Transcript Edgebreaker (EB) Second generation 3D compression Faster, simpler, more effective “Edgebreaker: Connectivity compression for triangle meshes,” J.

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 228, 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
SM, June 2002
05 Edgebreaker , 1
T-mesh primitives
vertex
• Vertex:
corner
– Location of a sample
triangle
• Triangles:
– Decompose approximating surface
border
• Edge:
– Bounds one or more triangles
– Joins two vertices
edge
• 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
SM, June 2002
05 Edgebreaker , 2
Classes of T-meshes
• Triangle soup
– Any collection of triangles (may intersect each other)
• 2D simplicial complex
– Collection of edges, vertices, faces that join but do not intersect
• Orientable manifold with boundary
– Each edge has 1 or 2 incident triangles. One incident cone per vertex.
• Boundary of a (regularized) solid
– Each edge has 2k incident triangles (non-manifold). Orientable. Handles.
• Zero-genus boundary of a manifold solid (simple mesh)
– Orientable. Manifold. Connected. No holes. No handles.
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 3
First, the case of a simple mesh
• A simple mesh is a deformed triangulated sphere
–
–
–
–
Orientable
2-manifold
No boundary (no holes)
No handles (no throu-holes)
• Properties
– Each edge has exactly 2 incident triangles
– Each vertex has a single cycle of incident triangles
– May be drawn as a planar graph
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 4
Simple meshes and patches
• A patch is a simple portion of a simple mesh
– Simply connected
• Bounded by a single manifold edge-loop
• Its boundary is a connected manifold loop
– Cycle of border edges
• It may be obtained from a simple mesh
– by removing one or more triangles
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 5
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
SM, June 2002
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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
SM, June 2002
05 Edgebreaker , 7
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
Triangle 2
Triangle 3
xyzxyzxyz
xyzxyzxyz
xyzxyzxyz
Very verbose! Not good for traversal.
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 8
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 plus 1 strip-end bit per triangle
• Send one vertex per triangle
– Plus 2 vertices per strip to start it
• Each vertex is transmitted twice on average
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 9
Corner table: data structure for T-meshes
“3D compression made simple: Edgebreaker on a corner-table.” J. Rossignac, A. Safonova, and A. Szymczak. In
Proceedings of the Shape Modeling International Conference, 2001.
• Table of corners, for each corner c store:
– c.v : integer reference to vertex table
– c.o : integer reference to opposite corner
c.v
c.l
• The 3 corners of each triangle are consecutive
c.n
– 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
vo
2
Triangle 0 corner 0 1 7
Triangle 0 corner 1 2 8
Triangle 0 corner 2 3 5
Triangle 1 corner 3 2 9
Triangle 1 corner 4 1 6
3
2
c.p
c.o
vertex 1 x y z
1 3
0
c.r
c
4
1
vertex 2 x y z
5
4
vertex 3 x y z
vertex 4 x y z
Triangle 1 corner 5 4 2
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 10
c.t
Using adjacency table for T-mesh traversal
• Visit T-mesh (triangle-spanning tree)
– Mark triangles as you visit
– Start with any corner c and call Visit(c)
– Visit(c)
• mark c.t;
• IF NOT marked(c.r.t) THEN visit(c.r);
• IF NOT marked(c.l.t) THEN visit(c.l);
• Label vertices
– Label vertices with consecutive integers
– Label(c.n.v); Label(c.n.n.v); Visit(c);
– Visit(c)
•
•
•
•
IF NOT labeled(c.v) THEN Label(c.v);
mark c.t;
IF NOT marked(c.r.t) THEN visit(c.r);
IF NOT marked(c.l.t) THEN visit(c.l);
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 11
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, …
3
2
2
Triangle 1 corner 0 1
a
1 3
Triangle 1 corner 1 2
b
Triangle 1 corner 2 3
c
Triangle 2 corner 3 2
c
Triangle 2 corner 4 1
d
Triangle 2 corner 5 4
e
0
• Sort (bins, linear cost):
4
5
4
– 122, 125 ...131... 143 ...230...244 … 1
• Pair-up consecutive entries 2k and 2k+1
voa
– (122, 125)...131... 143...230...244…
• Their corners are opposite
2
– (122,125)...131...143...230...244…
3
2
4
1
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
a
Triangle 1 corner 1 2
b
Triangle 1 corner 2 3 5 c
1 3
0
Triangle 1 corner 0 1
5
4
Triangle 2 corner 3 2
c
Triangle 2 corner 4 1
d
Triangle 2 corner 5 4 2 e
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Connectivity/geometry discrepancy
• Connectivity of T-mesh may conflict with actual geometry
– Vertices with different names may be coincident
– Edges with different names may be coincident
– Triangles, edges, and vertices may intersect
• T-mesh with consistent geometry
– Triangles, edges, vertices are pairwise disjoint
• We consider edges and triangles to be open
– I.e., not containing their boundary
• Manifold graphs may be used with invalid geometry
– Coincident edges and vertices: Non-manifold singularities
– Self-intersecting surfaces
Manifold graph
Non-manifold shape
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 13
Edgebreaker: A simple,
fast, and effective second
generation 3D compression
Jarek Rossignac
GVU Center and College of Computing
Georgia Tech, Atlanta
http://www.gvu.gatech.edu/~jarek
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 14
Edgebreaker encodes construction steps
Area not yet covered
Decompress
Compress
Specification of the next triangle
Compress
Binary
format
Decompress
Sequence of specification for adding triangles
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 15
Edgebreaker is a state machine
C?
