v0,…,~vi,…,vp] • e.g.: t = (e2+e3+e4) • e.g.: e0+ e1+ e4 = (v0+v2)+(v0+v1)+(v1+v2) = 0 Department of Computer Science and Engineering 10/52 Cycle, Boundary and Homology • p-cycle: if pc = 0. • p-boundary: if there’s (p+1)-chain d, s.t. c=pd . • p-cycle group Zp = kernel (p). • p-boundary group Bp = image (p+1). • Hp= Zp / Bp • Each element of Hp represents an equivalent class. • c1 and c2 are in the same class if c1 + c2 is in Bp. • Trivial elements in Hp bounds (p+1)-chain Department of Computer Science and Engineering 11/52 Handle and tunnel loops – definitions [Dey-Li-Sun 07] • Let M be a connected, closed, surface in R3. M separates R3 into inside I and outside O. Both I and O have M as boundary. • Handle: A loop on M whose homology class is trivial in H1(I) and non-trivial in H1(O). • Tunnel: A loop on M whose homology class is trivial in H1(O) and non-trivial in H1(I). Department of Computer Science and Engineering 16/52 Nasty knots • Consider a thickened trefoil • handle loop (green) • tunnel loop (red) Department of Computer Science and Engineering 17/52 Handle and tunnel loops – existence • Theorem (Dey, Li, Sun[2007]) • For any connected closed surface M of genus g in R3, there exist g handle loops {hi}i=1…g forming a basis for H1(O) and g tunnel loops {ti}i=1...g forming a basis for H1(I). • Furthermore, {hi}i=1...g and {ti}i=1...g form a basis for H1(M). Department of Computer Science and Engineering 18/52 First algorithm for GR models [Dey-Li-Sun 07] • Graph retractability assumption : A surface is graph retractable if both I and O deformation retract to a graph, denoted I and O, respectively. Department of Computer Science and Engineering 19/52 Inner and outer cores • I and O are disjoint core graphs, each with g loops : {KjI}j=1g and {KjO}j=1g and {Kj}j=12g all together. • How to compute I and O ? Department of Computer Science and Engineering 20/52 Linking number • J and K are two disjoint loops in R3. • In a regular projection, there are two ways in which J crosses under K. • The linking number, lk(K, J), is the sum of these signed crossings. Department of Computer Science and Engineering 21/52 Linking number Theorem • A loop on M is a handle iff lk(, KiI) 0 for at least one inner core and lk(, KiO)=0 for all outer cores. • A loop on M is a tunnel iff lk(, KiO) 0 for at least one outer core and lk(, KiI)=0 for all inner cores. Department of Computer Science and Engineering 22/52 Minimally linked basis • A loop is minimally linked if it links once only with one core loop. • Theorem : There exist 2g minimally linked loops, denoted {Jj}j=12g, such that lk(Ki, Jj)=ij. • Half of them linked with KiI’s, denoted {JjI}j=1g, are handle loops. • Other half linked with KiO’s, denoted {JjO}j=1g, are tunnel loops. • {[JjI]} j=1g form a basis for H1(I) and {[JjO]}j=1g form a basis for H1(O). Hence {[Jj]}j=12g form a basis for H1(M). Department of Computer Science and Engineering 23/52 Topological algorithm • Assume I and O are given. • Step1: Compute {Ki}i=12g using the spanning tree of I and O. • Step2: Compute a system of 2g loops on M, denoted {j}j=12g. (say by Dey-Schipper or Vegter-Yap linear time algorithms) • Step3: Compute lk(Ki, j) for all i and j. Let A be the 2gx2g matrix {lk(Ki, j)}. • A is the transform matrix from a minimally linked basis {[Jj]}j=12g to basis {[j]} j=12g. • A-1 = {aji} exists and has integer entries. • [Jj] = i=12g aji[i]. • Step4: Obtain Jj by concatenating i’s according to the above expression. Department of Computer Science and Engineering 24/52 An implementation to compute system of handle and tunnel loops with small size • Compute I and O • Basic idea: Collapse the inside (outside) Voronoi diagram to obtain I (O). • The curve-skeleton [Dey-Sun06] captures the geometry better. • Each skeleton edge (e) gets associated with an additional value called geodesic size (g(e)) indicating the local size of M. • Establish graph structure on the curve skeleton. • The geodesic size for a graph edge, g(E) = min{g(e): e is a skeleton edge in E}. • Issue: not always work for any graph retractable surface, e.g., a thickening of a house with two room. Department of Computer Science and Engineering 25/52 Core graphs by curve-skeletons[DS05] Department of Computer Science and Engineering 26/52 An implementation to compute system of handle and tunnel loops with small size • Compute core loops {Kj}j=12g • Compute the maximal spanning tree for I and O using geodesic sizes as weight. • Add the remaining edges, Ei’s, to form Ki’s. • Compute minimally linked {Jj}j=12g • Basic idea: compute Ji’s at different location indicated by Ei’s. • Let e be the skeleton edge with the smallest geodesic size in Ei. Let p be one of the vertices of the dual Delaunay triangle of e. • Compute an optimal system of loops by geodesic exploration and apply topological algorithm. • In our experiments, one of the loop in the system of loops itself satisfies the condition to be Ji’s. Department of Computer Science and Engineering 27/52 Results Department of Computer Science and Engineering 28/52 A Persistence based algorithm [Dey-Li-Sun-Cohen-Steiner 08] • A fast algorithm based on persistent homology introduced by Edelsbrunner, Letscher, Zomorodian[2002]. • The algorithm does not require any extra structure such as medial axis, curve skeleton, or Reeb graphs. • It works on a larger class of models. Both surface mesh and isosurface with volume grids can be input. Department of Computer Science and Engineering 29/52 Filtration in persistent homology • Filtration of a simplicial complex K is a nested sequence of complexes: • 0 = K-1 < K0 < K1 < …< Kn = K • Assume Ki - Ki-1 = i • Persistence homology studies how the homology groups change over the filtration. • In our algorithm, we use the pairing concept in the persistent homology. Department of Computer Science and Engineering 30/52 Positive and Negative simplices • Once a p-simplex is added to Ki-1, it is • Positive if it creates a non-boundary p-cycle. • Negative if it kills an existing (p-1)-cycle. • A negative p-simplex is always paired with a unique positive (p-1)-simplex ' where kills a (p-1)-cycle created by '. Department of Computer Science and Engineering 31/52 Positive and negative simplices Department of Computer Science and Engineering 32/52 Pairing algorithm – pair () 1) c=p 2) d is youngest positive (p-1)-simplex in c 3) while (d is paired and c Φ) do • Let c' be the cycle killed by the simplex paired with d • c=c'+c • d is the youngest positive (p-1)-simplex in c end while 4) if c is not empty, then is negative p-simplex and paired with d else is positive p-simplex. Department of Computer Science and Engineering 33/52 Computing handle and tunnel loops • Topological algorithm • Two refinements for improving loop quality. • Assume M, I, and O denote the simplicial complexes of surface, inside, and outside respectively; g is the genus of M. Department of Computer Science and Engineering 35/52 Topological algorithm • 3 Steps: • 1) Simplices of M are added to the filtration, generating 2g unpaired positive edges; • 2) Simplices of I are added, g previously unpaired edges get paired, each pair corresponds to a handle loop; • 3) Simplices of O are added, the rest of g unpaired edges get paired, each pair corresponds to a tunnel loop. Department of Computer Science and Engineering 36/52 Topological algorithm cont. Department of Computer Science and Engineering 37/52 Refinement 1 • For a negative triangle , a series of homologous loops are obtained. • Let L1, L2,…,Lk be those that lie on M, sorted by the time when they are found. Set the handle (or tunnel) loop L() for as L1. Department of Computer Science and Engineering 38/52 Refinement 2 • A handle (or tunnel) loop is found when a cross section of M get filled. • Cross sections of small size will get filled first if we add triangles in I and O to the filtration in increasing order of their geodesic sizes. Department of Computer Science and Engineering 39/52 Computing geodesics – for refinement 2 • Geodesic size of an edge E =(A1, A2): geodesic distance between points A1' and A2' on M, where Ai'=Ai if Ai is on M; otherwise, Ai' is the closet point on M for Ai • Geodesic size size of a triangle t: max {g(e)|e is in t} Department of Computer Science and Engineering 40/52 Entire algorithm • 1) Compute geodesic size for edges and triangles in I and O; • 2) For each simplex on M, pair(); • 3) E = unpaired positive edges on M; • 4) In the order of increasing geodesic size, pair() for each triangle in I; • 5) if ( is negative and its paired positive edge is in E), then output a handle: hi = L(); • 6) Do similarly as 4) and 5) to get tunnels. Department of Computer Science and Engineering 41/52 Entire algorithm Department of Computer Science and Engineering 42/52 Gearbox (genus 78) Department of Computer Science and Engineering 43/52 Isosurface Fuel (genus 8) Department of Computer Science and Engineering 44/52 Isosurface (Engine, genus 20) Department of Computer Science and Engineering 45/52 Aneurysm Department of Computer Science and Engineering 46/52 Knotty cup and noise Department of Computer Science and Engineering 47/52 Experiments • We implemented our algorithm in C++. All experiments were done on a Dell PC with 2.8GHz Intel Pentium D CPU and 1GB RAM. Department of Computer Science and Engineering 48/52 Feature Detection • Compute handle (tunnel) features based on handle (tunnel) loops. • Sweep the handle (tunnel) loop until the length become too long. Department of Computer Science and Engineering 49/52 Topological simplification Department of Computer Science and Engineering 50/52 Simplification (Buddha) Department of Computer Science and Engineering 51/52 Conclusions and future work • A simple, topologically correct algorithm for handles and tunnels. • Mathematical proof of geometric qualities of the loops? • What about surfaces with boundaries? • Handletunnel software available from http://www.cse.ohio-state.edu/~tamaldey/handle.html Department of Computer Science and Engineering 52/52 Pegasus (genus 5) Department of Computer Science and Engineering 53/52 2-torus (genus 2) Department of Computer Science and Engineering 54/52