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					                                                   3D Compression




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

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002     3D Compression Challenges , 1
How should one measure shape complexity?




•     Number of components, handles and holes
•     Algebraic degree of bounding surfaces
•     Number of vertices in mesh
•     Stabbing number
•     Area3/volume2
•     Curvature integral
•     Number of branches in skeleton
•     …
•     Number of bits required to store or transmit the model
Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 2
                                Storage size depends on
•     The shape, topology, and attributes of the model
•     Choice of representation
•     Acceptable accuracy loss
•     Compression used



                                                              Error


                                                                Storage




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002               3D Compression Challenges , 3
       Storage size depends on representation

                                                                   
        2D                  1D                    0D




Jarek Rossignac, CoC & GVU Center, Georgia Tech        May, 2002       3D Compression Challenges , 4
                          Overall objective and issues
• Find the most compact representation for each object
• If you want loss-less encoding
        – Not much freedom
                • Triangle mesh or CSG?
                • Sphere(C,r) or NURBS
• If you accept lossy encoding
        – How do you define the error?
        – Can you compute it?
        – How to convert between representations?
                • Boundary-to-CSG
        – How to compress each representation?
• We focus on triangle meshes
        – Popular, supported, simplest
        – Representing polyhedra, Bsplines, and Subdivision surfaces

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002       3D Compression Challenges , 5
    Focus on explicit representation (T-mesh)
• Samples: Location and attributes (color, mass)
• Connectivity: Triangle/vertex incidence
• Fit: Rule for bending triangles (subdivision surfaces, NURBS)
                    Samples
                    (vertices):
             vertex 1 x y z c
             vertex 2 x y z c
             vertex 3 x y z c

                                       V(3B+k) bits         Triangle/vertex
                                                            incidence:      v4
                                                            Triangle 1   1   2 3
                                                           Triangle 2
                                                                         3   24
                                                            Triangle 3   4   5 2      t3
                                                           Triangle 4    7   5 6 v5
                                                                                           v2
                                                            Triangle 5   6   5 8
                T = 2V                   V(6log2V) bits Triangle 6       8   51
Jarek Rossignac, CoC & GVU Center, Georgia Tech        May, 2002                           3D Compression Challenges , 6
                 Storage size depends on accuracy
                                                                                          (7,7)

• Depends on vertex data quantization
        – Represent coordinates as normalized integers
                • Coordinates relative to bounding rectangle
                • Select unit for desired resolution [0..2B]  (0,0)
                • Vertex coordinates = B-bit integers (6<B<14) Error EB
• Depends on the sampling density (LOD)                                    ET
        – Storage grow with the density of samples
        – Sub-sampled or simplified down to T triangles: Error ET                            T
                • Estimate ET = K/T (where K approximates shape complexity)




• Optimal choice of B and T (King&Rossignac)
        – Reduce error with a storage cap: F = T+3BcT/2
        – Reduce storage with a cap on error bound: E=EB+ET
Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002               3D Compression Challenges , 7
 Triangle count reduction techniques (LOD)
• Quantize & cluster vertex data (Rossignac&Borrel’92)
        – remove degenerate triangles (that have coincident vertices)




        – Adapted by P. Lindstrom for out-of-core simplification
• Repeatedly collapse best edge (Ronfard&Rossignac96)
        – while minimizing maximum error bound
        – Adapted by M. Garland for least square error




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002         3D Compression Challenges , 8
                              Different Error Measures
                                                                                    screen

• Image (color) fidelity (view dependent)
                                                                                              eye
        – Error on the color of pixels
        – Sensitive to view direction and surface orientation                    Silhouette
        – Too constraining for most 3D applications                              has moved
                • Shading models are approximate
                • Users can’t detect shading inaccuracies
• Screen space geometric error (view-dependent)                                               eye
        – Measures silhouette displacement
                                                                                     Hidden part
        – Must preserve depth order                                                  appeared
        – Bounded by projection of 3D deviation                     A
• Geometric 3D deviation (view-independent)                                H(A,B)
        –    Bound defined by model/application tolerance
                                                                B
        –    Hausdorff: H(A,B)=max(d(a,B),d(b,A)), aA, bB                  dev(A,B)
        –    Expensive to compute: (F1,F2,F3)
        –    Poor measure of discrepancy
        –    Often approximated conservatively


Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002             3D Compression Challenges , 9
      Storage may be reduced by compression
                                                                                               A
• Vertex coordinates (compress to 2 Bytes/vertex)                                         B
        – Normalize/quantize coordinates
                                                                                       C
        – Predict using decoded neighbors
                                                                                                    D
        – Store corrections using entropy compression
                                                                                                       V
                                                                                 V=aA+bB+cC+dD+X



                                                                         C
                                                                  C     R
• Incidence (< 2 bits/vertex)                                     R     C
                                                              C              C
        – Depth-first triangle-tree (spiral)
                                                                  C R        C
        – Encode as string of symbols                                   C

                         CCCCRCCRCRC…

• Fit rule and parameters (constant cost?)
Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                         3D Compression Challenges , 10
Complexity of a shape = Storage/Error curve
Storage




                                        Error of the approximating model
   Curve depends on representation and compression scheme used
                                                               Estimate ET = K/T
 Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002          3D Compression Challenges , 11
                                                  Terminology
                                                                 vertex
• Vertex:
        – Location of a sample                                            corner
• Triangles:                                                                              triangle
        – Decompose approximating surface
                                                                                          border
• Edge:
        – Bounds one or more triangles                                                edge
        – 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
Jarek Rossignac, CoC & GVU Center, Georgia Tech      May, 2002            3D Compression Challenges , 12
                    First, the case of a simple mesh
• A simple mesh is homeomorphic to a triangulated sphere
        –    Orientable
        –    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




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002      3D Compression Challenges , 13
                    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



Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                3D Compression Challenges , 14
                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

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002     3D Compression Challenges , 15
     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)
                              vertex 1 vertex 2 vertex 3
                 Triangle 1   xyzxyzxyz
                 Triangle 2   xyzxyzxyz
                 Triangle 3   xyzxyzxyz



  Very verbose! Not good for traversal.

Jarek Rossignac, CoC & GVU Center, Georgia Tech       May, 2002          3D Compression Challenges , 16
                 Representation as Triangle strips
• Continue a strip by attaching a new triangle to an edge of
  the previous one                   L




                                                              R   R


• Need only indicate which edge and when to start a new strip
        – 1 Left/Right bit per triangles
        – 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

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002           3D Compression Challenges , 17
                             36-to-1 compression ratio
Independent triangles: 576V bits                               Triangle Strips: 214V bits
 (coordinates as floats 9x32T bits)                            (1.1x3x4B/T + 1B/S + 1b/T)
   Each vertex encoded ~ 6 times                               Each vertex encoded ~ twice
              vertex 1 vertex 2 vertex 3                                vertex 1   xyz      Strip 1   10
 Triangle 1   xyzxyzxyz                                                 vertex 2   xyz      Strip 2   7
 Triangle 2   xyzxyzxyz                                         L       vertex 3   xyz
 Triangle 3   xyzxyzxyz

                                                                R   R

   V and T tables                                                V(6log V) bits
                                                   V(3B+k) bits Triangle 1 2
   Example: V=1000, B=10, k=0                                              1 2 3
                                              vertex 1 x y z c Triangle 2 3 2 4
   Samples+incidence storage costs:           vertex 2 x y z c Triangle 3 4 5 2
   (3B+6log2V)V = (30+60)V = 98V bits         vertex 3 x y z c Triangle 4 7 5 6
                                                                Triangle 5 6 5 8

