1. Rate-Distortion Optimization for Geometry Compression of Triangular Meshes
Frédéric Payan
PhD Thesis
Supervisor : Marc Antonini
I3S laboratory - CReATIVe Research Group
Université de Nice - Sophia Antipolis
Sophia Antipolis - FRANCE

2. Motivations
Goal :
propose an efficient compression algorithm for highly detailed triangular meshes
Objectives :
High compression ratio
Rate-Quality Optimization
Multiresolution approach
Fast algorithm

3. Summary Background Distortion criterion for multiresolution meshes
Optimization of the Rate-Distorsion trade-off
Experimental results
Conclusions and perpectives

4. Summary Background Triangular Meshes Remeshing
I. Background
Summary
Background
Triangular Meshes
Remeshing
Multiresolution analysis
Compression
Bit allocation

5. Triangular Meshes 3D modeling Applications : Medecine CAD Map modeling
I. Background
Triangular Meshes
3D modeling
Applications :
Medecine
CAD
Map modeling
Games
Cinema
Etc.

6. Irregular meshes valence different of 6 => 2 informations :
I. Background
Irregular meshes
valence different of 6
=> 2 informations :
Geometry (vertices)
Connectivity (edges)
4 neighbors
5 neighbors
9 neighbors

7. More than 380 millions of triangles => several Gigabytes (Michelangelo Project, 1999)
Examples
40,000 triangles => Mb
99,732 triangles => Mb

8. => Considered solution : Semi-regular remeshing
I. Background
Irregular meshes (2)
Multiresolution Analysis :
Without connectivity modification => wavelet transform for irregular meshes (S.Valette et R.Prost, 2004)
A mesh is only one instance of the surface geometry => Remeshing
goal : regular and uniform geometry sampling
=> Considered solution : Semi-regular remeshing

9. Summary Background Triangular Meshes Remeshing
I. Background
Summary
Background
Triangular Meshes
Remeshing
Multiresolution analysis
Compression
Bit allocation

10. Semi-regular remeshing
I. Background
Semi-regular remeshing
Simplification
Irregular mesh
Coarse mesh
Subdivision
Semi-regular mesh
Coarse mesh
Finest semi-regular version
Subdivised mesh (1)
Original mesh

11. Semi-regular remesher
I. Background
Semi-regular remesher
MAPS (A. Lee et al. , 1998)
Coarse mesh (geometry+connectivity)
N sets of 3D details (geometry) => 3 floating numbers
Normal Meshes (I. Guskov et al., 2000)
N’ sets of 3D details (geometry) => 1 floating number

12. => More compact representation
I. Background
Normal Meshes
Known direction: normal at the surface
Surface to remesh
=> More compact representation

13. Summary Background Triangular Meshes Remeshing
I. Background
Summary
Background
Triangular Meshes
Remeshing
Multiresolution analysis
Compression
Bit allocation

14. Multiresolution analysis
I. Background
Multiresolution analysis …
Details
Details
Details
Details
Multiresolution Representation:
Low frequency (LF) mesh (geometry + topology)
N sets of wavelet coefficients (3D vectors) (geometry)

15. Summary Background Triangular Meshes Remeshing
I. Background
Summary
Background
Triangular Meshes
Remeshing
Multiresolution analysis
Compression
Bit allocation

16. I. Background
Compression
Objective : reduce the information quantity useful for representing numerical data
2 approachs : Lossy or lossless compression
High compression ratii
=> Lossy compression

17. Target bitrate or distortion
I. Background
Compression scheme
Semi-regular
Wavelet coefficients
Entropy Coding Q
1010…
Transform
Bit Allocation
Target bitrate or distortion
Remeshing
Optimize the Rate-Distortion (RD) tradeoff
Preprocessing

18. Summary Background Triangular Meshes Remeshing
I. Background
Summary
Background
Triangular Meshes
Remeshing
Multiresolution analysis
Compression
Bit allocation

19. I. Background
Bit allocation: goal
Optimization of the tradeoff between bitstream size and reconstruction quality:
minimize D(R) or
minimize R(D) D R

20. Bit allocation and meshes
I. Background
Bit allocation and meshes
Related Works (geometry compression):
Zerotree coding
PGC : Progressive Geometry Compression (A. Khodakovsky et al., 2OOO)
NMC :
Normal Mesh Compression ( A. Khodakovsky et I. Guskov, 2002).
=> Stop coding when bitstream given size is reached.
Estimation-quantization (EQ) coding
MSEC : Geometry Compression of Normal Meshes Using Rate-Distortion Algorithms (S. Lavu et al., 2003)
=> Local RD optimization.

21. Proposed bit allocation
I. Background
Proposed bit allocation
Low computational complexity
Improve the quantization process
Maximize the quality of the reconstructed mesh according to a given target bitrate
=> Which distortion criterion for evaluating the losses?

