1. 05 Edgebreaker, 1 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebreaker (EB) Second generation 3D compression Faster, simpler, more effective “Edgebreaker: Connectivity compression for triangle meshes,” J. Rossignac, IEEE Transactions on Visualization and Computer Graphics, vol. 5, no. 1, pp. 47–61, 1999. “Optimal Bit Allocation in Compressed 3D Models”. D. King and J. Rossignac. Computational Geometry, 14:91– 118, 1999. “Wrap&Zip decompression of the connectivity of triangle meshes compressed with Edgebreaker,” J. Rossignac and A. Szymczak. Computational Geometry: Theory and Applications, 14(1-3):119-135, 1999. “Connectivity compression for irregular quadrilateral meshes,” D. King, J. Rossignac, and A. Szymczak, Technical Report TR–99–36, GVU, Georgia Tech, 1999. “An Edgebreaker-based efficient compression scheme for regular meshes,” A. Szymczak, D. King, and J. Rossignac, in Proceedings of 12th Canadian Conference on Computational Geometry, 20(2):257–264, 2000. “3D Compression and progressive transmission,” J. Rossignac. Lecture at the ACM SIGGRAPH conference July 2- 28, 2000. “3D compression made simple: Edgebreaker on a corner-table.” J. Rossignac, A. Safonova, and A. Szymczak. In Proceedings of the Shape Modeling International Conference, 2001. “Edgebreaker on a Corner Table: A simple technique for representing and compressing triangulated surfaces”, J. Rossignac, A. Safonova, A. Szymczak, in Hierarchical and Geometrical Methods in Scientific Visualization, Farin, G., Hagen, H. and Hamann, B., eds. Springer-Verlag, Heidelberg, Germany, 2002. “Guess Connectivity: Delphi Encoding in Edgebreaker”, V. Coors and J. Rossignac, GVU Technical Report. June 2002. “A Simple Compression Algorithm for Surfaces with Handles”, H. Lopes, J. Rossignac, A. Safanova, A. Szymczak and G. Tavares. ACM Symposium on Solid Modeling, Saarbrucken. June 2002.

2. 05 Edgebreaker, 2 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Vertex: –Location of a sample Triangles: –Decompose approximating surface Edge: –Bounds one or more triangles –Joins two vertices Corner: –Abstract association of a triangle with a vertex –May have its own attributes (not shared by corners with same vertex) Used to capture surface discontinuities Border (oriented half-edge, dart): –Association of a triangles with a bounding edge. –Orientation cycle around triangle, inverse of opposite border A triangle has 3 borders and 3 corners T-mesh primitives triangle border vertex corner edge

3. 05 Edgebreaker, 3 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Classes of T-meshes Triangle soup –Any collection of triangles (may intersect each other) 2D simplicial complex –Collection of edges, vertices, faces that join but do not intersect Orientable manifold with boundary –Each edge has 1 or 2 incident triangles. One incident cone per vertex. Boundary of a (regularized) solid –Each edge has 2k incident triangles (non-manifold). Orientable. Handles. Zero-genus boundary of a manifold solid (simple mesh) –Orientable. Manifold. Connected. No holes. No handles.

4. 05 Edgebreaker, 4 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 First, the case of a simple mesh A simple mesh is a deformed triangulated sphere –Orientable –2-manifold –No boundary (no holes) –No handles (no throu-holes) Properties –Each edge has exactly 2 incident triangles –Each vertex has a single cycle of incident triangles –May be drawn as a planar graph

5. 05 Edgebreaker, 5 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Simple meshes and patches A patch is a simple portion of a simple mesh –Simply connected Bounded by a single manifold edge-loop Its boundary is a connected manifold loop –Cycle of border edges It may be obtained from a simple mesh –by removing one or more triangles

6. 05 Edgebreaker, 6 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Dual graph: –Nodes represent triangles –Links represent edges That join adjacent triangles Vertex Spanning Tree (VST) –Edge-set connecting all vertices –No cycles –Cuts mesh into simply connected polygon with no interior vertices Triangle-Spanning Tree (TST) –Graph of remaining edges –No loops –Connects all triangles Dual graphs and spanning trees From Bosen VST TST

