Extensions to Edgebreaker

1
Extensionsto Edgebreaker

• Jarek Rossignac
• GVU Center and College of Computing
• Georgia Tech, Atlanta
• http//www.gvu.gatech.edu/jarek

2
Edgebreaker extensions and improvements

• Better connectivity compression
• Tighter guaranteed upper bound (KingRossignac,
  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)

3
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
Isenburg (UCS) Reversi
Attene (Italy) retiling
Coors (Germany) Prediction
Lopes (Brasil) Handles
Gotsman (Israel) Polygons
4
Guaranteed 1.84T bit (KingRossignac 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

5
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 (gt2)
• Not convenient for treating non-manifolds (See
  later)
• CE is impossible
• Was at least 3, C increased it to at least 4,
  cant have an E
• CCRE is impossible
• Was at least 3, CC increased it to at least 5, R
  reduced it by 1, cant have an E
• These constraints impact the probability of the
  next symbol and improve coding

6
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 Tuttes lowest bound
• Fewer Q-meshes than T-meshes
• With same vertex count
• Theoretical proof
• Extended to polygons
• Fan boundaries

FaceFixer, IsenburgSnoeyink
7
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

8
Encoding polygon meshes, 5P bits

• D. King, J. Rossignac, and A. Szymczak,
  Connectivity compression for irregular
  quadrilateral meshes, Technical Report TR9936,
  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. 263270.
• B. Kronrod and C. Gotsman,Efficient Coding of
  Non-Triangular Meshes, Technical Report,
  Computer Science Department, Technion-Israel
  Institute of Technology, 1999.

9
Face Fixer (IsenburgSnoeyink 00)

• Face Fixer Compressing Polygon Meshes with
  Properties
• Siggraph 2000
• Encodes a label per edge
• Extends to polygons
• Represents polygon-spanning tree and its dual
  vertex-tree
• Encodes partition of mesh into regions
  (superfaces)
• Discusses how to encode attributes
• Needs 2.5T bits for simple triangle meshes
• Compared to 1.83T bits guaranteed by Edgebreaker
• Needs 4V bit encoding for simple quad-meshes
• Compared to measured 0.24V to 1.14V bits,
  guaranteed 2.67V bits of KingRossignacSzymczak

10
Face fixer results
11
Khodakowsky et al

• Near-Optimal Connectivity Encoding of 2-Manifold
  Polygon Meshes
• Khodakovsky, P. Alliez, M. Desbrun, P. Schroder,
  http//multires.caltech.edu/pubs/ircomp.pdf
• Encodes vertex valences and face valences
• Achieves Tuttes optimal limit if you ignore
  splits

12
Khodakowsky vs FaceFixer
13
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
• T2V4(H-S)
• T triangles, E edges, V vertices, H handles, S
  shells
• Euler T-EV2S -2H
• 2 borders per edge and 3 borders per triangle
  2E3T
• HS-(T-EV)/2
• Shared edges E3T/2
• 3 borders per triangle, 2 borders per edge

disk
14
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

15
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

16
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

17
Non-Manifolds

• Solid models have non-manifold edges and vertices
• Compression exploits manifold data structures
• Matchmaker Manifold BReps for non-manifold
  r-sets
• RossignacCardoze, ACM Symposium on Solid
  Modeling, 1999.
• Match pairs of incident faces for each NME
• Respects surface orientation minimizes number
  of NMVs

18
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

19
Delphi correct guesses
Depending on the model, between 51 and 97 of
guesses are correct.
83 correct guesses 1.47bpv 0.74T bits
20
Delphi Wrong non-C guesses
21
Delphi wrong C-guesses
28 of wrong guesses are Rs mistaken for Cs.
22
Apollo sequence encoding of Delphi
23
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)

24
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

25
PRM error lt0.02 with same V-count
26
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
  ToumaGotsman
• 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)

