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Advances and challenges in the design of compact representations of meshes and complexes

Jarek Rossignac School of Interactive Computing Georgia Tech. Advances and challenges in the design of compact representations of meshes and complexes. We want Constant Amortized Time ( CAT ) cost operators for Random Access and Traversal ( RAT ) of the mesh

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Advances and challenges in the design of compact representations of meshes and complexes

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  1. Jarek Rossignac School of Interactive Computing Georgia Tech Advances and challenges in the design ofcompact representations of meshes and complexes

  2. We want Constant Amortized Time (CAT) cost operators for Random Access and Traversal (RAT) of the mesh Access vertices and faces using (nearly) consecutiveIDs For (int v = 0; v<V; v++) processVertex(v); For (int f = 0; f<F; f++) processFace(f); Given vertex, v, obtain an incident face, f(v) Given face, f, obtain a bounding vertex, v(f) Given tuple (f,v) of a face f and a bounding vertex v, obtain: Next tuple (“swing”clockwise) around v Next tuple (“walk”cw) around f Want the time cost of each operator to be constant (on average) and small CATRAT v(f,v) v f(f,v) f

  3. For each face f, store a reference to one of its bounding edges For each vertex v, store a reference to one incident edge For each edge e, store references to Its 2 bounding vertices Its 2 incident faces Its 2 following edges along each face Cost = F + V + 6E references (32-bit integers) In a planar triangulation: F = 2V – 4 E = 3V Cost = 2V + V + 18 V = 21V = 672 bpv (bits per vertex) Zipper reduces this cost to 12 bpv Example of an edge-based scheme

  4. Theoretical limit [Tutte’62]. Achievable in theory: Succinct Reps [Aleardi, Devillers, Schaefffer’06]: 3.25 bpv Prior art: Mesh connectivity storage cost N-tuples [Brisson’89]: 63V Stable catalogs [Castelli, Devillers’06]: 7.6V RAT Star vertices [Kallmann, Thalmann’01]: 15V [Blandford, Blelloch, Cardoze, Kadow,’03]: 2.5V RAT Half-edge [Mantyle’88, Kettner’99] : 33V Directed-edge [Campagna, Kobbelt, Seidel’98]: 13V Tripod [Snoeyink, Speckmann’99]: 7V Winged-edge [Baumgart’72]: 27V r V = r × 32 bits per vertex Surface-Mesh [Sieger, Botsch‘11] : 21V ECT [Rossignac, Safonova, Szymczak‘01]: 13V SOT [Gurung, Rossignac’09 SPM]: 6V SQuad [Gurung, Laney, Lindstrom, Rossignac’11 Eurographics]: 4V LR / BELR [Gurung, Luffel, Lindstrom, Rossignac’11 SIGGRAPH]: 2V / 55 bpv Zipper [Gurung, Luffel, Lindstrom, Rossignac’12 SPM] : 12 bpv Locally decompressible RAT [Yoon, Lindstrom’07 TVCG]: 8 bpv T T ESQ (editable SQuad) [Castelli, Devillers, Rossignac’12 SIBGRAPI ]: 4V b bpv = b bits per vertex

  5. Access: v(f): an arbitrary vertex bounding given face f (f = 0,1,2…F-1) f(v): an arbitrary face incident upon given vertex (v = 0,1,2…V-1) Traversal: n(f,v): next vertex after v (“walking” cw) around face f s(f,v): next face afterf(“swinging” cw) around vertex v SIMPLIFIED NOTATION: Cornerc=(f,v) as place marker for RAT n(f,v) is now written c.n s(f,v) is now written c.s Bijections between corners and half-edges, edge-uses, oriented-edges, darts… Corner-based operators v c.n c c.s f

  6. Assign consecutive integer IDs to the 3 corners of each triangle Core operators c.v : the vertex of corner c c.t : the triangle of corner c v.c: an arbitrary corner of vertex v t.c: an arbitrary corner of triangle t c.n: nextcorner around c.t c.o : opposite (in adjacent triangle) Derived operators c.s = c.n.o.nswingaround c.v c.p= c.n.n (previous in c.t) c.l = c.n.o (left) c.r = c.p.o (right) c.u = c.r.p (unswing around c.v) Corners operatorsfor triangle meshes