              Can be compressed to: (14+2)V = 16V bits          Triangle 6 8 5 1

 Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                         3D Compression Challenges , 18
Connectivity compression: An old challenge
•     Tutte: Theoretical lower bound (Tutte 62): can’t guarantee < 1.62 bits per triangle
•     Itai,Rodeh: Representation of graphs, Acta Informatica, 82
•     Turan: On the succinct representation of graphs, Discrete Applied Math, 84: 6Tb
•     Naor: Succinct representation of unlabeled graphs, Discrete Applied Math, 90
•     Keeler,Westbrook: Encoding planar graphs, Discrete Applied Math, 93: 2.3Tb
•     Deering: Geometry Compression, Siggraph, 95
•     Denny,Sohler: Encode 2D triangulation as permutation of points, CCCG, 97: 0Tb

                                              Topological Surgery
                                              Taubin&Rossignac

Valence-based                                     Edgebreaker         Cutborder
Gotsman&team                                      Rossignac&team      Gumhold&Strasser

                                                  Spirale Reversi
                                                  Snoeyink&Isenburg
Jarek Rossignac, CoC & GVU Center, Georgia Tech         May, 2002           3D Compression Challenges , 19
                   A simple and elegant solution (?)
Given the left and right boundaries
                                                               L
of a triangle strip (corridor), we                                     L L     L

need T (left/right) bits to encode its
triangulation. ex: LRRLLRLRR                                   R   R
                                                                         R R
                                                                                   R


 Connecting vertices into a single
 spiral (Hamiltonian walk) defines
 the left and the right boundaries
 (walls) of a long corridor.


Store vertices in their order along the wall.
   (Can use former vertices to predict location of new ones.)
Encode connectivity using only 1 (left/right) bit per triangle !
 Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                   3D Compression Challenges , 20
      But..but..wait a minute! It doesn’t work!
                                             The corridor may
        L                                        have warts

                    ?

                  R                                     Warts are        ?
       R                                                hard to avoid

                                                                        The spiral may
                                                                        bifurcate




The corridor
may bifurcate
Jarek Rossignac, CoC & GVU Center, Georgia Tech     May, 2002                3D Compression Challenges , 21
       3Tb encoding of VST and TST suffices

                                         Guaranteed 3Tb connectivity
                                         (2Tb T-tree + 2Vb V-tree)
                                                                                                                       4                        5

                                                                                       1
                                                                                                               3                                        6
                                                                                                                                   1
                                                                                               2

                                                                                                                       2                2                    7
                                                                                           3




                                              +
                                                                                                                               3
                                                                                                                                                        12
                                                                                   4

                                                                                                                   4
                                                                           5                                                                7


                                                        12                                                 5                       11
                                                                   6           9                                                                    7

                                                  11
                                                                                                       6
                                                             7         8
                                                                                                                                        10
                                                                                                                   9
                                                       10
                                                                                                   7
                                                                                                           8               8                        7

Jarek Rossignac, CoC & GVU Center, Georgia Tech                  May, 2002                                         3D Compression Challenges , 22
                                                  Topological Surgery (TS)

                                                                  Taubin-Rossignac

                                                   IBM’95, VRML, MPEG-4

             “Geometric compression through Topological Surgery,” G. Taubin and J.
             Rossignac, ACM Transactions on Graphics, vol. 17, no. 2, pp. 84–115, 1998.
             “Geometry coding and VRML,” G. Taubin, W. Horn, F. Lazarus, and J.
             Rossignac, Proceeding of the IEEE, vol. 96, no. 6, pp. 1228–1243, June 1998.

Jarek Rossignac, CoC & GVU Center, Georgia Tech       May, 2002                      3D Compression Challenges , 23
              Run-length encoding of TST and VST

                             Most nodes have a single child.
                              - Group them into runs
                              - Store structure of each tree
                             (2 bits per run)
                              - Store left/right bit per triangle




                                                                                  5


                                                                        5
                                                                                           4
                                                                    2       3


Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                 3D Compression Challenges , 24
                                                  TS example




Jarek Rossignac, CoC & GVU Center, Georgia Tech     May, 2002   3D Compression Challenges , 25
   Spiral is better than other traversal orders




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 26
                 TS spiral joins topological layers




