Bilateral Mesh Denoising Shachar Fleishman Iddo Drori Daniel Cohen-Or Tel Aviv University

Denoising Input (scanned) model –Additive noise +=

Denoising Input (scanned) model –Additive noise Noise free model –Preserve features += filter

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

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

TV filtering: 50 iterations 10 iterations (MSE= ) (MSE= ) Images courtesy of Michael Elad

Wavelet Denoising (soft) Using DB5 Using DB3 (MSE= ) (MSE= ) Images courtesy of Michael Elad

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

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]

Bilateral mesh denoising Fast Simple Intuitive parameter selection

Bilateral filtering Gaussian filter *=

Bilateral filter Denoise Feature preserving Normalization Bilateral filtering *=

Bilateral filtering of meshes

Height above surface is equivalent to the gray level values in images

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

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

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

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-free surface is unknown

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

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

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

DenoisePoint(Vertex v, Normal n) {q i } = neighborhood(v) K=|{q i }| sum=0 normalizer=0 for i := 1 to K t = ||v-q i || h = W c =exp(-t 2 /(2σ c 2 )) W s =exp(-h 2 /(2σ s 2 )) Sum +=(w c *w s )h Normalizer += w c *w s End Return v+n*(sum/normalizer) v

DenoisePoint(Vertex v, Normal n) {q i } = neighborhood(v) K=|{q i }| sum=0 normalizer=0 for i := 1 to K t = ||v-q i || h = W c =exp(-t 2 /(2σ c 2 )) W s =exp(-h 2 /(2σ s 2 )) Sum +=(w c *w s )h Normalizer += w c *w s End Return v+n*(sum/normalizer) iterate over neighborhood v

DenoisePoint(Vertex v, Normal n) {q i } = neighborhood(v) K=|{q i }| sum=0 normalizer=0 for i := 1 to K t = ||v-q i || h = W c =exp(-t 2 /(2σ c 2 )) W s =exp(-h 2 /(2σ s 2 )) Sum +=(w c *w s )h Normalizer += w c *w s End Return v+n*(sum/normalizer) closeness v q

DenoisePoint(Vertex v, Normal n) {q i } = neighborhood(v) K=|{q i }| sum=0 normalizer=0 for i := 1 to K t = ||v-q i || h = W c =exp(-t 2 /(2σ c 2 )) W s =exp(-h 2 /(2σ s 2 )) Sum +=(w c *w s )h Normalizer += w c *w s End Return v+n*(sum/normalizer) height – similarity v q

DenoisePoint(Vertex v, Normal n) {q i } = neighborhood(v) K=|{q i }| sum=0 normalizer=0 for i := 1 to K t = ||v-q i || h = W c =exp(-t 2 /(2σ c 2 )) W s =exp(-h 2 /(2σ s 2 )) Sum +=(w c *w s )h Normalizer += w c *w s End Return v+n*(sum/normalizer) weights v

DenoisePoint(Vertex v, Normal n) {q i } = neighborhood(v) K=|{q i }| sum=0 normalizer=0 for i := 1 to K t = ||v-q i || h = W c =exp(-t 2 /(2σ c 2 )) W s =exp(-h 2 /(2σ s 2 )) Sum +=(w c *w s )h Normalizer += w c *w s End Return v+n*(sum/normalizer) Move the vertex in the normal direction v

Parameters The two parameters to the weight function: σ c, σ s –Interactively select a point p and the neighborhood radius ρ –σ c = 1 / 2 ρ –σ s = stdv(Nbhd(p, ρ)) Number of Iterations

Robustness Sharp features are treated as outliers

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

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

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

Results Source Two iterations Five iterations

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

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).

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

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

Non-iterative, Feature Preserving Mesh smoothing Bilateral mesh denoising

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

New results Bilateral mesh denoising Extended Bilateral mesh denoising