  7. Based on previously proposed Corner Table (CT): “Edgebreaker on a Corner Table: A simple technique for representing and compressing triangulated surfaces”, J. Rossignac, A. Safonova, A. Szymczak. Book chapter in Hierarchical and Geometrical Methods in Scientific Visualization, Farin, G., Hagen, H. and Hamann, B., eds. Springer-Verlag, Heidelberg, Germany. Pages 41-50, January 2003. ECT stores back pointer C[v] from vertex to one incident corner Extended Corner Table (ECT): 13V

  8. Geometry: short int X[V], Y[V], Z[V] 16-bit coordinates relative to AABB Yields 1/32 mm accuracy over a 2 m aircraft engine Connectivity: int C[V], V[3T], O[3T] v.c = C[v], c.v = V[c], c.o = O[c] stored in look-up tables c.t = c/3, t.c = 3t Since corners of t have consecutive IDs c.n = c.t.c + (c+1)%3 Since corners of t are sorted (orientation) Storage cost (14.5V = 464 bpv) Geometry: 3*0.5 = 1.5V Connectivity ECT= 6T+V = 13.0V (90% of total storage) For simplicity, assume low-genus, manifold mesh: T ≈ 2V Extended Corner Table (ECT)

  9. “SCT: Simplicial Corner Table”, J. Rossignac & X. Zhu, in preparation. Same data structure as ECT Extends topology to support Non-manifold edges Non-manifold vertices Wires (dangling edges) 6 corners per triangle 3 on each side Side-preserving crawl (spider) c.s and c.o do not cross surface New operator To cross surface SCT: Corner Table for Simplicial Complexes

  10. “TetStreamer: Compressed Back-to-Front Transmission of Delaunay Tetrahedra Meshes”, U. Bischoff and J. Rossignac, Proceedings of the Data Compression Conference, IEEE Computer Society, 2005. . Same data structure as CT V[] and O[] 4 corners per tet Use wedge as place marker for RAT Wedge operators (mimic corner ones) Swing around edge Next wedge in tet from vertex Cross wedge Use test to compute twist Between face-adjacent tets CT extended to tetrahedra

  11. “SOT: Compact Representation for Triangle and Tetrahedral Meshes”, T. Gurung and J. Rossignac, Georgia Institute of Technology, SIC Technical Report GT-IC-10-01, 2010 Start with ECT: C, V, O tables Match each v with unique t Sort triangles so that matched t has same ID as its matching vertex Discard C & V tables (no longer needed)! To compute c.v, swing (c=c.s) until c < 3V SOT (Sorted O Table) for triangle meshes: 6V

  12. Start with V & O tables Traversal matches each vertex with unique triangle Linear cost construction of SOT

  13. “SOT: Compact representation for Tetrahedral Meshes”, T. Gurung and J. Rossignac. ACM Symposium on Solid and Physical Modeling (SPM), 79-88. 2009. 4 references per tet Trivial combination of the extension of CT to tet meshes and of the SOT idea Extending SOT to tetrahedron meshes: 4T

  14. Start with V & O tables (for tet meshes) Depth-first traversal to match each vertex with unique tet Renumber tets so that matched tet has same ID as its vertex Discard the V table To compute the ID of vertex v from a corner c, we may need to visit all tets incident on v Same as visiting all triangles on a sphere (boundary of star(v)) SOT Construction

  15. “SQuad: Compact Representation for Triangle Meshes”, T. Gurung, D. Laney (LLNL), P. Lindstrom (LLNL), and J. Rossignac. Eurographics 2011. Published as a journal paper in the Computer Graphics Forum Journal (CGF) 30(2): 355-364. Match most v with two adjacent t’s (quad), sort quads, store only 4 (external) c.s swings per quad (the internal ones are implicit) SQuad (sorted quads): 4V

  16. Starts with depth-first traversal (as for SOT) But try to pair each matched triangle with an unmatched adjacent neighbor that is also incident on the matching vertex (only 2 candidates, try right first) Store only 4 outer swings per quad Single triangle groups stored using same structure Unmatched triangles stored at the end of the table Do not store V[]: instead swing as in SOT SQuad construction and representation 97% triangles are paired