22. Summary Background Distortion criterion for multiresolution meshes
Optimization of the Rate-Distorsion trade-off
Experimental results
Conclusions and perpectives

23. Target bitrate or distortion
II. Distortion criterion for multiresolution meshes
Coding/Decoding
Semi-regular
Entropy coding Q
1010…
Transform
Remeshing
Bit Allocation
Target bitrate or distortion
Preprocessing
Inverse
transform
Entropy Decoding Q*
Quantized semi-regular

24. Considered distorsion criterion
II. Distortion criterion for multiresolution meshes
Considered distorsion criterion
MSE due to quantization of the semi-regular mesh
Number of vertices
semi-regular vertices
quantized semi-regular vertices
MSE for one subband
Wavelet => ?

25. II. Distortion criterion for multiresolution meshes
Related works
K.Park and R.Haddad (1995)
M-channel scheme
quantization model : “noise plus gain”
B.Usevitch (1996)
quantization model : “additive noise”
N decomposition levels
Sampled on square grids
Filter bank
Problem : - non adapted for lifting scheme ! usable for any sampling grid ?

26. Lifting scheme for meshes
II. Distortion criterion for multiresolution meshes
Lifting scheme for meshes
3 prédiction operators P
=> wavelet coefficients
3 update operators U
=> LF mesh
Triangular grid => 4 channels

27. Triangulaire sampling
II. Distortion criterion for multiresolution meshes
Triangulaire sampling
1 triangular grid => 4 cosets n1 n2 2 3
LF subband (0) 1
HF subband 1
HF subband 2
HF subband 3

28. 4-channel lifting scheme: analysis
II. Distortion criterion for multiresolution meshes
4-channel lifting scheme: analysis + + + LF
-P1 U1 +
HF 1
split
-P2 U2 +
HF 2
Semi-regular mesh
-P3 U3 +
HF 3

29. 4-channel lifting scheme: synthesis
II. Distortion criterion for multiresolution meshes
4-channel lifting scheme: synthesis LF + + + P -U
HF 1 + -U P
Merge
HF 2 +
Semi-regular mesh -U P
HF 3 +
=> Derivation of the MSE of the quantized mesh according to the quantization error of each 4 subband

30. Proposed Method Input signal :
II. Distortion criterion for multiresolution meshes
Proposed Method
Input signal :
Quantization error model : « additive noise »
S is one realization of a stationar and ergodic random process => deterministic quantity
=> MSE of the input signal

31. Proposed Method: Hypothesis
II. Distortion criterion for multiresolution meshes
Proposed Method: Hypothesis
Uncorrelated error in each subband
Subband errors mutually uncorrelated
Synthesis filter energy
Quantization error energy

32. Proposed Method: principle
II. Distortion criterion for multiresolution meshes
Proposed Method: principle
Synthesis filter energy
Polyphase components of the filters
Cauchy theorem
Quantization error energy
Uncorrelated error in each subband

33. Proposed Method: solution
II. Distortion criterion for multiresolution meshes
Proposed Method: solution
For 1 decomposition level
MSE of the subband i
Weights relative to the non-orthogonal filters
with
Polyphase component

34. polyphase representation
II. Distortion criterion for multiresolution meshes
polyphase representation
Lifting scheme:
=> Polyphase components depend on only the prediction and update opérators
New formulation : => can be applied easily to lifting scheme

35. Proposed Method : solution
II. Distortion criterion for multiresolution meshes
Proposed Method : solution
For N decomposition levels
avec et

36. Outline This formulation can be applied to lifting scheme
II. Distortion criterion for multiresolution meshes
Outline
This formulation can be applied to lifting scheme
Global formulation of the weights for any :
Grid and related subsampling
number of channels M
Number of decomposition levels N

37. => PSNR Gain : up to 3.5 dB
II. Distortion criterion for multiresolution meshes
Experimental Results
=> PSNR Gain : up to 3.5 dB

38. Visual impact Without the weights Original With the weights
II. Distortion criterion for multiresolution meshes
Visual impact
Without the weights
Original
With the weights

39. Target bitrate or distortion
II. Distortion criterion for multiresolution meshes
Coding/Decoding
Semi-regular
Entropy coding Q
1010…
Transform
Remeshing
Bit Allocation
Target bitrate or distortion
Preprocessing
Inverse
transform
Entropy Decoding Q*
Quantized semi-regular

40. => Is the MSE suitable to control the quality?
II. Distortion criterion for multiresolution meshes
MSE and irregular mesh
Quality of the reconstructed mesh :
Reference : irregular mesh
Used metric:
geometrical distance between two surfaces:
the «surface-to-surface distance (s2s) »
=> Is the MSE suitable to control the quality?

41. Quality of the reconstructed mesh
II. Distortion criterion for multiresolution meshes
Quality of the reconstructed mesh
Forward distance:
distance between one point and one surface:
Quantized mesh
(semi-regular)
Input mesh (irregular)

42. Simplifying approximations
II. Distortion criterion for multiresolution meshes
Simplifying approximations
Normal meshes:
=> infinitesimal remeshing error
=> uniform and regular geometry sampling
Highly detailed meshes:
=> densely sampled geometry
Relation with the quantization error?

