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Rate-Distortion Optimization for Geometry Compression of Triangular Meshes

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. Motivations. Goal :

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Rate-Distortion Optimization for Geometry Compression of Triangular Meshes

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

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

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

  7. 40,000 triangles=> + 0.45 Mb 99,732 triangles=> + 1.1 Mb More than 380 millions of triangles => several Gigabytes (Michelangelo Project, 1999) Examples

  8. I. Background Irregular meshes (2) • MultiresolutionAnalysis : • 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. I. Background Summary • Background • Triangular Meshes • Remeshing • Multiresolution analysis • Compression • Bit allocation

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

  11. 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) • Coarse mesh (geometry+connectivity) • N’ sets of 3D details (geometry) => 1 floating number

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

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

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

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

  18. 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. 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. I. Background Proposed bit allocation • Low computational complexity • Improve the quantization process • Maximize the quality of the reconstructed meshaccording 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. II. Distortion criterion for multiresolution meshes Bit Allocation Inverse transform Coding/Decoding Semi-regular Entropy coding Q 1010… Transform Remeshing Target bitrate or distortion Preprocessing Entropy Decoding Q* Quantized semi-regular

  24. II. Distortion criterion for multiresolution meshes Wavelet => ? 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

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

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

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

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

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

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

  40. 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-surfacedistance (s2s) » => Is the MSE suitable to control the quality?

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

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

  44. II. Distortion criterion for multiresolution meshes Approximating formulation: Proposed heuristic Asymptotical case+ normal meshes => MSE : suitable criterion to controlthe 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. 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. III.Optimization of the Rate-Distorsion trade-off z z x x z z Global frame x x Local frame Local frames • Normal at the surface: z-axis of the local frame => Coefficient : • Tangential components (x and y-coordinates) • Normal components (z-coordinates)

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

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

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