Compression opportunities using progressive meshes

1
Compression opportunities using progressive meshes

• Hugues Hoppe
• Microsoft Research
• SIGGRAPH 98 course 3D Geometry compression

2
Triangle Meshes
3
Triangle Meshes
v2,f1 (nx,ny,nz) (u,v)v2,f2 (nx,ny,nz)
(u,v)
corner attrib.
4
Complex meshes
43,000 faces
lots of faces!
Challenges - rendering - storage - transmission
geometrycompression
5
Talk outline

• Progressive mesh (PM) representation
• Analysis of PM compression
• Improved PM compression
• Progressive simplicial complex (PSC) repr.

6
Progressive mesh representation

• Basic idea
• Simplify arbitrary mesh through sequence of
  transformations.
• Record
• simplified meshsequence of inverse
  transformations

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Simplification Edge collapse
ecol(vs ,vt , vs )

vt
vl
vl
vr
vr
vs

vs
13,546
500
152
150 faces
M0
M1
M175
Mn
ecol0
ecoln-1
ecoli
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Invertible! Vertex split transformation
attributes
vspl(vs ,vl ,vr , vs ,vt ,)


vt

vl
vr
vl
vr
vs
vs

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Reconstruction process
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PM benefits
PM
Mn
Vn
lossless
M0
Fn
vspl
attributes

• progressive transmission
• continuous-resolution
• smooth LOD
• geometry compression

• single resolution

11
Application Progressive transmission

• Transmit records progressively

time
M0
Receiver displays
M0
( progressive GIF JPEG)
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Application Continuous-resolution LOD

• From PM, extract Mi of any desired complexity.

3,478 faces?
M0
vspl0
vspl1
vspli-1
vspln-1
Mi
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Property Vertex correspondence
Mc
Mf
M0
v1
v1
Mn
v2
v2
v3
v3
v4
v5
v6
v7
v8
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Application Smooth transitions
Mc
Mf
M0
v1
v1
Mn
v2
v2
v3
v3
v4
v5
Mfc
v6
v7
V
F
V
v8
can form a smooth visual transition geomorph
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BUT, geometry compression?
M0
vspl0
vspl1
vspli-1
vspln-1
Mn

• M0 is typically small
• key is encoding of vspli

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Vertex split encoding
Record
vspli (vs ,vl ,vr ,vs ,vt ,)

• vs (log2i bits)

• vl vr (5 bits)

vt

vl
vl
vr
vr
vs
vs

Analysis

• connectivity n(log2n4) bits vs. n(6log2n) bits

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Vertex split encoding
Record
vspli (vs ,vl ,vr ,vs ,vt ,)

• vs (log2i bits)

• vl vr (5 bits)

vt

vl
vl
vr
vr
vs
vs

Analysis

• connectivity n(log2n4) bits vs. n(6log2n) bits

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Vertex split encoding
Record
vspli (vs ,vl ,vr ,vs ,vt ,)

• vs (log2i bits)

• vl vr (5 bits)

vt

vl
vl
vr
vr
vs
vs

• predict face attrib.

Analysis

• connectivity n(log2n4) bits vs. n(6log2n) bits

• geometry 40n bits vs. 96n bits

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Summary of vsplit encoding
?
(n is vertices, 2n is faces)
42,712 facesn21,373151 Kbytes (ignoring
corners)
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Improved PM compression

• Connectivity
• group vsplits ? forest splits Taubin etal98
• permute vsplits
• Geometry
• apply smoothing Taubin etal98
• local prediction single delta
• Face attributes
• already negligible
• Corner attributes
• wedge data structure

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Compression of connectivity

• Detail flclw , nrot

vl
vr
vs

• Problem locating vsplit on mesh (using either
  flclw or vs ) requires log2i bits.

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Progressive Forest Split (PFS)
Taubin,Gueziec,Horn,Lazarus98
M0
vspl0
vspl1
vspl5
vspln-1
vspl2
vspl3
vspl4

• PM n(4log2n) bits

• PFS n(8..10) bits

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Other solution permutation of vsplits
M0
flclw,1
flclw,2
M0
?flclw

• We record ?flclw and minimize ?flclw by permuting
  vsplits.

