INFORMATIKINFORMATIK

Mesh Smoothing by Adaptive and Mesh Smoothing by Adaptive and Anisotropic Gaussian Filter Anisotropic Gaussian Filter

Applied to Mesh NormalsApplied to Mesh Normals

Max-Planck-Institut fMax-Planck-Institut füür Informatik Saarbrr Informatik Saarbrücken, ücken, GermanyGermany

Max-Planck-Institut fMax-Planck-Institut füür Informatik Saarbrr Informatik Saarbrücken, ücken, GermanyGermany

Yutaka Yutaka OhtakeOhtakeYutaka Yutaka OhtakeOhtake

Alexander Alexander BelyaevBelyaev

Alexander Alexander BelyaevBelyaev

Hans-PeterHans-PeterSeidelSeidel

Hans-PeterHans-PeterSeidelSeidel

INFORMATIKINFORMATIKNoise on MeshesNoise on Meshes

Meshes obtained from digitalizing real world Meshes obtained from digitalizing real world objects often contain undesirable noise.objects often contain undesirable noise.Meshes obtained from digitalizing real world Meshes obtained from digitalizing real world objects often contain undesirable noise.objects often contain undesirable noise.

From range image From range image of Stanford Bunnyof Stanford Bunny

Angel modelAngel modelfrom shadow scanningfrom shadow scanning

INFORMATIKINFORMATIKMesh SmoothingMesh Smoothing

Mesh smoothing is required Mesh smoothing is required for removing the noise. for removing the noise.Mesh smoothing is required Mesh smoothing is required for removing the noise. for removing the noise.

MeshMeshSmoothingSmoothing

INFORMATIKINFORMATIKMesh Smoothing MethodsMesh Smoothing Methods•Laplacian, Bilaplacian smoothing flowsLaplacian, Bilaplacian smoothing flows

•Taubin’s signal processing (Taubin’s signal processing (| | approachapproach

•Mean curvature flowMean curvature flow

•Anisotropic diffusionAnisotropic diffusion

•Laplacian, Bilaplacian smoothing flowsLaplacian, Bilaplacian smoothing flows

•Taubin’s signal processing (Taubin’s signal processing (| | approachapproach

•Mean curvature flowMean curvature flow

•Anisotropic diffusionAnisotropic diffusion

Iteration

D oldnew PP

PoldPnew

INFORMATIKINFORMATIK

Conventional Conventional Smoothing Approaches Smoothing Approaches

We have to specify several parameters.We have to specify several parameters.• Number of iterationsNumber of iterations• A threshold deciding geometric featuresA threshold deciding geometric features

We have to specify several parameters.We have to specify several parameters.• Number of iterationsNumber of iterations• A threshold deciding geometric featuresA threshold deciding geometric features

Iterations Iterations

Best Best Over-smoothingOver-smoothingNoisyNoisy

INFORMATIKINFORMATIKOur ObjectiveOur Objective

Developing fully automatic smoothing methodDeveloping fully automatic smoothing method no parameter is requiredno parameter is requiredDeveloping fully automatic smoothing methodDeveloping fully automatic smoothing method no parameter is requiredno parameter is required

Taubin’s Taubin’s smoothingsmoothing

Developed Developed methodmethod

INFORMATIKINFORMATIKKey TechniqueKey Technique

Amount of smoothing is decided adaptively.Amount of smoothing is decided adaptively.Amount of smoothing is decided adaptively.Amount of smoothing is decided adaptively.

Noisy meshNoisy meshWhite: large smoothing is neededWhite: large smoothing is neededBlack: small smoothing is neededBlack: small smoothing is needed

INFORMATIKINFORMATIKContentsContents

•Adaptive Gaussian Filter on 2D ImageAdaptive Gaussian Filter on 2D Image

•Mesh Smoothing via Diffusion of NormalsMesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Adaptive and Anisotropic Gaussian Filter on Normal field Gaussian Filter on Normal field

•Adaptive Gaussian Filter on 2D ImageAdaptive Gaussian Filter on 2D Image

•Mesh Smoothing via Diffusion of NormalsMesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Adaptive and Anisotropic Gaussian Filter on Normal field Gaussian Filter on Normal field

INFORMATIKINFORMATIKAdaptive Gaussian FilterAdaptive Gaussian Filter

Proposed by G.GProposed by G.Góómez, 2000mez, 2000Proposed by G.GProposed by G.Góómez, 2000mez, 2000

fully automaticfully automatic

Noisy imageNoisy image

Local scale map (Size of Gaussian kernel)Local scale map (Size of Gaussian kernel)

Smoothed imageSmoothed image

INFORMATIKINFORMATIKScale SpaceScale Space

The best smoothing amount The best smoothing amount is adaptively found in scale space. is adaptively found in scale space.The best smoothing amount The best smoothing amount is adaptively found in scale space. is adaptively found in scale space.