?
x
Not marked
L?
?
x
?
x
Marked (visited)
x
Last visited
?
Next to be encoded
To-do stack
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 ?
?
x
Encode sequence of codes
C: 0, L:110, R: 101, S:100, E:111
Only 2T bits (because |C|=V=T/2)
?
x
?
Jarek Rossignac, CoC & GVU Center, Georgia Tech
and vertices
E
as encountered by C operations
SM, June 2002
05 Edgebreaker , 16
Edgebreaker compression
C ?
?
x
L ?
?
x
?
x
R
C
C R
Jarek Rossignac, CoC & GVU Center, Georgia Tech
C
RR R
LC
L E
S R
E
CR
S ?
?
C
C
CCCCRCCRCRC…
? R
?
x
?
x
C
C
C
R
E
…CRSRLECRRRLE
SM, June 2002
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EB re-numbering of vertices
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 18
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
T1 c1 2 8
v2 x y z
T1 c2 3 5
T2 c3 2 9
T2 c4 1 6
v3 x y z
v4 x y z
T2 c5 4 2
c.v
c.l
c
c.t
c.o
RR R
b
LC
L E
E a S R
CR
repeat {
c.t.m:=1;
if c.v.m == 0
then {
write(vertices, c.v);
write(clers, C);
c.v.m:= 1;
c:=c.r }
else if c.r.t.m==1
then if c.l.t.m== 1
c.r
then {write(clers, E);
return }
else {write(clers, R);
c:=c.l }
else if c.l.t.m == 1
then {write(clers, L);
c:=c.r }
else {write(clers, S);
compress(c.r);
c:=c.l } }
# mark the triangle as visited
# test whether tip vertex was visited
# append vertex index to “vertices”
# append encoding of C to “clers”
# mark tip vertex as visited
# continue with the right neighbor
# test whether right triangle was visited
# test whether left triangle was visited
# append encoding of E to clers string
# exit (or return from recursive call)
# append encoding of R to clers string
# move to left triangle
# test whether left triangle was visited
# append encoding of L to clers string
# move to right triangle
# append encoding of S to clers string
# recursive call to visit right branch first
# 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
SM, June 2002
05 Edgebreaker , 19
Edgebreaker decompression
• How does it work?
– No problem, except at S
– Can you recover where the tip of each S is from the CLERS string alone?
• Three solutions:
– Count changes of border length in CLERS string (Rossignac)
• Read CLERS string backwards to compute there the tip of each S is
– Wrap&Zip (Rossignac&Szymczak)
• Build TST polygon and then fold it
– Spirale Reversi (Isenberg&Snoeyink)
• Read CLERS string backward and build mesh in reverse order
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 20
EB decompression: how come it works?
Receive the CLERS sequence
Decode it
Construct the TST polygon
RR R
LC
L E
S R
E
CR
Decode&reconstruct vertices
…CRSRLECRRRLE
How to fold the polygon?
“Wrap&Zip decompression of the connectivity of triangle meshes compressed with Edgebreaker,”
J. Rossignac and A. Szymczak. Computational Geometry: Theory and Applications, 14(1-3):119-135, 1999.
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 21
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:
C
L
E
R
S
seed
Then ZIP all pairs of adjacent bounding edges when both point away
from their common vertex.
CRSRLECRRRLE
R RR
L C
L E
S R
E
R
C
R RR
L C
L E
S R
E
R
C
RR R
LC
L E
E RS R
C
Linear time complexity. Zip only after L and E.