43. Hypothesis: asymptotical case
II. Distortion criterion for multiresolution meshes
Hypothesis: asymptotical case
=>
Preservation of the LF subbands
=> normal orientations slightly modified => errors lie in the normal direction (normal meshes) θ
ε(v2 ) n’ n n’ n θ
ε(v2 )

44. II. Distortion criterion for multiresolution meshes
Proposed heuristic
Approximating formulation:
Asymptotical case + normal meshes
=> MSE : suitable criterion to control the quality of the reconstructed mesh

45. Summary Background Distortion criterion for multiresolution meshes
Optimization of the Rate-Distorsion trade-off
Experimental results
Conclusions and perpectives

46. Optimization of the Rate-Distorsion trade-off
III.Optimization of the Rate-Distorsion trade-off
Optimization of the Rate-Distorsion trade-off
Objective :
find the quantization steps that maximize the quality of the reconstructed mesh
Scalar quantization (less complex than VQ)
3D Coefficients => data structuring?

47. Local frames Normal at the surface: z-axis of the local frame
III.Optimization of the Rate-Distorsion trade-off
Local frames
Normal at the surface: z-axis of the local frame
=> Coefficient :
Tangential components (x and y-coordinates)
Normal components (z-coordinates) z x
Global frame x z
Local frame

48. Histogram of the polar angle
III.Optimization of the Rate-Distorsion trade-off
Histogram of the polar angle
Local frame: θ x y z 0°
90°
180°
=> Components treated separately (2 scalar subbands)
=> Most of coefficients have only normal components

49. MSE of one subband i MSE relative to the tangential components
III.Optimization of the Rate-Distorsion trade-off
MSE of one subband i
MSE relative to the tangential components
MSE relative to the normal components

50. How solving the problem?
III.Optimization of the Rate-Distorsion trade-off
How solving the problem?
Find the quantization steps and lambda that minimize the following lagrangian criterion:
Method:
=> first order conditions
Distortion
Constraint relative to the bitrate

51. Solution Need to solve (2N + 4) equations with (2N + 4) unknowns
III.Optimization of the Rate-Distorsion trade-off
Solution
Need to solve (2N + 4) equations with (2N + 4) unknowns
PDF of the component sets: Generalized Gaussian Distribution (GGD) => model-based algorithm (C. Parisot, 2003)

52. Model-based algorithm
III.Optimization of the Rate-Distorsion trade-off
Model-based algorithm
compute the variance and α for each subband
compute the bitrates for each subband λ
Complexity : 12 operations / semi-regular
Example : 0.4 second (PIII 512 Mb Ram)
=> Fast process.
Target bitrate reached?
Look-up tables
new λ
compute the quantization step of each subband

53. Summary Background Distortion criterion for multiresolution meshes
Optimization of the Rate-Distorsion trade-off
Experimental results
Conclusions and perpectives

54. Connectivity coding* (coarse mesh connectivity)
IV. Experimental results
Compression scheme
1010…
Connectivity coding* (coarse mesh connectivity)
MPX
Unlifted Butterfly SQ
3D-CbAC*
Normal meshes
Bit Allocation
Target Bitrate
Preprocessing
* Context-based Bitplane Arithmetic Coder (EBCOT-like)
* Touma-Gotsman coder

55. Input mesh (irregular)
IV. Experimental results
Visual results
Input mesh (irregular)
CR = bits/iv
CR = bits/iv
CR = bits/iv
Compression ratio:

56. Comparison Quality criterion : State-of-the-art methods:
IV. Experimental results
Comparison
Quality criterion :
State-of-the-art methods:
NMC (Normal meshes + Butterfly NL + zerotree)
EQMC (Normal meshes + Butterfly NL + EQ)
PGC (MAPS + Loop)
Bounding box diagonal
s2s between the irregular input mesh and the quantized semi-regular one

57. PSNR-bitrate curve: Rabbit
IV. Experimental results
PSNR-bitrate curve: Rabbit

58. PSNR-bitrate curve: Feline
IV. Experimental results
PSNR-bitrate curve: Feline
=> PSNR Gain: up to 7.5 dB

59. PSNR-bitrate curve: Horse
IV. Experimental results
PSNR-bitrate curve: Horse

60. Geometrical comparison
IV. Experimental results
Geometrical comparison
NMC (62.86 dB)
Proposed algorithm (65.35 dB)
Bitrate = 0.71 bits/iv

61. Summary Background Distortion criterion for multiresolution meshes
Optimization of the Rate-Distorsion trade-off
Experimental results
Conclusions and perpectives

62. => Better results than the state-of-the-art methods.
V. Conclusions and perspectives
Conclusions
New shape compression method:
Contributions :
Weighted MSE : suitable distortion criterion
Original formulation of the weights (suitable in case of lifting scheme)
Bit alllocation of low computational complexity that optimizes the quality of a quantized mesh.
An original Context-based Bitplane Arithmetic Coder
=> Better results than the state-of-the-art methods.

63. V. Conclusions and perspectives
Take into account some visual properties the human eye appreciates (local curvature, volume, smoothness…) Reference : Z.Karni and C.Gotsman, 2000
Algorithm for huge meshes: « on the flow » compression Reference : A. Elkefi et al., 2004

64. End