7. 05 Edgebreaker, 7 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Euler formula for Simple Meshes Mesh has V vertices, E edges, and T triangles E = (V-1)+(T-1) –VST has V nodes and thus V-1 links –TST has T nodes and thus T-1 links E = 3T/2 –There are 3 borders (edge-uses) per triangle –There are twice more edge-uses then edges Therefore: T = 2V - 4 –Because (V-1)+(T-1) = 3T/2 –we have V-2 = 3T/2-T = T/2 –There are about twice as many triangles as vertices The number C of corners (vertex-uses) is about 6V –C=3T=6V-12 –On average, a vertex is used 6 times

8. 05 Edgebreaker, 8 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 For each triangle: –For each one of its 3 corners, store: Location Attributes (may be the same for neighboring corners) Each vertex location is repeated (6 times on average) –geometry = 36 B/T (float coordinates: 9x4 B/T) –Plus 3 attribute-sets per triangle (6 per vertex) Representation as independent triangles x y z x y z x y z vertex 1 vertex 2 vertex 3 Triangle 2 Triangle 1 Triangle 3 Very verbose! Not good for traversal.

9. 05 Edgebreaker, 9 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Representation as Triangle strips Continue a strip by attaching a new triangle to an edge of the previous one Need only indicate which edge and when to start a new strip –1 Left/Right bit per triangles plus 1 strip-end bit per triangle Send one vertex per triangle –Plus 2 vertices per strip to start it Each vertex is transmitted twice on average L RR

10. 05 Edgebreaker, 10 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 “3D compression made simple: Edgebreaker on a corner-table.” J. Rossignac, A. Safonova, and A. Szymczak. In Proceedings of the Shape Modeling International Conference, 2001. Table of corners, for each corner c store: –c.v : integer reference to vertex table –c.o : integer reference to opposite corner The 3 corners of each triangle are consecutive –List them according to ccw orientation of triangles –Trivial access to triangle ID: c.t = INT(c/3) –c.n = 3c.t + (c+1)MOD 3, c.p = c.n.n, c.l = c.p.o, c.r = c.n.o Corner table: data structure for T-meshes vertex 1 x y z vertex 2 x y z vertex 3 x y z vertex 4 x y z Triangle 0 corner 0 1 7 Triangle 0 corner 1 2 8 Triangle 0 corner 2 3 5 Triangle 1 corner 3 2 9 Triangle 1 corner 4 1 6 Triangle 1 corner 5 4 2 v o 1 2 3 4 0 1 2 3 4 5 c.t c c.v c.o c.n c.p c.l c.r

11. 05 Edgebreaker, 11 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Using adjacency table for T-mesh traversal Visit T-mesh (triangle-spanning tree) –Mark triangles as you visit –Start with any corner c and call Visit(c) –Visit(c) mark c.t; IF NOT marked(c.r.t) THEN visit(c.r); IF NOT marked(c.l.t) THEN visit(c.l); Label vertices –Label vertices with consecutive integers –Label(c.n.v); Label(c.n.n.v); Visit(c); –Visit(c) IF NOT labeled(c.v) THEN Label(c.v); mark c.t; IF NOT marked(c.r.t) THEN visit(c.r); IF NOT marked(c.l.t) THEN visit(c.l);

12. 05 Edgebreaker, 12 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 c.o can be derived from c.v (needs not be transmitted): Build table of triplets { min(c.n.v, c.n.n.v), max(c.n.v, c.n.n.v), c} –230, 131, 122, 143, 244, 125, … Sort (bins, linear cost): –122, 125...131... 143...230...244 … Pair-up consecutive entries 2k and 2k+1 –(122, 125)...131... 143...230...244… Their corners are opposite –(122,125)...131...143...230...244… Computing adjacency from incidence Triangle 1 corner 0 1 a Triangle 1 corner 1 2 b Triangle 1 corner 2 3 c Triangle 2 corner 3 2 c Triangle 2 corner 4 1 d Triangle 2 corner 5 4 e v o a Triangle 1 corner 0 1 a Triangle 1 corner 1 2 b Triangle 1 corner 2 3 5 c Triangle 2 corner 3 2 c Triangle 2 corner 4 1 d Triangle 2 corner 5 4 2 e v o a 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 1 2 3 4