0

INFORMATIKINFORMATIK

How to Choose How to Choose Optimal Local Scale Optimal Local Scale

2best )(minarg

II

c

Minimum is found Minimum is found in scale space.in scale space.

Homogeneous Homogeneous regionregion Near edge regionNear edge region

ConstantConstant 　　 independent of inputindependent of input

kernel sizekernel size originaloriginal smoothedsmoothed

Only one iteration is required.Only one iteration is required.

INFORMATIKINFORMATIKContentsContents

•Adaptive Gaussian Filter on 2D ImageAdaptive Gaussian Filter on 2D Image •Mesh Smoothing via Diffusion of NormalsMesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Adaptive and Anisotropic Gaussian Filter on Normal field Gaussian Filter on Normal field

•Adaptive Gaussian Filter on 2D ImageAdaptive Gaussian Filter on 2D Image •Mesh Smoothing via Diffusion of NormalsMesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Adaptive and Anisotropic Gaussian Filter on Normal field Gaussian Filter on Normal field

INFORMATIKINFORMATIKExtension to Triangle MeshesExtension to Triangle Meshes

Instead of the intensity of 2D images, Instead of the intensity of 2D images, the field of normals on meshes is smoothed. the field of normals on meshes is smoothed.Instead of the intensity of 2D images, Instead of the intensity of 2D images, the field of normals on meshes is smoothed. the field of normals on meshes is smoothed.

2D Image Triangle mesh

),( yxI )(Tn

INFORMATIKINFORMATIKWorks Exploring Similar Idea Works Exploring Similar Idea

•Karbacher and HKarbacher and Häusleräusler, 1998 , 1998 • Smoothing vertex normals Smoothing vertex normals

•Ohtake, Belyaev, and Bogaevski, CAD2000Ohtake, Belyaev, and Bogaevski, CAD2000• Diffusion of face normals for crease enhancementDiffusion of face normals for crease enhancement

•Taubin, 2001Taubin, 2001• Analysis of integrability of smoothed face normalsAnalysis of integrability of smoothed face normals

•Tasdizen, Whitaker, Burchard, Tasdizen, Whitaker, Burchard, and Osher, Vis’02 and Osher, Vis’02• Anisotropic diffusion of normals for smoothing implicitAnisotropic diffusion of normals for smoothing implicit

s (level set approach)s (level set approach)

•Karbacher and HKarbacher and Häusleräusler, 1998 , 1998 • Smoothing vertex normals Smoothing vertex normals

•Ohtake, Belyaev, and Bogaevski, CAD2000Ohtake, Belyaev, and Bogaevski, CAD2000• Diffusion of face normals for crease enhancementDiffusion of face normals for crease enhancement

•Taubin, 2001Taubin, 2001• Analysis of integrability of smoothed face normalsAnalysis of integrability of smoothed face normals

•Tasdizen, Whitaker, Burchard, Tasdizen, Whitaker, Burchard, and Osher, Vis’02 and Osher, Vis’02• Anisotropic diffusion of normals for smoothing implicitAnisotropic diffusion of normals for smoothing implicit

s (level set approach)s (level set approach)

INFORMATIKINFORMATIK

Mesh Smoothing viaMesh Smoothing via Diffusion of Normal Field Diffusion of Normal Field

Our mesh smoothing = Our mesh smoothing = smoothing normals + integration of normals smoothing normals + integration of normals (in a least-square sense)(in a least-square sense)

Our mesh smoothing = Our mesh smoothing = smoothing normals + integration of normals smoothing normals + integration of normals (in a least-square sense)(in a least-square sense)

Smoothing normals

Integration of normals

Adaptive Gaussian filterAdaptive Gaussian filter I will explain first.I will explain first.

INFORMATIKINFORMATIK

Integration of Integration of Face Normal Filed Face Normal Filed

Minimizing squared differences Minimizing squared differences of triangle normals and smoothed normals of triangle normals and smoothed normalsMinimizing squared differences Minimizing squared differences of triangle normals and smoothed normals of triangle normals and smoothed normals

2

trianglesallfit )()()(area)(

T

TTTME mn

Conjugate gradient descent method is used.Conjugate gradient descent method is used.

INFORMATIKINFORMATIK

Result of Result of Integration of Normals Integration of Normals

100K triangle, takes about 10 sec.100K triangle, takes about 10 sec.100K triangle, takes about 10 sec.100K triangle, takes about 10 sec.

