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  1. Second generation 3D compressionFaster, simpler, more effective Edgebreaker (EB) “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. vertex corner triangle border edge T-mesh primitives • 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

  3. 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. 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. 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. Dual graphs and spanning trees From Bosen • 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 TST VST

  7. 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. Triangle 1 vertex 1 vertex 3 vertex 2 Triangle 2 x y z x y z x y z Triangle 3 x y z x y z x y z x y z x y z x y z Representation as independent triangles • 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) Very verbose! Not good for traversal.

  9. L R R 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

  10. c.v c c.l 2 c.n c.p 3 1 3 2 4 5 0 4 c.o 1 c.t v o Triangle 0 corner 0 1 7 Triangle 0 corner 1 2 8 Triangle 0 corner 235 Triangle 1 corner 3 2 9 Triangle 1 corner 4 1 6 Triangle 1 corner 542 vertex 1 x y z vertex 2 x y z vertex 3 x y z vertex 4 x y z Corner table: data structure for T-meshes “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 c.r

  11. 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. v o a 2 Triangle 1 corner 0 1 a Triangle 1 corner 1 2 b Triangle 1 corner 23 c Triangle 2 corner 3 2 c Triangle 2 corner 4 1 d Triangle 2 corner 54 e 3 1 3 2 4 5 0 4 1 v o a Triangle 1 corner 0 1 a Triangle 1 corner 1 2 b Triangle 1 corner 235 c Triangle 2 corner 3 2 c Triangle 2 corner 4 1 d Triangle 2 corner 542 e 2 3 1 3 2 4 5 0 4 1 Computing adjacency from incidence • 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…

  13. Manifold graph Non-manifold shape 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

  14. 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. Edgebreaker encodes construction steps Area not yet covered Decompress Compress Specification of the next triangle Decompress Compress Binary format Sequence of specification for adding triangles

  16. ? Marked (visited) x Not marked x Last visited ? Next to beencoded ? x To-do stack ? x ? Encode sequence of codes x C: 0, L:110, R: 101, S:100, E:111 ? ? and vertices x as encountered by C operations Edgebreaker is a state machine ? C ? L • 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 ? R ? S Only 2T bits (because |C|=V=T/2) E

  17. Edgebreaker compression ? ? C x C R C ? ? C L R C C x R C C C CCCCRCCRCRC… ? ? R x R R R ? ? S L C L E x S R E R C ? ? E …CRSRLECRRRLE x

  18. EB re-numbering of vertices

  19. Edgebreaker compression algorithm v o T1 c0 1 7 T1 c1 2 8 T1 c23 5 T2 c3 2 9 T2 c4 1 6 T2 c54 2 v1 x y z v2 x y z v3 x y z v4 x y z R R R b L C L E a S R E R C c.v c.r c.l c c.t c.o Source code, examples: http://www.gvu.gatech.edu/~jarek/edgebreaker/eb • recursive procedurecompress (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 • elseif c.r.t.m==1 # test whether right triangle was visited • thenif 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 • elseif 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)

  20. 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. EB decompression: how come it works? Receive the CLERS sequence Decode it Construct the TST polygon Decode&reconstruct vertices R R R L E L C S R E R C …CRSRLECRRRLE 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. C L E R S seed R R R L E C L S R E R C R R R R R R L L C L E E C L S R S R E E R R C 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. CRSRLECRRRLE Linear time complexity. Zip only after L and E.

  23. Wrap&Zip more complex example

  24. C R L S E Spirale Reversi decompression for EB 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. compression clers = …CCRRCCRRRCRRCRCRRCCCRRCRRCRCRRRCRCRCRRSCRRSLERERLCRRRSEE reversi = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC… decompression


  26. Edgebreaker Results • Compression results for connectivity information • Guaranteed2T 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 <http://www.gvu.gatech.edu/~jarek/papers> • 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. 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. 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. Edgebereaker compression contributors King (Atlanta): 1.84Tbits, quads Gumhold (Germany): 1.80T bits Rossignac (Atlanta): Edgebreaker Safonova (CMU): Holes, code Szymczak (Atlanta): regularity, resampling Shikhare (India): translation Attene (Italy): retiling Isenburg (UCS): Reversi Coors (Germany): Prediction Lopes (Brasil): Handles Gotsman (Israel): Polygons

  30. 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. 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. Triangulated quad1.34T 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. ? x Quad meshes (King,Rossignac,Szymczak 99) • “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

  34. C S E L Encoding polygon meshes, 5P bits 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.

  35. 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. S* 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

  37. 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. 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. 2 3 1 2 5 0 4 3 4 1 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

  40. Already traversed covered area Active loop c c c c d g(c) X c.v v c Vr c.p Vl c.n GE c.o Figure 2: Connectivity guessed by parallelogram prediction 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

  41. X g(c) g(c) g(c) X X Guess C Guess L Guess R X g(c) X g(c) Guess S Guess E Figure 3: Guess clers Symbol based on geometry prediction. Delphi correctguesses Depending on the model, between 51% and 97% of guesses are correct. 83% correct guesses: 1.47bpv = 0.74T bits

  42. Guess wrong L Guess wrong R g(c) g(c) g(c) g(c) g(c) g(c) X X X X X X Situation R Situation C Situation S Situation L Situation C Situation S Guess wrong S Guess wrong E X X X X X g(c) g(c) g(c) g(c) g(c) Half of the wrong guesses are Cs mistaken for Rs Situation C Situation R Situation L Situation C Situation S Figure 5 Wrongly guessed non-C triangles. They grey triangle shows the actual situation. The yellow triangle visualizes the parallelogram prediction. Delphi: Wrongnon-C guesses

  43. Guess C in X X g(c) g(c) Situation R Situation L X X g(c) g(c) Situation E Situation S Figure 4: Wrongly guesses C triangles Delphi wrong C-guesses 28% of wrong guesses are Rs mistaken for Cs.

  44. 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. Apollo sequence encoding of Delphi

  45. 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. 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. 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. L L 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

  49. Swing-Wrapper resolution control

  50. SwingWrapper results: 0.4Tb total (0.01%) 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 134,074T WRL=4,100,000B