  17. Squad preserves vertex order So it mostly preserves memory coherence Except for unmatched triangles (which are stored at the end) SQuad preserves memory coherence

  18. “ESQ: Editable SQuad representation for triangle meshes”, L. Castelli Aleardi, O. Devillers, J. Rossignac. IEEE sponsored Conference on Graphics, Patterns and Images (SIBGRAPI), August 22, 2012, Brazil. Extend the different types of groups: V, VT, VTT, T, VVT…? Use a separate array for each type (fixe size) Support constant time local edits: Valence-3 Vertex deletion Edge flip Triangle split Reform groups locally Many configurations Update arrays (no holes) ESQ: Editable SQuad

  19. “LR: Compact connectivity representation for triangle meshes”, Topraj Gurung, Peter Lindstrom, Mark Luffel, Jarek Rossignac. ACMSIGGRAPH2011. Published as a journal paper in the ACM Transactions on Graphics (TOG), 30(4): 67, August 2011. Arrange quad diagonals along quasi-Hamiltonian cycle (ring), store v.L / v.R, + exceptions LR : 2V

  20. Ring Expander (LR ring construction) 0.005% of vertices are not in the ring Compute a nearly-Hamiltonian cycle in a mesh

  21. Subscripts: number of ring edges T0 have no ring edges (splits) T1 have one ring edge (most common) T2 have two ring edge (dead-ends) Superscripts: warts & exceptions Wart: adjacent (T0w,T2w) pair Non-wart T0’s and their T1ineighbors are exceptions Triangle types (relative to ring) Triangles in warts: 2.35% Non-wart T0: 1.05%

  22. Vertices: numbered along ring v.P previous ring vertex (implicit) v.N next ring vertex (implicit) v.L left neighbor (stored as L[v] in L array) v.R right neighbor (stored in R array) Triangles: Match v with 2 triangles sharing edge (v , v.N) Implicitly numbered 2v and 2v+1 Corners: Numbered around quad: v.i = 8v+i Skip .3 to avoid cost of division by 3 Numbering elements and storing data in LR 2v+1 2v

  23. Computing c.o A few cases. Here are two examples: Case 2: test: v.L.P.L == v.N Case 1: test: v.L == v.N.L

  24. Each T2w is associated with two ring edges (has 2 slots) (v.P,v) and (v,v.N) LR stores T0win the first slot u = tip vertex of matching T0w L[v.P]=uand L[v]=v.P Saves 15 refs per wart 20% storage cost reduction, resulting in cost =1.08T Wart “skipping”(LR)

  25. Relative indexing: Stores delta (v.L – v)%V instead of v.L Can often be represented using 16 bits Uses a ring-expander that makes shorter corridors Compromise between corridor length and (T0 count) BELR: 52 bpv Breadth first Depth first Optimized

  26. “Zipper: A compact connectivity data structure for triangle meshes”, T. Gurung, M. Luffel, P. Lindstrom, and J. Rossignac. ACM Symposium on Solid and Physical Modeling, October 2012. Journal of Computer-Aided Design. 45(2): 262-269 (2013). Store most v.L and v.R as 2-bit deltas (gaps) Use only 6 refs per T0 exception An improved ring construction leading to fewer exceptions Zipper: 12 bpv • Stores connectivity using only 6 bits per triangle • 57× better than Surface-Mesh [Sieger’11] • 35× better than Directed-Edge [Campagna’98] • 4.4× better than BELR [GLLR SIGGRAPH’11] • Linear time cost construction • Local connectivity changes: non constant cost • Constant time access and traversal operators • 2×-3× faster than BELR, 3× slower than LR • 15% slower in app than Surface-Mesh [Sieger‘11]

  27. Zipper stores a 3-bit code per ring triangle 2-bit delta: v.ΔL= v.P.L–v.L(when itis 0, 1, 2, or 3) All deltas are ≥ 0, except at T0’s If v has valence ≤ 6, then 0 ≤ ΔL ≤ 3 Stats: v.ΔL = 0 (29%), 1 (44%), 2 (19%), 3 (4%) 1-bit wart marker Code ‘111’ identifies an exception: v.L stored explicitly (The rare warts with delta=3 are treated as exceptions.) 2-bit deltas