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Legal vsplit permutations

• Determine dependencies between vsplits
• Xia Varshney 96
• Hoppe 97
• vsplit is candidate if it has no dependencies.
• Greedy algorithm
• Maintain candidate vsplits in balanced tree,
  sorted by flclw .
• Remove vsplit with smallest ?flclw and update
  candidate tree.

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Result of permuting vsplits
9.7n
(log2n4)n
9.7n10.3n
(now linear)

• Drawback intermediate meshes Mi (0ltiltn) lose
  geometric accuracy.
• O(n log n) bits to undo permutation.

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Layered permutations
(mesh complexity increasing exponentially)
checkpoints
M0
vspl0
vspl1
vspln-1
vspl2
vspl3
vspl4
vspl5
vspl6
vspl7
vspl8
vspl9
vspl10
M0
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Results using layered permutations
checkpoints
growth factor
connectivitybits (nverts)
1
549
9.7n
0.1 bit/vert
9
2.00
9.8n
13
1.63
9.8n
visuallyidentical tooriginal PM !
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1.40
9.9n
24
1.30
10.0n
35
1.20
10.2n
66
1.10
10.5n
20,373
1.00
16.4n
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Geometry local prediction delta
vt

vs

vs

• Predict position of vt .

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Result of predicted delta
9.7n
20.9n
?

• Intermediate meshes Mi (0ltiltn) have minor loss
  in geometric accuracy.

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Corner attributes
vertex
face
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Wedge data structure
vertex
wedge
face
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Vsplit encoding of wedges
(6 new corners)
1..6 wedge attribute deltas
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Results using wedges
9.7n
20.9n
23.7n
10 corner continuity booleans 2.5n bits wedge
attribute deltas 21.2n bits
(16-bit nx,ny,nz at corners)
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Estimating normals from wedges
9.7n
20.9n
2.5n
88 Kbytes
original
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Progressive Simplicial Complexes

• SIGGRAPH 97
• (Joint work with Jovan Popovic)

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PM restrictions

• Supports only meshes (orientable,
  2-dimensional manifolds)
• Preserves topological type

M0
Mn
Mi
167,744
8,000
2,522
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Progressive Simplicial Complexes

• Represent arbitrary triangulations
• any dimension,
• non-orientable,
• non-manifold,
• non-regular,
• Progressively encode both geometry and topology.

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Generalization
PM
PSC
edge collapse(ecol)
vertex split(vspl)
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LOD sequence
M1
M22
M116
Mn
gvspl1
gvspli
gvspln-1
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Generalized vertex split

• Connectivity
• PM (log2n4)n bits
• PSC (log2n7)n bits

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Space analysis
geometry!
connectivity
connectivity
materials
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PSC analysis
PSC
Mn
Vn
lossless
M1
gvspl
Kn
single vertex
arbitrarysimplicial complex
() progressive geometry and topology () no
base mesh () 3 bit / vertex overhead ()
slower decompression
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Summary Progressive geometry

• Connectivity
• group vsplits ? forest splits Taubin etal98
• permute vsplits
• Geometry
• apply smoothing Taubin etal98
• local prediction single delta
• Face attributes
• already negligible
• Corner attributes
• wedge data structure

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Conclusions

• Geometry storage overwhelms connectivity,
  particularly for simplified meshes.
• Progressive representations
• reasonable compression
• benefits LOD
• Texture coordinates?

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Beyond Gouraud shading
44,000 triangles

• Future
• bump mapping
• environment mapping

Texture mapping!
200n bits (JPEG)
Cohen-etal98
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Simultaneous streaming
progressivegeometry
progressivetexture
network
runtime tradeoffof geometry texture(platform-d
ependent)
viewer / application
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References

• H. Hoppe. Progressive meshes. Computer Graphics
  (SIGGRAPH 96), pages 99-108.
• J. Popovic, H. Hoppe. Progressive simplicial
  complexes. Computer Graphics (SIGGRAPH 97),
  pages 217-224.
• H. Hoppe. Efficient implementation of
  progressive meshes. Computers Graphics, Vol.
  22, pages 27-36, 1998.
• http//research.microsoft.com/hoppe/