bilateral mesh denoising shachar fleishman iddo drori daniel cohen-or tel aviv university

Bilateral Mesh DenoisingBilateral Mesh Denoising

Shachar FleishmanIddo Drori

Daniel Cohen-Or

Tel Aviv University

DenoisingDenoising

• Input (scanned) model– Additive noise

+ =

DenoisingDenoising

• Input (scanned) model– Additive noise

• Noise free model– Preserve features

+ = filter

Image denoisingImage 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=146.3339( )=131.5013MSE)

Images courtesy of Michael Elad

Wavelet Denoising (soft) Using DB5 Using DB3

(MSE=144.7436( )=150.7006MSE)

Images courtesy of Michael Elad

Filtering via the Bilateral2 iterations with 1111 Sub-gradient based 55

(MSE=89.2516( )=93.4024MSE)

Images courtesy of Michael Elad

Mesh denoising, smoothing and fairing 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 denoisingBilateral mesh denoising

• Fast

• Simple

• Intuitive parameter selection

Bilateral filteringBilateral filtering

• Gaussian filter

2

22' ( )u p

p N u

I u e I p

* =

• Bilateral filter

2

2 2

2

2 2

2 2

2 2

( )

'

c s

c s

I u I pu p

p N u

I u I pu p

p N u

e e I p

I u

e e

Denoise Feature preserving

Normalization

Bilateral filteringBilateral filtering

* =

Bilateral filtering of meshesBilateral filtering of meshes

Bilateral filtering of meshesBilateral filtering of meshes

Bilateral filtering of meshesBilateral filtering of meshes

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

Bilateral filtering of meshesBilateral filtering of meshes

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

• Apply the bilateral filter to heights

Bilateral filtering of meshesBilateral 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 meshesBilateral 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 meshesBilateral 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

SolutionSolution

• A plane that passes through the point is the estimator to the smooth surface

• Plane L=(p,n)

SolutionSolution

• A plane that passes through the point is the estimator to the smooth surface

• Plane L=(p,n)

P

n

L

Similaritycloseness

Computing the planeComputing 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){qi} = neighborhood(v)K=|{qi}|sum=0normalizer=0for i := 1 to K

t = ||v-qi||h = <n,v-qi>Wc=exp(-t2/(2σc

2))Ws=exp(-h2/(2σs

2))Sum +=(wc*ws)hNormalizer += wc*ws

EndReturn v+n*(sum/normalizer)

v

DenoisePoint(Vertex v, Normal n){qi} = neighborhood(v)K=|{qi}|sum=0normalizer=0for i := 1 to K

t = ||v-qi||h = <n,v-qi>Wc=exp(-t2/(2σc

2))Ws=exp(-h2/(2σs

2))Sum +=(wc*ws)hNormalizer += wc*ws

EndReturn v+n*(sum/normalizer)

iterate over neighborhood

v

DenoisePoint(Vertex v, Normal n){qi} = neighborhood(v)K=|{qi}|sum=0normalizer=0for i := 1 to K

t = ||v-qi||h = <n,v-qi>Wc=exp(-t2/(2σc

2))Ws=exp(-h2/(2σs

2))Sum +=(wc*ws)hNormalizer += wc*ws

EndReturn v+n*(sum/normalizer)

closeness v

q

DenoisePoint(Vertex v, Normal n){qi} = neighborhood(v)K=|{qi}|sum=0normalizer=0for i := 1 to K

t = ||v-qi||h = <n,v-qi>Wc=exp(-t2/(2σc

2))Ws=exp(-h2/(2σs

2))Sum +=(wc*ws)hNormalizer += wc*ws

EndReturn v+n*(sum/normalizer)

height – similarityv

q

DenoisePoint(Vertex v, Normal n){qi} = neighborhood(v)K=|{qi}|sum=0normalizer=0for i := 1 to K

t = ||v-qi||h = <n,v-qi>Wc=exp(-t2/(2σc

2))Ws=exp(-h2/(2σs

2))Sum +=(wc*ws)hNormalizer += wc*ws

EndReturn v+n*(sum/normalizer)

weights

v

DenoisePoint(Vertex v, Normal n){qi} = neighborhood(v)K=|{qi}|sum=0normalizer=0for i := 1 to K

t = ||v-qi||h = <n,v-qi>Wc=exp(-t2/(2σc

2))Ws=exp(-h2/(2σs

2))Sum +=(wc*ws)hNormalizer += wc*ws

EndReturn v+n*(sum/normalizer)

Move the vertex in the normal direction

v

ParametersParameters

• 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

RobustnessRobustness

• Sharp features are treated as outliers

RobustnessRobustness

• Sharp features are treated as outliers

• The bilateral filter does not recover smoothed signal

ResultsResults

Source Anisotropic denoising of height fields - Desburn ’00

Bilateral mesh denoising

ResultsResults

SourceAnisotropic Geometric

Diffusion in Surface Processing - Clarenz ‘00

Bilateral mesh denoising

ResultsResults

Source

Two iterations

Five iterations

Future WorkFuture Work

• Adapt the algorithm to point sets

• Robust estimator of normals

AcknowledgementsAcknowledgements

• 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 - predictorsComparison - predictors

Non-iterative, Feature Preserving Mesh smoothing

Bilateral mesh denoising

New resultsNew results

Bilateral mesh denoising

Extended Bilateral mesh denoising