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 27
                  Results TS (MPEG-4) vs VRML




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 28
      Topological Surgery: Results and Impact
• Results:
     – Compresses to less than 12 bits per triangle
             • Connectivity: about 2.0 T bits (increases for smaller meshes)
             • Geometry: 6 bits per coordinate (decreases with tessellation&quantization)


• Publications and impact of Topological Surgery:
     – Interfaced to VRML 2.0 offered by IBM (1995)
     – Core of the MPEG-4 standard for 3D Compression
     – Geometric compression through topological surgery, Taubin&Rossignac
             • ACM Transactions on Graphics, 17(2):84-116, April 1998 (Best Paper Award)
     – Geometry coding and VRML, Taubin, Horn, Lazarus, & Rossignac
             • Proceedings of the IEEE, 96(6):1228-1243, June 1998
     – Inspired several approaches
             • Touma,Gotsman: Triangle Mesh Compression, GI, 98
             • Gumbold,Straßer: Realtime Compression of Triangle Mesh Connectivity, Siggraph, 98
             • Taubin,Gueziec,Horn,Lazarus: Progressive forest split compression, Siggraph, 98

 Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                  3D Compression Challenges , 29
                                              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 .
Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                              3D Compression Challenges , 30
        Edgebereaker compression contributors
                                                                Gumhold                        King
                                  Rossignac                                                    (Atlanta):
                                  (Atlanta):                    (Germany):
     Safonova                                                   1.80T bits                     1.84Tbits,
                                  Edgebreaker                                                  quads
     (CMU):
     Holes, code

                                                                                                Szymczak
                                                                                                (Atlanta):
                                                                                                regularity,
                                                                                                resampling
Shikhare
(India):
translation




Attene (Italy):                                                                               Isenburg
                                                                                              (UCS):
retiling                                                                     Coors            Reversi
                             Lopes (Brasil):
                             Handles         Gotsman (Israel):               (Germany):
                                             Polygons                        Prediction
  Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                         3D Compression Challenges , 31
             Edgebreaker is a state machine
                  ?                          C?                       Marked (visited)
                  x                                                   Not marked
                                                               x      Last visited

                  ?                          L?                ?      Next to be encoded
                  x                                                  To-do stack

                                                             if tip vertex not marked then C
                  ?                    ? R                   else if left neighbor marked
                  x                                                then if right neighbor marked then E else L
                                                                   else if right neighbor marked then R else S


                  ?                               S ?                Encode sequence of codes
                  x                                                     C: 0, L:110, R: 101, S:100, E:111
                                                                   Only 2T bits (because |C|=V=T/2)
                 ?                       ?          E                and vertices
                 x                                                      as encountered by C operations
Jarek Rossignac, CoC & GVU Center, Georgia Tech         May, 2002                          3D Compression Challenges , 32
                             Edgebreaker compression
                  ?                          C ?
                  x
                                                                                C
                                                                         C     R

                  ?                          L ?                     C
                                                                         R     C
                                                                                    C
                  x                                                      C R        C
                                                                               C


                                       ? R                                              CCCCRCCRCRC…
                  ?
                  x

                                                                               RR R
                  ?                                S ?                         LC     L E
                  x
                                                                               E   S R
                                                                                 CR
                 ?                       ?           E                                  …CRSRLECRRRLE
                 x
Jarek Rossignac, CoC & GVU Center, Georgia Tech          May, 2002                        3D Compression Challenges , 33
                        EB re-numbering of vertices