27
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

28
Swing-Wrapper resolution control
29
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 WRL4,100,000B
30
SwingWrapper vs AliezDesbrun
Original268K triangles WRL file 8.5 Mbytes
? 0.4 62K triangles encoded with 37072 bytes
? 1.6 18K triangles encoded with 10314 bytes
? 4.1 9K triangles encoded with 5624 bytes
31
Summary

• Topological Surgery (MPEG-4) RLE of TST and VST
• Edgebreaker connectivity (CLERS)
• Efficient WrapZip 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 (EK/V)
• Piecewise Regular Meshes (reliefs) 1Tb total
  geometryconnectivity (0.02 error)
• SwingWrapper Isosceles Cs, 0.36Tb total (
  0.007 error), 0.06Tb (0.15 error)

32
Publications on Edgebreaker

• Geometric compression through Topological
  Surgery, G. Taubin and J. Rossignac, ACM
  Transactions on Graphics, vol. 17, no. 2, pp.
  84115, 1998.
• Geometry coding and VRML, G. Taubin, W. Horn,
  F. Lazarus, and J. Rossignac, Proceeding of the
  IEEE, vol. 96, no. 6, pp. 12281243, June 1998.
• Edgebreaker Connectivity compression for
  triangle meshes, J. Rossignac, IEEE Transactions
  on Visualization and Computer Graphics, vol. 5,
  no. 1, pp. 4761, 1999.
• Optimal Bit Allocation in Compressed 3D Models.
  Davis King and Jarek Rossignac. Computational
  Geometry, 1491118, 1999.
• WrapZip 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.
• GrowFold Compression of Tetrahedral Meshes,
  A. Szymczak and J. Rossignac. Proc. ACM Symposium
  on Solid Modeling, pp. 54-64, June 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)257264, 2000.
• Compressing the connectivity of tetrahedral
  meshes, A. Szymczak and J. Rossignac.
  Computer-Aided Design, 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, to appear
  in 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.

33
Papers on Lossy Compression

• Multi-resolution 3D approximations for rendering
  complex scenes, J. Rossignac and P. Borrel.
  Geometric Modeling in Computer Graphics, pp.
  455-465, Springer Verlag, Eds. B. Falcidieno and
  T.L. Kunii, Genova, Italy, June 28-July 2, 1993.
• The IBM 3D Interaction Accelerator (3DIX), P.
  Borrel, K.S. Cheng, P. Darmon, P. Kirchner, J.
  Lipscomb, J. Menon, J. Mittleman, J. Rossignac,
  B.O. Schneider, and B. Wolfe, RC 20302, IBM
  Research, 1995.
• Geometric Simplification, J. Rossignac, in
  Interactive Walkthrough of Large Geometric
  Databases (ACM Siggraph Course Notes 32), pp.
  D1-D11, Los Angeles, 1995.
• Full-range approximations of triangulated
  polyhedra, R. Ronfard and J. Rossignac.
  Proceedings of Eurographics96, Computer Graphics
  Forum, pp. C-67, Vol. 15, No. 3, August 1996.
• Simplification and Compression of 3D Scenes, J.
  Rossignac, Eurographics Tutorial, 1997.
• Geometric Simplification and Compression, J.
  Rossignac, in Multiresolution Surface Modeling
  Course, ACM Siggraph Course notes 25, Los
  Angeles, 1997.
• Compressed Progressive Meshes, R. Pajarola and
  J. Rossignac. IEEE Transactions on Visualization
  and Computer Graphics, vol. 6, no. 1, pp, 79-93,
  2000.
• Squeeze Fast and progressive decompression of
  triangle meshes, R. Pajarola and J. Rossignac,
  in Proceedings of Computer Graphics International
  Conference, 2000, pp. 173182. Switzerland, June
  2000.
• Implant Sprays Compression of Progressive
  Tetrahedral Mesh Connectivity, R. Pajarola, J.
  Rossignac, A. Szymczak. Proceedings of IEEE
  Visualization, San Francisco, October 24-29,
  1999.
• An Unequal Error Protection Method for
  Progressively Compressed 3-D Meshes, G.
  Al-Regib, Y. Altunbasak and J. Rossignac. Proc.
  IEEE International Conf. on Acoustics, Speech and
  Signal Processing ICASSP'02. Orlando, May 2002.
• 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.
• SwingWrapper Retiling Triangle Meshes for
  Better Compression, M. Attene, B. Falcidieno, M.
  Spagnuolo and J. Rossignac, Technical Report.
  March 2002.
• A joint source and channel coding approach for
  progressiv ely compressed 3D mesh transmission,
  G. Al-Regib, Y. Altunbasak and J. Rossignac,
  ICIP, 2002.
• Protocol for streaming compressed 3D animations
  over lossy channels, G. Al-Regib, Y. Altunbasak,
  J. Rossignac. and R. Mersereau, IEEE Int. Conf.
  on Multimedia and Expo (ICME), Lausanne, August.
  2002.

