1. INFORMATIK Mesh Smoothing by Adaptive and Anisotropic Gaussian Filter Applied to Mesh Normals Max-Planck-Institut für Informatik Saarbrücken, Germany Yutaka Ohtake Alexander Belyaev Hans-Peter Seidel

2. INFORMATIK Noise on Meshes Meshes obtained from digitalizing real world objects often contain undesirable noise. From range image of Stanford Bunny Angel model from shadow scanning

3. INFORMATIK Mesh Smoothing Mesh smoothing is required for removing the noise. Mesh Smoothing

4. INFORMATIK Mesh Smoothing Methods Laplacian, Bilaplacian smoothing flowsLaplacian, Bilaplacian smoothing flows Taubin’s signal processing (l | m) approachTaubin’s signal processing (l | m) approach Mean curvature flowMean curvature flow Anisotropic diffusionAnisotropic diffusion Laplacian, Bilaplacian smoothing flowsLaplacian, Bilaplacian smoothing flows Taubin’s signal processing (l | m) approachTaubin’s signal processing (l | m) approach Mean curvature flowMean curvature flow Anisotropic diffusionAnisotropic diffusion Iteration P old P new

5. INFORMATIK Conventional Smoothing Approaches 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. Number of iterationsNumber of iterations A threshold deciding geometric featuresA threshold deciding geometric features Iterations Best Over-smoothing Noisy

6. INFORMATIK Our Objective Developing fully automatic smoothing method no parameter is required Taubin’s smoothing Developed method

7. INFORMATIK Key Technique Amount of smoothing is decided adaptively. Noisy mesh White: large smoothing is needed Black: small smoothing is needed

8. INFORMATIK Contents 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 Gaussian Filter on Normal fieldAdaptive and Anisotropic 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 Gaussian Filter on Normal fieldAdaptive and Anisotropic Gaussian Filter on Normal field

9. INFORMATIK Adaptive Gaussian Filter Proposed by G.Gómez, 2000 fully automatic Noisy image Local scale map (Size of Gaussian kernel) Smoothed image

10. INFORMATIK Scale Space The best smoothing amount is adaptively found in scale space.

11. INFORMATIK How to Choose Optimal Local Scale Minimum is found in scale space. Homogeneous region Near edge region Constant independent of input kernel size original smoothed Only one iteration is required.

12. INFORMATIK Contents 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 Gaussian Filter on Normal fieldAdaptive and Anisotropic 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 Gaussian Filter on Normal fieldAdaptive and Anisotropic Gaussian Filter on Normal field

13. INFORMATIK Extension to Triangle Meshes Instead of the intensity of 2D images, the field of normals on meshes is smoothed. 2D ImageTriangle mesh

14. INFORMATIK Works Exploring Similar Idea Karbacher and Häusler, 1998Karbacher and Häusler, 1998 Smoothing vertex normalsSmoothing 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, and Osher, Vis’02Tasdizen, Whitaker, Burchard, and Osher, Vis’02 Anisotropic diffusion of normals for smoothing implicits (level set approach)Anisotropic diffusion of normals for smoothing implicits (level set approach) Karbacher and Häusler, 1998Karbacher and Häusler, 1998 Smoothing vertex normalsSmoothing 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, and Osher, Vis’02Tasdizen, Whitaker, Burchard, and Osher, Vis’02 Anisotropic diffusion of normals for smoothing implicits (level set approach)Anisotropic diffusion of normals for smoothing implicits (level set approach)

15. INFORMATIK Mesh Smoothing via Diffusion of Normal Field Our mesh smoothing = smoothing normals + integration of normals (in a least-square sense) Smoothing normals Integration of normals Adaptive Gaussian filter I will explain first.

16. INFORMATIK Integration of Face Normal Filed Minimizing squared differences of triangle normals and smoothed normals Conjugate gradient descent method is used.

17. INFORMATIK Result of Integration of Normals 100K triangle, takes about 10 sec. Original mesh Flat shaded by smoothed normals (100 times averaged) Result of integrating smoothed normals

18. INFORMATIK Contents 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 Gaussian Filter on Normal fieldAdaptive and Anisotropic 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 Gaussian Filter on Normal fieldAdaptive and Anisotropic Gaussian Filter on Normal field

19. INFORMATIK Gaussian Filter on Mesh Normals Dual mesh Primal mesh Smoothed normal : Geodesic distance found via Dijkstra’s algorithm Weight : (weighted average)

20. INFORMATIK Scale Space 10 scales

21. INFORMATIK Adaptive Gaussian Filter on Mesh Normals Constant (independent to noise size) variance Flat region High curvature region Average of edge length

22. 410K triangles 5 min. Golf club (Cyberware)

23. INFORMATIK Problem near Sharp Features Smoothed Scale map Minimum support size Under-smoothing

24. INFORMATIK Anisotropy Averaging regions should be adjusted to geometric features. Desired Averaging region

25. INFORMATIK Anisotoropic Neighborhood Penalty of changing normals

26. Anisotropic Adaptive Gaussian Filter Isotropic Adaptive Gaussian Filter Large smoothing is achieved near sharp edges

27. Noisy mesh (50K triangles) Taubin’s smoothing Proposed method (small features are well preserved)

28. Error Analysis Compare Ideal Smoothed after adding noise L2 vertex-based error L2 normal-based error Our method, Desbrun’s mean curvature flow, Taubin’s smoothing with various weights

29. INFORMATIK Conclusion Fully automatic smoothing method;Fully automatic smoothing method; produces good results if noise is not so large (natural noise).produces good results if noise is not so large (natural noise). preserves sharp features.preserves sharp features. It is time-consuming in comparison with conventional mesh smoothing methods.It is time-consuming in comparison with conventional mesh smoothing methods. Fast averaging normals on large ring neighorhoods is required.Fast averaging normals 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 if noise is not so large (natural noise).produces good results if noise is not so large (natural noise). preserves sharp features.preserves sharp features. It is time-consuming in comparison with conventional mesh smoothing methods.It is time-consuming in comparison with conventional mesh smoothing methods. Fast averaging normals on large ring neighorhoods is required.Fast averaging normals 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.