13. 05 Edgebreaker, 13 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Connectivity/geometry discrepancy Connectivity of T-mesh may conflict with actual geometry –Vertices with different names may be coincident –Edges with different names may be coincident –Triangles, edges, and vertices may intersect T-mesh with consistent geometry –Triangles, edges, vertices are pairwise disjoint We consider edges and triangles to be open –I.e., not containing their boundary Manifold graphs may be used with invalid geometry –Coincident edges and vertices: Non-manifold singularities –Self-intersecting surfaces Non-manifold shape Manifold graph

14. 05 Edgebreaker, 14 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebreaker: A simple, fast, and effective second generation 3D compression Jarek Rossignac GVU Center and College of Computing Georgia Tech, Atlanta http://www.gvu.gatech.edu/~jarek

15. 05 Edgebreaker, 15 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebreaker encodes construction steps Binary format Compress Decompress Compress Decompress Area not yet covered Specification of the next triangle Sequence of specification for adding triangles

16. 05 Edgebreaker, 16 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 ? ? x ? x ? x ? x ? x ? ? ? Edgebreaker is a state machine C L R S ? E if tip vertex not marked then C else if left neighbor marked then if right neighbor marked then E else L else if right neighbor marked then R else S if tip vertex not marked then C else if left neighbor marked then if right neighbor marked then E else L else if right neighbor marked then R else S Encode sequence of codes C: 0, L:110, R: 101, S:100, E:111 and vertices as encountered by C operations Marked (visited) Not marked ? Next to be encoded To-do stack x Last visited Only 2T bits (because |C|=V=T/2)

17. 05 Edgebreaker, 17 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebreaker compression C ? ? L ? ? R ? ? ? S ? ? E ? x x x x x C C C C C C C C R R R CCCCRCCRCRC… R LEC R R R L E …CRSRLECRRRLE C R S

18. 05 Edgebreaker, 18 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 EB re-numbering of vertices

19. 05 Edgebreaker, 19 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 recursive procedure compress (c) repeat { c.t.m:=1;# mark the triangle as visited if c.v.m == 0 # test whether tip vertex was visited then {write(vertices, c.v);# append vertex index to “vertices” write(clers, C);# append encoding of C to “clers” c.v.m:= 1;# mark tip vertex as visited c:=c.r }# continue with the right neighbor else if c.r.t.m==1# test whether right triangle was visited then if c.l.t.m== 1 # test whether left triangle was visited then {write(clers, E);# append encoding of E to clers string return }# exit (or return from recursive call) else {write(clers, R);# append encoding of R to clers string c:=c.l }# move to left triangle else if c.l.t.m == 1 # test whether left triangle was visited then {write(clers, L);# append encoding of L to clers string c:=c.r }# move to right triangle else {write(clers, S);# append encoding of S to clers string compress(c.r);# recursive call to visit right branch first c:=c.l } } # move to left triangle vertices=…ab, clers =...CRSRLECRRRLE (2T bit code: C=0, L=110, R=101, S=100, E=111) T1 c0 1 7 T1 c1 2 8 T1 c2 3 5 T2 c3 2 9 T2 c4 1 6 T2 c5 4 2 v o v1 x y z v2 x y z v3 x y z v4 x y z C R SR LEC R R R L E a b c c.t c.v c.l c.r c.o Edgebreaker compression algorithm Source code, examples: http://www.gvu.gatech.edu/~jarek/edgebreaker/eb

20. 05 Edgebreaker, 20 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebreaker decompression How does it work? –No problem, except at S –Can you recover where the tip of each S is from the CLERS string alone? Three solutions: –Count changes of border length in CLERS string (Rossignac) Read CLERS string backwards to compute there the tip of each S is –Wrap&Zip (Rossignac&Szymczak) Build TST polygon and then fold it –Spirale Reversi (Isenberg&Snoeyink) Read CLERS string backward and build mesh in reverse order

21. 05 Edgebreaker, 21 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 EB decompression: how come it works? C R SR L E C R R R L E …CRSRLECRRRLE Receive the CLERS sequence Decode it Construct the TST polygon Decode&reconstruct vertices How to fold the polygon? “Wrap&Zip decompression of the connectivity of triangle meshes compressed with Edgebreaker,” J. Rossignac and A. Szymczak. Computational Geometry: Theory and Applications, 14(1-3):119-135, 1999.