34
Encoding the corner attributes

• Jarek Rossignac
• GVU Center and College of Computing
• Georgia Tech, Atlanta
• http//www.gvu.gatech.edu/jarek

35
Common attributes

• Attributes One per corner
• parameters for color and texture calculations
• Could be the same for all 3 corners (flat
  triangle)
• Could be the same for two adjacent corners
  (smooth half-edge)
• Could be the same for opposite corners of an edge
  (smooth edge)
• Could be the same for all coincident corners
  (smooth vertex)
• Linear interpolation of shape and attributes over
  triangle

36
Corner tabledata 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.a index to table of corner attributes
• Table of vertex locations
• Table of attributes

c.t
37
Reducing attribute references

• Store a discontinuity bit for every corner
• Says if corner has same attribute than previous
  corner in cw order
• Total cost 3T bits (in fact, stores 2 bits per
  edge)
• One bit for ever edge/vertex pair
• Geometry coding and VRML
• Taubin, Horn, Lazarus, Rossignac
• Proc. IEEE, 86(6), 1998
• Use additional 1 bit per vertex
• Identify vertices for which all corners have same
  attribute
• Avoiding sending bits for these corners (saves
  20 to 70)
• Face Fixer
• IsenburgSnoeyink
• Siggraph 00

38
Compressing attribute values

• Quantize to the minimum resolution
• Less than 24-bit color
• Discretized normals
• 1 bit selects hemisphere
• 2 bits select quadrant
• 2 more bits select sub-quadrant
• Predict from previously decoded neighbors
• Average, Extrapolate
• Encode the difference

39
Attribute prediction

• Estimate vertex normal from geometry
• For example, use sum of cross-products UxV
• Estimate texture coordinates
• 2-D prediction over T-mesh
• In Texture coordinate space

40
Single-resolution compression of Tetrahedral
MeshesGrowFold

• Jarek Rossignac
• (with Andrzej Szymczak)
• GVU Center and College of Computing
• Georgia Tech, Atlanta
• http//www.gvu.gatech.edu

41
Representing triangle and tetrahedra meshes
T 6.5V
Connectivity dominates storage cost!
T 2V
42
Compressing tetrahedral meshes

• Non-uniform physical properties through space
• 3D samples with scalar value(s)
• Irregular tetrahedral mesh
• Finite elements resulting from automatic meshing
• Could be result of simplifying regular grids
• Incidence table dominates storage cost 4Tlog(V)
  bits
• T6V (usually between 4 and 7 times more
  tetrahedra than vertices)
• V may be very large (2563)
• Need to compress connectivity
• Want linear cost in T

43
GrowFold

• SzymczakRossignac
• GrowFold Compression of tetrahedral meshes,
• ACM Symposium on Solid Modeling 99
• Encode tetra-tree (3bits per tetrahedron)
• Has internal and external triangle-faces
• Mark fold edges on external faces (4b/tet)
• 2 bits per face mark zero or one of the edges
• 2 free faces per tetrahedron
• Results 7 bits/tet
• Instead of 4log(V)