Multiresolution Analysis for Irregular Meshes Application to Surface Denoising

1. Multiresolution Analysis for Irregular MeshesApplication to Surface Denoising Michaël Roy

2. Outline • Introduction to Multiresolution Analysis • Multiresolution Analysis for Irregular Meshes • Scheme • Results • Surface Denoising • Scheme • Results • Conclusion and Future Work

3. What is Multiresolution Analysis? • Introduced by Stéphane Mallat in 1987 • Time (Space) / Frequency representation • Represents general functions in terms of simpler, fixed blocks at different scales • General framework • Wavelet transform • Sub-band coding • Quadrature mirror filters • Pyramid scheme S. Mallat, A Theory for Multiresolution Signal Decomposition: A Wavelet Representation, IEEE Transactions on PAMI, 11(7), 1989

4. What are the advantages? • Efficiency ! • Linear complexity O(n) • Time (space) / frequency localization • Scalability • Application in numerous areas • mathematics • engineering • computer science • statistics • physics • etc.

5. V3 V2 W2 Analysis (Decomposition) Synthesis (Reconstruction) V1 W1 V0 W0 Vi Approximation Wi Details How does it work?

6. V4 W3 V3 W2 V2 A simple example (1/2)

7. Initial data : 9 7 3 5 Average : 8 4 1 Difference with the average : 1 -1 Average : 6 2 Difference with the average : 2 Analysis (Decomposition) Synthesis (Reconstruction) Resolution Approximation Details 2 9 7 3 5 1 8 41 -1 0 62 Transformed data : 62 1 -1 A simple example (2/2)

8. Lifting Scheme • Introduced by Wim Sweldens in 1995 level m-1 even level m split predictor predictor merge odd details Decomposition Reconstruction W. Sweldens, The lifting scheme: A construction of second generation wavelets, SIAM Journal on Mathematical Analysis, 35(6), 1998

9. Semi-regular meshes Zorin (1997) Lounsbery (1995) Irregular meshes Guskov (1999) Kobbelt (1998) Literature Review

10. Multiresolution Analysis for Irregular Meshes • Introduced by Igor Guskov in 1999 Note: removes one vertex per step level m-1 Simplification Subdivision level m details I. Guskov, W. Sweldens, and P. Schröder, Multiresolution Signal Processing for Meshes, Proceedings of ACM SIGGRAPH, 1999

11. Improvement of the Subdivison • Guskov's Subdivision • Minimizes the second order differences • Requires the 1-ring neighborhood and flaps • Requires a local parametrization • Improved Subdivision • Uses new discrete differential-geometry operator • Minimizes the curvature • Requires the 1-ring neighborhood • No parametrization M. Meyer, M. Desbrun, and P. Schröder, Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, Proceedings of Visualization and Mathematics, 2002

12. Meshes with Attributes • Geometrical attributes • vertex position • normal vectors • curvature • Appearance attributes • colors • texture

13. Attribute Multiresolution Analysis • Geometrical Analysis • Attribute Analysis Note: Assumes the attributes are linked to the surface M. Roy, S. Foufou, A. Koschan, F. Truchetet, and M. Abidi, Multiresolution Analysis for Irregular Meshes with Appearance Attributes, Submitted to IEEE ICIP, 2003

14. level m level m-1 Improvement of the Decomposition • Guskov‘s analysis removes one vertex per level • Improvement using global downsampling Removes an independent set of vertices per level

15. Initial model (82 000 faces) Base level (16 faces) Uniform Laplacian subdivision Our subdivision Results (1/4) • Subdivision Convergence

16. Initial model Low-pass filter Stop-band filter Enhance filter Results (2/4) • Frequency Filtering

17. Initial model Geometric analysis Color analysis min max Results (3/4) • Attribute Analysis

18. Geometric analysis Impulse noise Noise detection Normal analysis Results (4/4) • Impulse Noise Detection

19. Surface Denoising • Scanned models contain measurement errors • Denoising or smoothing ? • Smoothing removes the high frequencies and retain the low • Denoising attempts to remove whatever noise is present and retains whatever signal is present regardless of the frequency content

20. Model of a Noisy Surface • Noisy surface  noisy detail coefficients • Assumptions for the denoising algorithm • Noise and detail coefficients are independent • The noise is an additive white Gaussian noise

21. Denoising Scheme • Estimate “clean” detail coefficient variance • Denoising using the Wiener filtering M. Roy, S. Foufou, A. Koschan, F. Truchetet, and M. Abidi, Surface Denoising for Irregular Meshes using Wiener Filtering, Submitted to ICCV, 2003

22. Laplacian smoothing Curvature smoothing Our method Results (1/2)

23. Initial model Our method Results (2/2)

24. Conclusion • Multiresolution analysis for irregular meshes • Improvement of the non-uniform subdivision • Improvement of the decomposition • Attribute analysis • Surface denoising • Future work • Formalization of the analysis (lifting scheme) • Feature detection using attribute analysis • Improvement of the denoising algorithm (anisotropic)

25. Michaël Roy Multiresolution Analysis for Irregular MeshesApplication to Surface Denoising