  28. To compute v.L without summing all preceding deltas: Force an exception every 32-vertex block Distribute 3-bit (delta+wart) codes into three 32-bit words (hi,mid,lo) Use bitwise AND to compute mask identifying exceptions (111) Use a population count (POPCNT ) instruction to count set bits (number of exceptions) and identify index of preceding exception Compute sum of deltas from it to v using POPCNT Using POPCNT instead of loop improves speed 10x Decoding v.L and v.R quickly (Zipper)

  29. From T1 to T1 (trivial when delta is 0 or 1) From T0 to a T1 or T2 With each T0 , westore 3 vertex refs (c.v, u, v) We use v.L/v.R & v.P to infer c.o in constant time Computing opposites in T1 and T2 (Zipper)

  30. We store T0’s in a 4-ary cuckoo hash [Fotakis&’03]) With each T0 we store only 6 references: 3 IDs of opposite corner (hash values) Can discard hash if c.o not required by application 3 vertex refs (c.v, u, v) To get from a corner c in a T1 to c.o in a T0 Use c as hash key, which returns 4 candidates for c.o For the correct c.o, c.o.t is incident upon c.p.v and c.n.v Computing opposites in T0 (Zipper)

  31. Reduce the number of T0 triangles by 2.3x Every T0 is connected to a T2 by a corridor of T1’s Rearrange ring to remove T1’s in T0:T1…T1:T2 corridors to make (T0,T2 ) adjacent (wart) Increases warts from 2.35% to 3.2% Reduces number of exceptions Saves about 1 bpt Ring Bender (Zipper)

  32. Storage: median 5.98 bpt Worst case models: Buddha (13.37 bpt) and Thai (7.97 bpt) Average (excluding Buddha & Thai): 5.89 bpt Fixed storage cost: 5 bpt (5 words for 32 triangles block) 3 words for codes, a reference into an exception table, and a reference for the forced exception. Variable cost: increase exception count by 3% adds 1 bpt Performance of c.v and c.o when mesh fits in memory: Zipper ops are 2-3x faster than BELR, Zipper ops are 1.8-3.6x slower than LR In applications (graph distance): 15% slower than Surface Mesh [Sieger’11] Storage and speed results (Zipper)

  33. Summary for Zipper Stores connectivity using only 6T bits 35x better than Directed-Edge [Campagna’98] 4.4x better than BELR [GLLR SIGGRAPH’11] Linear time cost construction No constant cost connectivity changes Constant time access and traversal operators 2x-3x faster than BELR, 3x slower than LR, In app (distance): 15% slower than Surface Mesh [Sieger’11] Interesting (or impossible?) challenges Zipper-64: 5 bpt? Application to general polygon mesh connectivity: 5 bpv? Support non-manifold complexes? Support streaming? Support connectivity changes? Apply to tetrahedron meshes? Summary & future challenges

  34. General position polygon mesh Assume manifold No 4 planes intersect Each vertex has valence 3 Use Zipper to represent its dual graph 12 bits per primal face Primal mesh has twice as many vertices as faces For more general polygon meshes Must split vertices Dual of triangulating polygonal faces Zipper extended to polygon Y-meshes: 6 bpv

  35. Extend ECT to support simplicial complexes SOT: 6V Match (VT groups), renumber matched triangles, store O Extend SOT to tetrahedron meshes SQuad: 4V Match (VTT groups) renumber matched triangles, store 2 swings per tri ESQ: Modify SQuad to support local edit at constant cost LR: 2V (BELR 1.6V) Build quasi-Hamiltonian ring, store v.L , v.R, exceptions (T0 and neighbors) BELR: store deltas (v.L - v / v.R – v), optimize ring: deltas fit in 16 bits Zipper: 0.19 V Store gaps (v.L – v.P.L / v.R – v.P.R), encode most of them with 2 bits, improve ring and reduce number and cost of exceptions Apply Zipper to Y-meshes (valence-3 polygon meshes) Recap

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