22. 05 Edgebreaker, 22 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 C Wrap&Zip EB decompression (with Szymczak) Orient bounding edges while building triangle tree at decompression. All oriented clockwise (up tree), except for C and the seed triangle: Then ZIP all pairs of adjacent bounding edges when both point away from their common vertex. S C R R L R CLERS seed R R E L CRSRLECRRRLE C R SR LEC R R R L E C R SR L E C R R R L E E Linear time complexity. Zip only after L and E.

23. 05 Edgebreaker, 23 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Wrap&Zip more complex example

24. 05 Edgebreaker, 24 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Spirale Reversi decompression for EB CRLSE clers = …CCRRCCRRRCRRCRCRRCCCRRCRRCRCRRRCRCRCRRSCRRSLERERLCRRRSEE reversi = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC… compression decompression M. Isenburg and J. Snoeyink. Spirale reversi: Reverse decoding of the Edgebreaker encoding. Technical Report TR-99-08, Department of Computer Science, University of British Columbia, October 4 1999.

25. 05 Edgebreaker, 25 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Reversi details O = EESRRRCLRERELSRRCS O = EESRRRCLREREL O = EESRRRCLRERELSRRC O = EES O = E 1 O = EE 1 O = EESRRRCLR 1 O = EESRRRCLRER 1 2 2 1 1 O = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC…

26. 05 Edgebreaker, 26 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebreaker Results Compression results for connectivity information –Guaranteed 2T bits for any simple mesh (improved later to 1.80T bits) –Entropy down to 0.9T bits for non-trivial large models Frequency: C=50%, R about 35%, S and E = 1-to-5% –Source code available: 3 page detailed pseudo-code, arrays of integers, fast http://www.gvu.gatech.edu/~jarek/edgebreaker/eb Publications –Rossignac, Edgebreaker Compression, IEEE TVCG’99 Sigma Xi Best Paper Award –Rossignac&Szymczak, Wrap&zip, CGTA’99 –King&Rossignac: Guaranteed 3.67V bit encoding..., CCCG’99 –Szymczak&King&Rossignac: Mostly regular meshes, CCCG’00 –….

27. 05 Edgebreaker, 27 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Spiraling solutions Several approaches visit the same spiraling TST a Itai,Rodeh: Representation of graphs, Acta Informatica, 82 Keeler,Westbrook: Short encoding of planar graphs and maps, Discrete Applied Math, 93 Gumbold,Straßer: Realtime Compression of Triangle Mesh Connectivity, Siggraph, 98 Rossignac: Edgebreaker: Compressing the incidence graph of triangle meshes, TVCG, 99 Touma,Gotsman: Triangle Mesh Compression, GI, 98 Taubin,Rossignac: Geometric compression through topological surgery, ACM ToG, 98 They encode how each new triangle is attached to previously restored ones

28. 05 Edgebreaker, 28 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebreaker extensions and improvements Better connectivity compression –Tighter guaranteed upper bound (King&Rossignac, Gumhold): 1.80T bits –Sufficiently regular meshes (with Szymczak and King): 0.81T bits guaranteed –Delphi Connectivity predictors (with Coors): between 0.2T and 1.5T bits Topological extensions –Quadrilateral meshes (with Szymczak and King): 1.34T bits –Handles/holes (with Safonova, Szymczak, Lopes, and Tavares) –Non manifold solids (with Cardoze) Implementation (with Safonova, Coors, Szymczak, Shikhare, Lopes) Retiling and loss optimization –Optimal quantization (with King and Szymczak): best B and T –Piecewise regular resampling (with Szymczak and King) 1T bits total –Uniform C-triangles (with Attene, Falcidieno, Spagnuolo): 0.4T bits total Higher dimension –Tetrahedra for FEM (with Szymczak): 7T bits (prior to entropy) –Pentatopes for 4D simulations (with Szymczak, and with Snoeyink)

