Bilateral Mesh Denoising

1. Bilateral Mesh Denoising Shachar FleishmanIddo DroriDaniel Cohen-Or Tel Aviv University

2. Denoising • Input (scanned) model • Additive noise + =

3. filter + = Denoising • Input (scanned) model • Additive noise • Noise free model • Preserve features

4. Image denoising • Wavelet denoising [Donoho ’95] • Anisotropic diffusion [Perona & Malik ’90] • Bilateral filter [Smith & Brady ’97], [Tomasi & Manduchi ’98] • [Black et al. ’98] • Anisotropic diffusion • Robust statistics • [Elad ’01], [Durand & Dorsey ’02] relate • Anisotropic diffusion • Robust statistics • Bilateral filter

5. Original and noisy ( 2=900) images Images courtesy of Michael Elad

6. TV filtering: 50 iterations 10 iterations (MSE=146.3339) (MSE=131.5013) Images courtesy of Michael Elad

7. Wavelet Denoising (soft) Using DB5 Using DB3 (MSE=144.7436) (MSE=150.7006) Images courtesy of Michael Elad

8. Filtering via the Bilateral 2 iterations with 1111 Sub-gradient based 55 (MSE=89.2516) (MSE=93.4024) Images courtesy of Michael Elad

9. Mesh denoising, smoothing and fairing • Adapt image denoising algorithms to meshes • Wiener filter [Peng et al. ’01] • Isotropic diffusion [Desbrun et al. ’99] • Anisotropic diffusion of height fields [Desbrun et al. ’00] • Anisotropic diffusion on meshes [Clarenz et al. ’00, Xu & Bajaj ’03] • Bilateral filter [Choudhury & Tumblin ’03] [Jones et al. ’03]

10. Bilateral mesh denoising • Fast • Simple • Intuitive parameter selection

11. * = Bilateral filtering • Gaussian filter

12. * = Bilateral filtering Denoise Feature preserving • Bilateral filter Normalization

13. Bilateral filtering of meshes

14. Bilateral filtering of meshes

15. Bilateral filtering of meshes • Height above surface is equivalent to the gray level values in images

16. Bilateral filtering of meshes • Height above surface is equivalent to the gray level values in images • Apply the bilateral filter to heights

17. Bilateral filtering of meshes • Height above surface is equivalent to the gray level values in images • Apply the bilateral filter to heights • Move the vertex to its new height

18. Bilateral filtering of meshes • Height above surface is equivalent to the gray level values in images • Apply the bilateral filter to heights • Move the vertex to its new height • In practice: • Sharp features

19. Bilateral filtering of meshes • Height above surface is equivalent to the gray level values in images • Apply the bilateral filter to heights • Move the vertex to its new height • In practice: • Sharp features • The noise-freesurface is unknown

20. n P L Solution • A plane that passes through the point is the estimator to the smooth surface • Plane L=(p,n)

21. n P L Similarity closeness Solution • A plane that passes through the point is the estimator to the smooth surface • Plane L=(p,n)

22. Computing the plane • The approximating plane should be: • A good approximation to the surface • Preserve features • Average of the normal to faces in the 1-ring neighborhood

23. DenoisePoint(Vertex v, Normal n) {qi} = neighborhood(v) K=|{qi}| sum=0 normalizer=0 for i := 1 to K t = ||v-qi|| h = <n,v-qi> Wc=exp(-t2/(2σc2)) Ws=exp(-h2/(2σs2)) Sum +=(wc*ws)h Normalizer += wc*ws End Return v+n*(sum/normalizer) v

24. DenoisePoint(Vertex v, Normal n) {qi} = neighborhood(v) K=|{qi}| sum=0 normalizer=0 for i := 1 to K t = ||v-qi|| h = <n,v-qi> Wc=exp(-t2/(2σc2)) Ws=exp(-h2/(2σs2)) Sum +=(wc*ws)h Normalizer += wc*ws End Return v+n*(sum/normalizer) iterate over neighborhood v

25. DenoisePoint(Vertex v, Normal n) {qi} = neighborhood(v) K=|{qi}| sum=0 normalizer=0 for i := 1 to K t = ||v-qi|| h = <n,v-qi> Wc=exp(-t2/(2σc2)) Ws=exp(-h2/(2σs2)) Sum +=(wc*ws)h Normalizer += wc*ws End Return v+n*(sum/normalizer) closeness v q

26. DenoisePoint(Vertex v, Normal n) {qi} = neighborhood(v) K=|{qi}| sum=0 normalizer=0 for i := 1 to K t = ||v-qi|| h = <n,v-qi> Wc=exp(-t2/(2σc2)) Ws=exp(-h2/(2σs2)) Sum +=(wc*ws)h Normalizer += wc*ws End Return v+n*(sum/normalizer) height – similarity v q

27. DenoisePoint(Vertex v, Normal n) {qi} = neighborhood(v) K=|{qi}| sum=0 normalizer=0 for i := 1 to K t = ||v-qi|| h = <n,v-qi> Wc=exp(-t2/(2σc2)) Ws=exp(-h2/(2σs2)) Sum +=(wc*ws)h Normalizer += wc*ws End Return v+n*(sum/normalizer) v weights

28. DenoisePoint(Vertex v, Normal n) {qi} = neighborhood(v) K=|{qi}| sum=0 normalizer=0 for i := 1 to K t = ||v-qi|| h = <n,v-qi> Wc=exp(-t2/(2σc2)) Ws=exp(-h2/(2σs2)) Sum +=(wc*ws)h Normalizer += wc*ws End Return v+n*(sum/normalizer) v Move the vertex in the normal direction

29. Parameters • The two parameters to the weight function: σc, σs • Interactively select a point p and the neighborhood radius ρ • σc =1/2 ρ • σs = stdv(Nbhd(p, ρ)) • Number of Iterations

30. Robustness • Sharp features are treated as outliers

31. Robustness • Sharp features are treated as outliers • The bilateral filter does not recover smoothed signal

32. Results Bilateral mesh denoising Anisotropic denoising of height fields - Desburn ’00 Source

33. Results Anisotropic Geometric Diffusion in Surface Processing - Clarenz ‘00 Bilateral mesh denoising Source

34. Results Two iterations Five iterations Source

36. Future Work • Adapt the algorithm to point sets • Robust estimator of normals

37. Acknowledgements • Models and images courtesy of Jean-Yves Bouguet, Mathieu Desbrun, Alexander Belyaev, Christian Rossl from Max Planck Insitut fur Informatik, Udo Diewald and Michael Elad • Israel Science Foundation funded by the Israel Academy of Sciences and Humanities • Israeli Ministry of Science • A grant from the German Israel Foundation (GIF).

38. Input Bilateral mesh denoising Non-iterative, Feature Preserving Mesh smoothing

39. Bilateral mesh denoising Non-iterative, Feature Preserving Mesh smoothing Source

40. Bilateral mesh denoising Non-iterative, Feature Preserving Mesh smoothing

41. Comparison - predictors Bilateral mesh denoising Non-iterative, Feature Preserving Mesh smoothing

42. New results Bilateral mesh denoising Extended Bilateral mesh denoising