Original meshOriginal mesh Flat shaded by Flat shaded by smoothed normalssmoothed normals

(100 times averaged)(100 times averaged)

Result of integrating Result of integrating smoothed normalssmoothed normals

INFORMATIKINFORMATIKContentsContents

•Adaptive Gaussian Filter on 2D ImageAdaptive Gaussian Filter on 2D Image •Mesh Smoothing via Diffusion of NormalsMesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Adaptive and Anisotropic Gaussian Filter on Normal field Gaussian Filter on Normal field

•Adaptive Gaussian Filter on 2D ImageAdaptive Gaussian Filter on 2D Image •Mesh Smoothing via Diffusion of NormalsMesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Adaptive and Anisotropic Gaussian Filter on Normal field Gaussian Filter on Normal field

INFORMATIKINFORMATIK

Gaussian Filter Gaussian Filter on Mesh Normals on Mesh Normals

jj

jj

w

w

n

nm

Dual meshDual meshPrimal meshPrimal mesh

)()(area dKTw jj

Smoothed normal :Smoothed normal :

Geodesic distanceGeodesic distancefound via found via

Dijkstra’s algorithmDijkstra’s algorithm

Weight :Weight :

(weighted average)(weighted average)

4

INFORMATIKINFORMATIKScale SpaceScale Space

10 scales10 scales)length edge of average(2.0

0

INFORMATIKINFORMATIK

Adaptive Gaussian Filter Adaptive Gaussian Filter on Mesh Normals on Mesh Normals

22

best minarg

ec

Constant Constant (independent to noise size)(independent to noise size)

22 )(1

jjj

ww

nm variancevariance

Flat Flat regionregion

High curvatureHigh curvatureregionregion

Average Average of edge lengthof edge length

410K triangles410K triangles5 min.5 min.

Golf clubGolf club(Cyberware)(Cyberware)

INFORMATIKINFORMATIKProblem near Sharp FeaturesProblem near Sharp Features

SmoothedSmoothed

Scale mapScale map

MinimumMinimumsupport sizesupport size

Under-smoothingUnder-smoothing

INFORMATIKINFORMATIKAnisotropyAnisotropy

Averaging regions should be Averaging regions should be adjusted to geometric features. adjusted to geometric features.Averaging regions should be Averaging regions should be adjusted to geometric features. adjusted to geometric features.

DesiredDesiredAveraging Averaging

regionregion

INFORMATIKINFORMATIKAnisotoropic NeighborhoodAnisotoropic Neighborhood

2

|)()(|

2

|)()(|

6

|)()(|1),(

222 QOPOQPkQPd

nnnnnn

Penalty of Penalty of changing normalschanging normals

),( QPd

point)(start O

P

Q

AnisotropicAnisotropicAdaptive Gaussian FilterAdaptive Gaussian Filter

Isotropic Isotropic Adaptive Gaussian FilterAdaptive Gaussian Filter

Large smoothing is achieved Large smoothing is achieved near sharp edgesnear sharp edges

Noisy meshNoisy mesh(50K triangles)(50K triangles)

Taubin’s Taubin’s smoothingsmoothing

Proposed methodProposed method(small features are (small features are

well preserved)well preserved)

Error AnalysisError Analysis

0.0036

0.0037

0.0038

0.0039

0.004

0.0041

0.0042

0.0043

0.0044

0.0045

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Our methodHntaubintaubin 1/ dtaubin cot

0.063

0.065

0.067

0.069

0.071

0.073

0.075

0.077

0.079

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

CompareCompare

IdealIdeal Smoothed after Smoothed after adding noiseadding noise

L2 vertex-based errorL2 vertex-based error L2 normal-based errorL2 normal-based error

Our method, Desbrun’s mean curvature flow, Our method, Desbrun’s mean curvature flow, Taubin’s smoothing with various weightsTaubin’s smoothing with various weights

INFORMATIKINFORMATIKConclusionConclusion

•Fully automatic smoothing method;Fully automatic smoothing method;• produces good results produces good results

if noise is not so large (natural noise). if noise is not so large (natural noise).• preserves sharp features.preserves sharp features.

•It is time-consuming in comparison with coIt is time-consuming in comparison with conventional mesh smoothing methods.nventional mesh smoothing methods.• Fast averaging normals Fast averaging normals

on large ring neighorhoods is required. on large ring neighorhoods is required.

•It is not capable to remove large noise.It is not capable to remove large noise.• Noise size is close to sampling interval.Noise size is close to sampling interval.

•Fully automatic smoothing method;Fully automatic smoothing method;• produces good results produces good results

if noise is not so large (natural noise). if noise is not so large (natural noise).• preserves sharp features.preserves sharp features.

•It is time-consuming in comparison with coIt is time-consuming in comparison with conventional mesh smoothing methods.nventional mesh smoothing methods.• Fast averaging normals Fast averaging normals

on large ring neighorhoods is required. on large ring neighorhoods is required.

•It is not capable to remove large noise.It is not capable to remove large noise.• Noise size is close to sampling interval.Noise size is close to sampling interval.