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
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Wrap&Zip more complex example
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 23
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
compression
clers = …CCRRCCRRRCRRCRCRRCCCRRCRRCRCRRRCRCRCRRSCRRSLERERLCRRRSEE
reversi = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC…
decompression
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 24
E
Reversi details
1
O=E
1
1
O = EES
O = EE
O = EESRRRCLR
1
2
O = EESRRRCLRER
1
1
2
O = EESRRRCLRERELSRRC
O = EESRRRCLREREL
O = EESRRRCLRERELSRRCS
O = EESRRRCLRERELSRRCSRRCRCRCRRRCRCRRCRRCCCRRCRCRRCRRRCCRRCC…
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 25
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
SM, June 2002
05 Edgebreaker , 26
Spiraling solutions
• Several approaches visit the same spiraling TST a
Itai,Rodeh: Representation of graphs, Acta Informatica, 82
Keeler,Westbrook: Short encoding of planar graphs and maps, Discrete Applied Math, 93
Gumbold,Straßer: Realtime Compression of Triangle Mesh Connectivity, Siggraph, 98
Rossignac: Edgebreaker: Compressing the incidence graph of triangle meshes, TVCG, 99
Touma,Gotsman: Triangle Mesh Compression, GI, 98
Taubin,Rossignac: Geometric compression through topological surgery, ACM ToG, 98
• They encode how each new triangle is attached to previously restored ones
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 27
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
SM, June 2002
05 Edgebreaker , 28
Edgebereaker compression contributors
Safonova
(CMU):
Holes, code
Rossignac
(Atlanta):
Edgebreaker
King
(Atlanta):
1.84Tbits,
quads
Gumhold
(Germany):
1.80T bits
Szymczak
(Atlanta):
regularity,
resampling
Shikhare
(India):
translation
Attene (Italy):
retiling
Lopes (Brasil):
Gotsman (Israel):
Handles
Polygons
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
Coors
(Germany):
Prediction
Isenburg
(UCS):
Reversi
05 Edgebreaker , 29
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
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 30
Guaranteed 1.80T bit(Gumhold 00)
“New bounds on the encoding of planar triangulations”, S. Gumhold,
Siggraph course notes on “3D Geometry Compression”
• 1.8T bits guaranteed for encoding CLERS string
• Exploits the length of the outer boundary of T-patch (>2)
– Not convenient for treating non-manifolds (See later)
• CE is impossible
– Was at least 3, C increased it to at least 4, can’t have an E
• CCRE is impossible
– Was at least 3, CC increased it to at least 5, R reduced it by 1, can’t have
an E
• These constraints impact the probability of the next symbol
and improve coding
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 31
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
SM, June 2002
05 Edgebreaker , 32
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
?
x
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 33
Encoding polygon meshes, 5P bits
D. King, J. Rossignac, and A. Szymczak, “Connectivity compression for irregular quadrilateral meshes,” Technical Report TR–99–36,
GVU, Georgia Tech, 1999.
• Triangulate each polygon as a fan and encode as CLERS
–
–
–
–
Record which edges are added (1 bit per triangle)
Guaranteed cost: min(5V, 5P) bits using primal or dual
Guaranteed cost: 2.5 bits per edge
Exploit planarity for geometry prediction
C
L
S
E
M. Isenburg and J. Snoeylink, “Face fixer: Compressing polygon meshes with properties,” in Siggraph 2000, Computer Graphics
Proceedings, 2000, pp. 263–270.
B. Kronrod and C. Gotsman,“Efficient Coding of Non-Triangular Meshes”, Technical Report, Computer Science Department,
Technion-Israel Institute of Technology, 1999.
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 34
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
SM, June 2002
05 Edgebreaker , 35
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
SM, June 2002
05 Edgebreaker , 36
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
SM, June 2002
05 Edgebreaker , 37
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
SM, June 2002
05 Edgebreaker , 38
Non-Manifolds
2
3
2
1 3
0 4
• Solid models have non-manifold edges and vertices
1
• 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
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 39
5
4
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
Vl
c.p
c.n
c
c.v
v
X
cGc
E
Vr
c.o
c
Figure 2: Connectivity guessed by
parallelogram prediction
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 40
Delphi correct guesses
g(c)
X
g(c)
g(c)
X
X
Guess C
Guess L
g(c)
Guess R
X
X
Guess S
Depending on the model,
between 51% and 97%
of guesses are correct.
g(c)
Guess E
Figure 3: Guess clers Symbol
based on geometry prediction.
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
83% correct guesses:
1.47bpv = 0.74T bits
05 Edgebreaker , 41
Delphi: Wrong non-C guesses
Guess wrong L
Guess wrong R
g(c)
g(c)
X
Situation L
g(c)
g(c)
X
Situation C
X
Situation S
X
g(c)
X
X
X
Situation R
Situation C
Situation S
Half of the wrong guesses are Cs mistaken for Rs
Guess wrong S
g(c)
g(c)
g(c)
Guess wrong E
X
g(c)
X
g(c)
X
g(c)
X
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
SM, June 2002
05 Edgebreaker , 42
Delphi wrong C-guesses
Guess C in
28% of wrong
guesses are Rs
mistaken for Cs.
X
g(c)
Situation R
X
g(c)
Situation S
X
g(c)
Situation L
X
g(c)
Situation E
Figure 4: Wrongly guesses C triangles
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 43
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
SM, June 2002
05 Edgebreaker , 44
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
SM, June 2002
05 Edgebreaker , 45
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
SM, June 2002
05 Edgebreaker , 46
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
SM, June 2002
05 Edgebreaker , 47
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
L
– 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
180¡
180¡+
x
x
Jarek Rossignac, CoC & GVU Center, Georgia Tech
L 3/2
SM, June 2002
C
05 Edgebreaker , 48

Swing-Wrapper resolution control
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 49
SwingWrapper results: 0.4Tb total (0.01%)
C
L
E
R
S
134,074T
WRL=4,100,000B
13,642T
1505T
L2 error 0.007%
L2 error 0.15%
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
SM, June 2002
05 Edgebreaker , 50
Triangles surrounded by valence 6 vertices
Jarek Rossignac, CoC & GVU Center, Georgia Tech
SM, June 2002
05 Edgebreaker , 51
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
SM, June 2002
05 Edgebreaker , 52