29. 05 Edgebreaker, 29 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Edgebereaker compression contributors Rossignac (Atlanta): Edgebreaker Szymczak (Atlanta): regularity, resampling King (Atlanta): 1.84Tbits, quads Safonova (CMU): Holes, code Coors (Germany): Prediction Lopes (Brasil): Handles Gotsman (Israel): Polygons Isenburg (UCS): Reversi Gumhold (Germany): 1.80T bits Shikhare (India): translation Attene (Italy): retiling

30. 05 Edgebreaker, 30 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Guaranteed 1.84T bit (King&Rossignac 99) “Guaranteed 3.67v bits encoding of planar triangle graphs” –Proc. 11th Canadian Conference on Computational Geometry, August 1999 Encoding of symbols that follow a C –C is 0, S is 10, R is 11 3 possible encoding systems for symbols that do not follow a C –Code I: C is 0, S is 100, R is 101, L is 110, E is 111 –Code II: C is 00, S is 111, R is 10, L is 110, E is 01 –Code III: C is 00, S is 010, R is 011, L is 10, E is 11 One of these 3 codes takes less than (2-1/6)T bits –Use a 2-bit switch to identify which code is used for each model

31. 05 Edgebreaker, 31 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Guaranteed 1.80T bit(Gumhold 00) “New bounds on the encoding of planar triangulations”, S. Gumhold, Siggraph course notes on “3D Geometry Compression” 1.8T bits guaranteed for encoding CLERS string Exploits the length of the outer boundary of T-patch (>2) –Not convenient for treating non-manifolds (See later) CE is impossible –Was at least 3, C increased it to at least 4, can’t have an E CCRE is impossible –Was at least 3, CC increased it to at least 5, R reduced it by 1, can’t have an E These constraints impact the probability of the next symbol and improve coding

32. 05 Edgebreaker, 32 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Triangulated quad 1.34T bits guaranteed "Connectivity Compression for Irregular Quadrilateral Meshes" D. King, J. Rossignac, A Szymczak. Triangulate quads as you reach them Always \, never / Consecutive in CLERS sequence Guaranteed 2.67 bits/quad –1.34T bits –Cheaper to encode that triangulation –Less than Tutte’s lowest bound Fewer Q-meshes than T-meshes –With same vertex count –Theoretical proof Extended to polygons –Fan boundaries FaceFixer, Isenburg&Snoeyink

33. 05 Edgebreaker, 33 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 “Connectivity Compression of Irregular Quad Meshes” Surfaces often approximated by irregular quad meshes Instead of triangulating, we encode quads directly –Measured 0.24V to 1.14V bits, guaranteed 2.67V bits (vs 3.67) Equivalent to a smart triangulation + Edgebreaker –Only \-splits (no /-split), as seen from the previous quad –Guarantees the triangle-pair is consecutive in triangle tree –First triangle of each quad cannot be R or E: 13 symbol pairs possible Quad meshes (King,Rossignac,Szymczak 99) x ?

34. 05 Edgebreaker, 34 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 D. King, J. Rossignac, and A. Szymczak, “Connectivity compression for irregular quadrilateral meshes,” Technical Report TR–99– 36, GVU, Georgia Tech, 1999. Triangulate each polygon as a fan and encode as CLERS –Record which edges are added (1 bit per triangle) –Guaranteed cost: min(5V, 5P) bits using primal or dual –Guaranteed cost: 2.5 bits per edge –Exploit planarity for geometry prediction M. Isenburg and J. Snoeylink, “Face fixer: Compressing polygon meshes with properties,” in Siggraph 2000, Computer Graphics Proceedings, 2000, pp. 263–270. B. Kronrod and C. Gotsman,“Efficient Coding of Non-Triangular Meshes”, Technical Report, Computer Science Department, Technion-Israel Institute of Technology, 1999. Encoding polygon meshes, 5P bits L CS E