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 34
    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.
                                                                                       c.l
• Table of corners, for each corner c store:                                                           c.v c.r
        – c.v : integer reference to vertex table                                                    c         c.p
        – c.o : integer reference to opposite corner
                                                                                               c.n
• Make the 3 corners of each triangle consecutive                                                                      c.t
        – List them according to ccw orientation of triangles                                            c.o
        – 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
                                 vo
        Triangle 0 corner 0 1 7                                 2
                                                                                vertex 1 x y z
        Triangle 0 corner 1 2 8
                                                             1 3                vertex 2 x y z
        Triangle 0 corner 2 3 5                   3    2
                                                            0       4   5   4   vertex 3 x y z
        Triangle 1 corner 3 2 9
        Triangle 1 corner 4 1 6                                 1               vertex 4 x y z
        Triangle 1 corner 5 4 2
Jarek Rossignac, CoC & GVU Center, Georgia Tech       May, 2002                                3D Compression Challenges , 35
          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}
                                                                     voa
          – 230, 131, 122, 143, 244, 125, …                           2               Triangle 1 corner 0 1          a
                                                                                      Triangle 1 corner 1 2          b
                                                                  1 3
                                                     3        2                       Triangle 1 corner 2 3          c
                                                                  0       4   5   4
  • Sort (bins, linear cost):                                                         Triangle 2 corner 3 2          c

          – 122, 125 ...131... 143 ...230...244 … 1                                   Triangle 2 corner 4 1          d
                                                                                      Triangle 2 corner 5 4          e
  • Pair-up consecutive entries 2k and 2k+1
          – (122, 125)...131... 143...230...244…                                                            voa
                                                                                      Triangle 1 corner 0 1         a
  • Their corners are opposite                                                        Triangle 1 corner 1 2         b
                                                                      2
          – (122,125)...131...143...230...244…                                        Triangle 1 corner 2 3 5 c
                                                                  1 3
                                                    3         2                       Triangle 2 corner 3 2         c
                                                                  0       4   5   4   Triangle 2 corner 4 1         d
                                                                                      Triangle 2 corner 5 4 2 e
                                                                      1
Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                                    3D Compression Challenges , 36
               Edgebreaker compression algorithm
  Source code, examples: http://www.gvu.gatech.edu/~jarek/edgebreaker/eb
         vo                               recursive procedure compress (c)
 T1 c0 1 7         v1 x y z               repeat {
 T1 c1 2 8                                        c.t.m:=1;                                  # mark the triangle as visited
                   v2 x y z
                                                  if c.v.m == 0                              # test whether tip vertex was visited
 T1 c2 3 5
                   v3 x y z                          then {      write(vertices, c.v);       # append vertex index to “vertices”
 T2 c3 2 9
                                                                 write(clers, C);            # append encoding of C to “clers”
                   v4 x y z
 T2 c4 1 6                                                       c.v.m:= 1;                  # mark tip vertex as visited
 T2 c5 4 2                                                       c:=c.r }                    # continue with the right neighbor
                      c.v                         else if c.r.t.m==1                         # test whether right triangle was visited
                                         c.r            then if c.l.t.m== 1                  # test whether left triangle was visited
         c.l
                         c                                          then {write(clers, E);   # append encoding of E to clers string
                         c.t                                               return }          # exit (or return from recursive call)
                                                                    else {write(clers, R);   # append encoding of R to clers string
                        c.o                                                c:=c.l }          # move to left triangle
    RR R                                                else if c.l.t.m == 1                 # test whether left triangle was visited
     b                                                              then {write(clers, L);   # append encoding of L to clers string
    LC     L E                                                              c:=c.r }         # move to right triangle
    E a S R                                                         else {write(clers, S);   # append encoding of S to clers string
     CR                                                                    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)
   Jarek Rossignac, CoC & GVU Center, Georgia Tech            May, 2002                                        3D Compression Challenges , 37
      EB decompression: how come it works?
     Receive the CLERS sequence
     Decode it
     Construct the triangle tree
     Decode&reconstruct vertices                                   RR R
                                                                   LC      L E
                …CRSRLECRRRLE                                      E    S R
                                                                       R
                                                                     C


  How to zip up the cracks?

   “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.

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                             3D Compression Challenges , 38
  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:



 seed         C             L          E          R             S
 Then ZIP all pairs of adjacent bounding edges when both point away
 from their common vertex.

