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Second generation geometry compression: Edgebreaker is fast, simple, and effective

Second generation geometry compression: Edgebreaker is fast, simple, and effective. Rossignac, Szymczak, King, Safonova Gumhold Isenberg&Snoeyink Coors Lopes. Edgebreaker (Rossignac98). Area not yet covered. Decompress. Compress. Specification of the next triangle. Decompress.

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Second generation geometry compression: Edgebreaker is fast, simple, and effective

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  1. Second generation geometry compression: Edgebreaker isfast, simple, and effective Rossignac, Szymczak, King, Safonova Gumhold Isenberg&Snoeyink Coors Lopes

  2. Edgebreaker (Rossignac98) Area not yet covered Decompress Compress Specification of the next triangle Decompress Compress Binary format Sequence of specification for adding triangles

  3. Same spiraling traversal as TS-MPEG4

  4. ? 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’s 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

  5. Examples of 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

  6. c.v c 2 c.n 3 1 3 2 4 5 0 4 c.o 1 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 • Table of corners, for each corner c store: • c.v : integer reference to vertex table • c.o : integer reference to opposite corner • c.o may be derived from c.v • Make the 3 corners of each triangle 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.l c.r c.p c.t

  7. 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 • recursive procedurecompress (c) • repeat{ • c.t.m:=visited; # mark the triangle as visited • if c.v.m==not visited # 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:= visited; # mark tip vertex as visited • c:=c.r} # continue with the right neighbor • elseif c.r.t.m==painted # test whether right triangle was visited • thenif c.l.t.m== visited # 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 == visited # 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)

  8. Decompression Receive the CLERS sequence Decode it Construct the triangle tree Decode&reconstruct vertices R R R L E L C …CRSRLECRRRLE S R E R C How to zip up the cracks?

  9. 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 decompression (Rossignac&Szymczak99) 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.

  10. R R R L E C L S R E R C Wrap&Zip properties • Only L and E require checking whether zipping is possible • No new zipping is made possible by C, R, S • L creates at most one zip (one pair of edges, no iteration) • E operation creates at least 2 zips • # of zipping operations = # of edges in vertex tree • An edge can only be tried and rejected for zipping only once • Wrap&Zip has linear time complexity (very fast in practice)

  11. Spirale Reversi (Isenburg&Snoeyink 99) • “Spirale Reversi: Reverse decoding of the Edgebreaker encoding” • Decompression for Edgebreaker similar to Wrap&Zip • Reads CLERS string backward • Reconstructs mesh backwards • Assigns vertices before the C operation defines them

  12. C R L S E Reversi Decompression (Isenburg&Snoeyink 99) compression clers = …CCRRCCRRRCRRCRCRRCCCRRCRRCRCRRRCRCRCRRSCRRSLERERLCRRRSEE reversi = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC… decompression

  13. 1 1 O = E O = EES O = EE 1 1 2 O = EESRRRCLR O = EESRRRCLRER 1 2 1 Reversi details O = EESRRRCLRERELSRRC O = EESRRRCLREREL O = EESRRRCLRERELSRRCS O = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC…

  14. Expected/measured results of Edgebreaker • Expected 1.7T bit and guaranteed 2.0T bit code • CL and CE combinations are impossible • Use a shorter code for S (11) and R (10) when they follow a C • On average 36% of symbols are R. Half of them follow a C • Expected 1.3T to 1.6T bits, guaranteed 2.0T bits • Encode CC, CS, CR pairs as single symbols • After even # of Cs: CR (01), CC (00), CS (1101), R (10), S(1111) • L (1110), E (1100) • Stanford Bunny: 1.3T bits, Delaunay triangulation: 1.6T bits • Measured entropy codes 0.91 to 1.26T bits • Huffman code: all words start with a consecutive sequence of C • Bunny: 173 words, 0.85T bits for Huffman codes, 0.91T bits total • Large model: 1,400 word dictionary (32Kb table, large part is fixed) • Gzip on the result improves them by 2%

  15. Edgebreaker + Wrap&Zip Results • Compress: Visit&mark triangles, encode CLERS ops, encode vertices • Decode: Build triangle tree, orient free edges, zip, label & decode vertices • Compression results for connectivity information • Guaranteed2T bitsfor all meshes homeomorphic to a sphere • C: 0, L:110, R: 101, S:100, E:111 (|C|=V=T/2) • Measured entropy code: about 1T bits (dictionary not included) • Publications (http://www.gvu.gatech.edu/~jarek/papers) • Rossignac, Edgebreaker Compression, IEEE TVCG’99 • Sigma Xi Award for Best Paper by Georgia Tech faculty published in 1999 • Rossignac&Szymczak, Wrap&zip, CGTA’99 • King&Rossignac:Guaranteed 3.67V bit encoding...,CCCG’99 • Szymczak&King&Rossignac:Compression of regular meshes, CCCG’00

  16. Edgebreaker Results • Compression results for connectivity information • Guaranteed2T bits, entropy down to 0.9T bits for large models • 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 • ….