35. 05 Edgebreaker, 35 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Manifold meshes may have handles Number of handles H –Is half the smallest number of closed curves cuts necessary to make the surface homeomorphic to a disk T=2V+4(H-S) –T triangles, E edges, V vertices, H handles, S shells –Euler: T-E+V=2S -2H –2 borders per edge and 3 borders per triangle: 2E=3T H=S-(T-E+V)/2 –Shared edges: E=3T/2 –3 borders per triangle, 2 borders per edge disk

36. 05 Edgebreaker, 36 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Simple encoding of handles in Edgebreaker “A Simple Compression Algorithm for Surfaces with Handles”, H. Lopes, J. Rossignac, A. Safanova, A. Szymczak and G. Tavares. ACM Symposium on Solid Modeling, Saarbrucken. June 2002. VST and TST miss 2 edges per handle Encode their adjacency explicitly –As corner pairs of “glue” edges –Additional connectivity cost 2Hlog(3T) Need to restart zipping –From each glue edge S*

37. 05 Edgebreaker, 37 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Example: EB compression of torus Each handle creates two S that will not be able to go left Encode the pair of opposite corner IDs

38. 05 Edgebreaker, 38 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Plug holes with dummy triangle fans C. Touma and C. Gotsman, “Triangle mesh compression,” in Graphics Interface, 1998. Encoder –Create a dummy vertex –Triangulate the hole as a star –Encode mesh with the holes filled –Encode the IDs of dummy vertices –Skip tip ID of biggest hole –RLE number of initial Cs Decoder –Receives filled mesh and IDs of dummy vertices –Reconstructs complete mesh –Removes star if dummy vertices What is a hole? –With Safonova, Szymczak

39. 05 Edgebreaker, 39 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Non-Manifolds Solid models have non-manifold edges and vertices Compression exploits manifold data structures Matchmaker: Manifold BReps for non-manifold r-sets –Rossignac&Cardoze, ACM Symposium on Solid Modeling, 1999. –Match pairs of incident faces for each NME –Respects surface orientation & minimizes number of NMVs 1 2 3 4 0 1 2 3 4 5

40. 05 Edgebreaker, 40 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Delphi: Guessed Connectivity = 0.74T bits “Guess Connectivity: Delphi Encoding in Edgebreaker”, V. Coors and J. Rossignac, GVU Technical Report. June 2002. Predict Edgebreaker code from decoded mesh Already traversed covered area Active loop c X d g(c) VrVr VlVl v GEGE c.v c c.n c c c c.p c.o Figure 2: Connectivity guessed by parallelogram prediction

41. 05 Edgebreaker, 41 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Delphi correct guesses Guess CGuess L X g(c) Guess R X g(c) Guess SGuess E X g(c) X X Figure 3 : Guess clers Symbol based on geometry prediction. Depending on the model, between 51% and 97% of guesses are correct. 83% correct guesses: 1.47bpv = 0.74T bits

42. 05 Edgebreaker, 42 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Delphi: Wrong non-C guesses Guess wrong R Situation L X g(c) Situation C X g(c) Situation S X g(c) Guess wrong L Situation R X g(c) Situation C X g(c) Situation S X g(c) Guess wrong S X g(c) Situation R X g(c) Situation L X g(c) Situation C X g(c) Situation S Guess wrong E Situation C g(c) X Figure 5 Wrongly guessed non-C triangles. They grey triangle shows the actual situation. The yellow triangle visualizes the parallelogram prediction. Half of the wrong guesses are Cs mistaken for Rs

43. 05 Edgebreaker, 43 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Delphi wrong C-guesses Guess C in X g(c) Situation S Situation R X g(c) Situation L X g(c) X Situation E Figure 4: Wrongly guesses C triangles 28% of wrong guesses are Rs mistaken for Cs.