                                 R RR                       R RR           RR R
                                 L C      L E               L C      L E   LC    L E
CRSRLECRRRLE                          S R                        S R
                                 E
                                     R
                                                            E
                                                                R
                                                                           E RS R
                                   C                          C             C

   Linear time complexity. Zip only after L and E.
  Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002               3D Compression Challenges , 39
               Wrap&Zip more complex example




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 40
            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.


                                                                  C           R          L             S             E

                                                                      compression


                 clers = …CCRRCCRRRCRRCRCRRCCCRRCRRCRCRRRCRCRCRRSCRRSLERERLCRRRSEE



reversi = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC…




           decompression



  Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                                3D Compression Challenges , 41
                                                  Reversi details

                                                                                                                            1
 1                          1
                                                                                1                           2
O=E                 O = EE                  O = EES             O = EESRRRCLR            O = EESRRRCLRER




                                              1
                             2                                                          1
                                                                           O = EESRRRCLRERELSRRC
            O = EESRRRCLREREL




       O = EESRRRCLRERELSRRCS

                                                  O = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC…

Jarek Rossignac, CoC & GVU Center, Georgia Tech            May, 2002                               3D Compression Challenges , 42
                                      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 <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
        –    ….



Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002             3D Compression Challenges , 43
  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)


 Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002             3D Compression Challenges , 44
    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
Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                   3D Compression Challenges , 45
             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


Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002         3D Compression Challenges , 46
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*


Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                                  3D Compression Challenges , 47
               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

 Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 48
            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

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002         3D Compression Challenges , 49
                                              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
          2
          1 3
3    2
         0 4        5     4
           1




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 50
  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




                             g(c) d                  c.v   X
                                             c
                                                      v
                                                 c
             Vl
                                  c.n
                                           c.p
                                                               Vr
                                 cGc
                                  E  c.o




                                        c
       Figure 2: Connectivity guessed by
       parallelogram prediction
Jarek Rossignac, CoC & GVU Center, Georgia Tech                     May, 2002                 3D Compression Challenges , 51
                                  Delphi correct guesses

                             X                                                     Depending on the model,
                  g(c)
                                          g(c)                          g(c)       between 51% and 97%
                                      X
                                                                               X   of guesses are correct.
       Guess C                    Guess L                          Guess R

                                      X                     X
                             g(c)                  g(c)




               Guess S                            Guess E



       Figure 3: Guess clers Symbol
       based on geometry prediction.                                                83% correct guesses:
                                                                                    1.47bpv = 0.74T bits

Jarek Rossignac, CoC & GVU Center, Georgia Tech                 May, 2002                    3D Compression Challenges , 52
                        Delphi: Wrong non-C guesses
  Guess wrong R                                                                 Guess wrong L



             g(c)                   g(c)                    g(c)                    g(c)         g(c)                   g(c)

                    X                      X                       X           X             X                      X

  Situation L            Situation C              Situation S                 Situation R   Situation C            Situation S

                           Half of the wrong guesses are Cs mistaken for Rs
 Guess wrong S                                                                              Guess wrong E
              X                      X                       X                        X                 X
     g(c)                    g(c)                    g(c)                    g(c)
                                                                                             g(c)



                                                                                                  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.
Jarek Rossignac, CoC & GVU Center, Georgia Tech                  May, 2002                                  3D Compression Challenges , 53
                               Delphi wrong C-guesses
                                                  Guess C in


     28% of wrong                                                     X                    X
     guesses are Rs                                   g(c)                    g(c)
     mistaken for Cs.

                                                  Situation R             Situation L

                                                                                        X
                                                                    X
                                                                g(c)        g(c)


                                                  Situation S             Situation E
                                                  Figure 4: Wrongly guesses C triangles
Jarek Rossignac, CoC & GVU Center, Georgia Tech           May, 2002                     3D Compression Challenges , 54
             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.