  17. Edgebreaker extensions and improvements • Better compression • Tighter guaranteed upper bound (King&Rossignac, Gumhold): 1.80T bits • Quadrilateral meshes (with Szymczak and King): 1.34T bits • Sufficiently regular meshes (with Szymczak and King): 0.81T bits guaranteed • Topological extensions • Non manifold solids (with Cardoze) • Handles (with Safonova, Szymczak, Coors, and Lopes) • Holes (with Safonova) • Geometry • Optimal quantization (with King and Szymczak): best B and T • Better predictors from connectivity and neighbors (with Coors) • Higher dimension • Tetrahedra for FEM (with Szymczak): 7T bits(prior to entropy) • Pentatopes for 4D simulations (with Szymczak, and with Snoeyink) • Progressive and error-resilient • Compressed batches of vertex-split upgrades (with Pajarola): 3.5T bits (total) • Tetrahedra (with Pajarola and Szymczak): 5T bits(after entropy)

  18. 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 • Further constraints (Gumhold 00): 1.80T bits guaranteed • E is impossible after C, CCR, … • Because they have increased the length of the bounding loop to more than 3

  19. ? 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

  20. Improved Edgebreaker code (Gumhold 00) • “New bounds on the encoding of planar triangulations” • Siggraph course notes on “3D Geometry Compression” • 1.8T bits guaranteed • 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

  21. Non-manifold boundaries (with Cardoze)

  22. Holes (with Safonova) • Many meshes have holes • Plug hole with a triangle fan (Touma&Gotsman’98) • Encode identifiers for dummy vertices • HLog(v) bits • The boundary of holes may by non-manfold • What is a hole?

  23. Handles (with Lopes and Safonova)

  24. Connectivity compression: An new problem? • Use vertex permutation to encode incidence • Denny,Sohler:Encoding a triangulation as a permutation of its point set,CCCG, 97 • Compression of the connectivity graph (planar triangle graph) • Itai,Rodeh: Representation of graphs, Acta Informatica, 82 • Turan: On the succinct representation of graphs, Discrete Applied Math, 84 • Naor:Succinct representation of general unlabeled graphs,Discrete Applied Math, 90 • Keeler,Westbrook: Short encoding of planar graphs and maps, Discrete Applied Math, 93 • Deering: Geometry Compression,Siggraph, 95 • Taubin,Rossignac: Geometric compression through topological surgery, ACM ToG, 98 • Taubin,Horn,Lazarus,Rossignac: Geometry coding and VRML, Proc. IEEE, 98 • Touma,Gotsman: Triangle Mesh Compression, GI, 98 • Gumbold,Straßer: Realtime Compression of Triangle Mesh Connectivity, Siggraph, 98 • Rossignac: Edgebreaker: Compressing the incidence graph of triangle meshes, TVCG, 99 • Rossignac,Szymczak: Wrap&Zip: Linear decompression of triangle meshes, CGTA, 99 • Szymczak,Rossignac: Grow&Fold: Compression of tetrahedral meshes, ACM SM, 99 • Compressed inverse of progressive simplification steps or batches • Hoppe: Progressive meshes, Siggraph, 96 • Taubin,Gueziec,Horn,Lazarus: Progressive forest split compression, Siggraph, 98 • Pajarola,Rossignac: Compressed Progressive Meshes, IEEE TVCG99 • Pajarola,Szymczak,Rossignac: ImplantSpray: Compressed Tetrahedral Meshes, VIS 99

  25. Edgebereaker compression contributors Szymczak: Decompression King: Improved code, quads Rossignac: Edgebreaker Safonova: Holes, implementation Isenburg (UCS): Spirale Reversi Gotsman (Israel): Polygons Lopes (Brasil): Handles Gumhold (Germany): Better bounds Coors (Germany): Prediction

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