44. 05 Edgebreaker, 44 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Apollo sequence encoding of Delphi Figure 6: Example Apollo encoding: Let us assume that we guessed the first triangle of the example correctly as type C. We than predict the tip of the right triangle at g(c) using the parallelogram rule. SinceBecause the distance of g(c) and the active border is too large, we guess again a type C triangle. Unfortunately, that guess was wrong. In fact, the right triangle, shown in gray color in the first picture, is of type R. In the Apollo sequence we encode this situation as (f,R) and continue the traversal with the left triangle of R. The prediction scheme is performed for all triangle in Edgebreaker sequence and leads to the following Apollo sequence: ((t), (f, R), (t), (t), (t), (t), (t), (t), (t), (f,R), (t), (t), (t)). With a trivial encoding scheme we can compress this sequence with 16 bits instead of 32 bits for the corresponding CLERS sequence.

45. 05 Edgebreaker, 45 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Remeshing techniques What if you do not need to preserve the exact model Allow discrepancy between original and received models –Imprecise vertex locations –Different connectivity –New selection of vertices on or near the surface –Simpler topology Now we can use other representations –Subdivision surface –Semi analytic (CSG) –Implicit (radial basis function interpolant) Or develop new ones designed for better compression –One parameter per sample (normal displacement, not tangential) Want most vertices to be regular elevation over 2D grid (PRM) Want mostly triangles to be isosceles (SwingWrapper)

46. 05 Edgebreaker, 46 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Piecewise Regular Meshes (PRM) “Piecewise Regular Meshes: Construction and Compression”. A. Szymczak, J. Rossignac, and D. King. To appear in Graphics Models, Special Issue on Processing of Large Polygonal Meshes, 2002. Split surface into terrain-like reliefs Resample each relief on a regular grid Merge reliefs and fill topological cracks Encode irregular part with Edgebreaker Compress with range coder (2 char context) Parallelogram prediction (x,y) & z

47. 05 Edgebreaker, 47 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 PRM results: 1T bits total, with 0.02% error Resampling chosen to limit surface error to less than 0.02% –Using 12-bit quantization on vertex location –Measured using Metro Decreases Entropy by 40% –80% storage savings when compared to Touma&Gotsman 0.6T - 1.8T bits total (geometry and connectivity) –89% Geometry – 8% Connectivity of the regular part of reliefs – 3% Irregular triangles Simple implementation –Re-sampling: 5 mns (not optimized) –Compression: 4 seconds –Simpler than MAPS (Lee, SIG98)

48. 05 Edgebreaker, 48 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 SwingWrapper: semi-regular retiling “SwingWrapper: Retiling Triangle Meshes for Better Compression”, M. Attene, B. Falcidieno, M. Spagnuolo and J. Rossignac, Technical Report. March 2002 Resample mesh to improve compression Try to form regular triangles –All C triangles are Isosceles –with both new edges of length L Fill cracks with irregular triangles Encode connectivity with Edgebreaker Encode one hinge angle per vertex L L

49. 05 Edgebreaker, 49 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Swing-Wrapper resolution control

50. 05 Edgebreaker, 50 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 SwingWrapper results: 0.4Tb total (0.01%) 134,074T WRL=4,100,000B 13,642T L2 error 0.007% 3.5Tb total 0.36Tb wrt original T 678-to-1 compression 1505T L2 error 0.15% 5.2Tb total 0.06Tb wrt original T 4000-to-1 compression

51. 05 Edgebreaker, 51 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Triangles surrounded by valence 6 vertices

52. 05 Edgebreaker, 52 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Summary Topological Surgery (MPEG-4): RLE of TST and VST Edgebreaker connectivity (CLERS): –Efficient Wrap&Zip or Reversi decompression –Guarantee 1.80Tb for simple meshes and 0.81T for mostly regular meshes –Simple extensions to handles, holes, and non-manifold boundaries –Delphi connectivity predictors: between 0.2Tb and 1.5Tb –Smart triangulation of quad-meshes: 1.34T bits –Encode vertex location using reordering and parallelogram prediction –Publicly available 2 page source code and examples Resampling and simplification –Simplification (vertex clustering and edge-collapse) –Optimal compromise between quantization and simplification (E=K/V) –Piecewise Regular Meshes (reliefs): 1Tb total geometry+connectivity (0.02% error) –Swing&Wrapper: Isosceles Cs, 0.36Tb total ( 0.007% error), 0.06Tb (0.15% error)