Transcript Talk slides
INFORMATIK Mesh Smoothing by Adaptive and Anisotropic Gaussian Filter Applied to Mesh Normals Yutaka Ohtake Alexander Belyaev Hans-Peter Seidel Max-Planck-Institut für Informatik Saarbrücken, Germany Noise on Meshes INFORMATIK Meshes obtained from digitalizing real world objects often contain undesirable noise. From range image of Stanford Bunny Angel model from shadow scanning Mesh Smoothing Mesh smoothing is required for removing the noise. Mesh Smoothing INFORMATIK Mesh Smoothing Methods INFORMATIK •Laplacian, Bilaplacian smoothing flows •Taubin’s signal processing (l | m) approach •Mean curvature flow •Anisotropic diffusion Pold P new P old D Pnew Iteration Conventional Smoothing Approaches INFORMATIK We have to specify several parameters. • Number of iterations • A threshold deciding geometric features Noisy Best Over-smoothing Iterations Our Objective INFORMATIK Developing fully automatic smoothing method no parameter is required Taubin’s l|m smoothing Developed method Key Technique INFORMATIK Amount of smoothing is decided adaptively. Noisy mesh White: large smoothing is needed Black: small smoothing is needed Contents INFORMATIK •Adaptive Gaussian Filter on 2D Image •Mesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Gaussian Filter on Normal field Adaptive Gaussian Filter INFORMATIK Proposed by G.Gómez, 2000 Noisy image Smoothed image fully automatic Local scale map (Size of Gaussian kernel) Scale Space INFORMATIK The best smoothing amount is adaptively found in scale space. 0 How to Choose Optimal Local Scale INFORMATIK Constant independent of input best argmin c ( I I ) kernel size original Homogeneous region 2 Minimum is found in scale space. smoothed Near edge region Only one iteration is required. Contents INFORMATIK •Adaptive Gaussian Filter on 2D Image •Mesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Gaussian Filter on Normal field Extension to Triangle Meshes INFORMATIK Instead of the intensity of 2D images, the field of normals on meshes is smoothed. 2D Image I ( x, y ) Triangle mesh n(T ) Works Exploring Similar Idea INFORMATIK •Karbacher and Häusler, 1998 • Smoothing vertex normals •Ohtake, Belyaev, and Bogaevski, CAD2000 • Diffusion of face normals for crease enhancement •Taubin, 2001 • Analysis of integrability of smoothed face normals •Tasdizen, Whitaker, Burchard, and Osher, Vis’02 • Anisotropic diffusion of normals for smoothing implicits (level set approach) Mesh Smoothing via Diffusion of Normal Field INFORMATIK Our mesh smoothing = smoothing normals + integration of normals (in a least-square sense) Smoothing normals Adaptive Gaussian filter Integration of normals I will explain first. Integration of Face Normal Filed INFORMATIK Minimizing squared differences of triangle normals and smoothed normals Efit ( M ) area (T ) n(T ) m(T ) 2 all trianglesT Conjugate gradient descent method is used. Result of Integration of Normals Original mesh Flat shaded by smoothed normals (100 times averaged) INFORMATIK Result of integrating smoothed normals 100K triangle, takes about 10 sec. Contents INFORMATIK •Adaptive Gaussian Filter on 2D Image •Mesh Smoothing via Diffusion of Normals •Adaptive and Anisotropic Gaussian Filter on Normal field Gaussian Filter on Mesh Normals wn Smoothed normal : m w n j j j j INFORMATIK (weighted average) Weight : w j area (T j ) K (d ) Geodesic distance found via Dijkstra’s algorithm 4 Primal mesh Dual mesh Scale Space INFORMATIK 0 0.2 (average of edge length ) 10 scales Adaptive Gaussian Filter on Mesh Normals Constant (independent to noise size) best argmin c e2 2 Average of edge length 1 wj 2 Flat region High curvature region INFORMATIK variance 2 w ( m n ) j j Golf club (Cyberware) 410K triangles 5 min. Problem near Sharp Features INFORMATIK Smoothed Scale map Minimum support size Under-smoothing Anisotropy INFORMATIK Averaging regions should be adjusted to geometric features. Desired Averaging region Anisotoropic Neighborhood INFORMATIK | n( P) n(Q) |2 | n(O) n( P) |2 | n(O) n(Q) |2 d ( P, Q) 1 k 6 2 2 Penalty of changing normals P O (start point) Q d ( P, Q ) Anisotropic Adaptive Gaussian Filter Isotropic Adaptive Gaussian Filter Large smoothing is achieved near sharp edges Noisy mesh (50K triangles) Taubin’s smoothing Proposed method (small features are well preserved) Error Analysis Compare Smoothed after adding noise Ideal Our method, Desbrun’s mean curvature flow, Taubin’s smoothing with various weights 0.0045 0.079 0.0044 0.077 0.0043 0.075 L2 vertex-based error L2 normal-based error 30 28 26 24 22 20 18 16 14 0 30 28 24 26 22 18 20 16 14 10 12 0.063 8 0.0036 6 0.065 4 0.0037 2 0.067 0 0.0038 12 0.069 10 0.0039 0.071 8 0.004 0.073 6 0.0041 4 Our method Hn taubin taubin 1/d taubin cot 2 0.0042 Conclusion INFORMATIK •Fully automatic smoothing method; • produces good results if noise is not so large (natural noise). • preserves sharp features. •It is time-consuming in comparison with conventional mesh smoothing methods. • Fast averaging normals on large ring neighorhoods is required. •It is not capable to remove large noise. • Noise size is close to sampling interval.