Jarek Rossignac, CoC & GVU Center, Georgia Tech                            May, 2002                                                          3D Compression Challenges , 55
                                   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)
Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                 3D Compression Challenges , 56
               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




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                           3D Compression Challenges , 57
        PRM error <0.02% with same V-count




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 58
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)

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002    3D Compression Challenges , 59
          SwingWrapper: semi-regular retiling
“SwingWrapper: Retiling Triangle Meshes for Better Compression”, M. Attene, B. Falcidieno, M. Spagnuolo and
    J. Rossignac, Technical Report. March 2002
                                                                                                         L
• Resample mesh to improve compression
• Try to form regular triangles
        – All C triangles are Isosceles                                                                  L
        – with both new edges of length L
• Fill cracks with irregular triangles
• Encode connectivity with Edgebreaker
• Encode one hinge angle per vertex

                                                  180¡            L 3/2    180¡+


                                                  x                           x
                                                                                          C                  


Jarek Rossignac, CoC & GVU Center, Georgia Tech       May, 2002                         3D Compression Challenges , 60
            Swing-Wrapper resolution control




Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 61
 SwingWrapper results: 0.4Tb total (0.01%)

                         C
                         L
                         E
                         R
                         S
                                             13,642T                 1505T
134,074T
                                             L2 error 0.007%         L2 error 0.15%
WRL=4,100,000B
                                             3.5Tb total             5.2Tb total
                                             0.36Tb wrt original T   0.06Tb wrt original T
                                             678-to-1 compression    4000-to-1 compression

Jarek Rossignac, CoC & GVU Center, Georgia Tech        May, 2002             3D Compression Challenges , 62
                 SwingWrapper vs Aliez&Desbrun
                                                  “Progressive Encoding for Lossless Transmission
                                                  of 3D Meshes”, P. Alliez and M. Desbrun, Proc.
                                                  of SIGGRAPH 2001.


                                                                                 Original:268K 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
Jarek Rossignac, CoC & GVU Center, Georgia Tech       May, 2002                                3D Compression Challenges , 63
   Grow&Fold: compression of Tetrahedra
“Grow&Fold: Compressing the connectivity of tetrahedral meshes,” A. Szymczak and J. Rossignac. CAD, 2000.
“Grow&Fold: Compression of Tetrahedral Meshes,” A. Szymczak and J. Rossignac ACM Symp. 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)

Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002                                 3D Compression Challenges , 64
                                                   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)




 Jarek Rossignac, CoC & GVU Center, Georgia Tech    May, 2002             3D Compression Challenges , 65
                                    Some Research Issues
Definition and measure of shape complexity
Good measure of discrepancy between two shapes
Efficient algorithm for measuring discrepancy
Definition of equivalence classes of “similar” shapes
Most compact encoding for a class (representative)
Visualization of variability in a class
Constant cost per handle, hole, and non-manifold singularity
Compact encoding of topology (handles, knots, braids)
Decomposing shape into topologically simple, natural features
Encoding cell complexes
Encoding higher-dimensional models and attributes


Jarek Rossignac, CoC & GVU Center, Georgia Tech   May, 2002   3D Compression Challenges , 66
                                                  Questions?
•     Progressive transmission
•     Selective decompression
•     Streaming
•     Animation
•     Optimal coding




Jarek Rossignac, CoC & GVU Center, Georgia Tech     May, 2002   3D Compression Challenges , 67
                                Progressive transmission
“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 Eurographics’96, 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. 173–182. 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.

Jarek Rossignac, CoC & GVU Center, Georgia Tech         May, 2002                                        3D Compression Challenges , 68
                          Publications on Edgebreaker
“Geometric compression through Topological Surgery,” G. Taubin and J. Rossignac, ACM Transactions on Graphics, vol.
    17, no. 2, pp. 84–115, 1998.
“Geometry coding and VRML,” G. Taubin, W. Horn, F. Lazarus, and J. Rossignac, Proceeding of the IEEE, vol. 96, no. 6,
    pp. 1228–1243, June 1998.
“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”. Davis King and Jarek 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.
“Grow&Fold: 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):257–264, 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.


Jarek Rossignac, CoC & GVU Center, Georgia Tech     May, 2002                                     3D Compression Challenges , 69

				
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