Dual/Primal Mesh Optimization for Polygonized Implicit Surfaces

1. Dual/Primal Mesh Optimization for Polygonized Implicit Surfaces Yutaka Ohtake Alexander G. Belyaev Max-Planck-Institut für Informatik, Germany University of Aizu, Japan.

2. Implicit Surfaces • Zero sets of implicit functions. • CSG operations. - =

3. Radial Basis Function Carr et al. “Reconstruction and Representation of 3D Objects with Radial Basis Functions”, SIGGRAPH2001 Visualization of f=0 RBF fitting

4. Visualization of Implicit Surfaces Polygonization (e.g. Marching cubes method) Ray-tracing

5. Problem of Polygonization 503 grid 1003 grid 2003 grid • Sharp features are broken

6. Reconstruction of Sharp Features Input Output and Rough Polygonization(Correct topology) Post- processing

7. Basic Idea of Optimization • Mesh tangent to implicit surface gives better reconstruction of sharp features. • dual mesh Marching cube method Our method

8. Related Works • Extension of Marching Cubes • Kobbelt, Botsh, Schwanecke, and Seidel “Feature Sensitive Surface Extraction from Volume Data”, SIGGRAPH 2001, August. • Post-processing approach. • Ohtake, Belyaev, and Pasko “Accurate Polygonization of Implicit Surfaces”, Shape Modeling Internatinal 2001, May. • Ohtake and Belyaev“Mesh Optimization for Polygonized Isosurfaces”, Eurographics 2001, September.

9. Previous work (1) • Kobbelt, Botsh, Schwanecke, and Seidel proposed • A new distance field representation for detecting accurate vertex positions. • Vertex insertion rule for reconstructing sharp features. (and edge flipping) newly inserted

10. Related work (2) • Our previous work • Mesh evolution for • fitting mesh normals to implicit surface normals. • keeping mesh vertices close to implicit surface. Can not estimate implicit surface normals at high curvature regions

11. Advantages of Proposed Method • Extremely good • in reconstruction of sharp features • Adaptive meshing • Works better than mesh evolution approach

12. Contents • Basic Optimization Method • Combining with Adaptive Remeshing and Subdivision • Discussion

13. estimated numerically Basic Optimization Algorithm • Triangle centroids are projected onto the implicit surface. • Mesh vertices are optimized according to tangent planes.

14. Dual sampling (face points are projected to f=0) Dual Sampling (fitting to tangent planes)

15. Projection of face points • Find a point at other side of surface. • Bisection method along the lines. f < 0 f > 0

16. Fitting to Tangent Planes • Minimize the sum of squared distance. distance m(P2) m(P1) x Same as Garland-Heckbert quadric error metric (SIG’97)

17. Minimization of the Error • Solving system of linear equations. • SVD is used (similar to Kobbelt et al. SIG’01). • The old primal vertex position is shifted to the origin of coordinates. • Small singular values are set to zero.

18. Thresholding of Small Singular Values

19. Contents • Basic Optimization Method • Combining with Adaptive Remeshing and Subdivision • Discussion

20. Improvement of Mesh Sampling Rate Curvature weighted resampling Input Dual/Primal mesh optimization output

21. Repeated Double Dual Resampling • Double dual sampling • improves mesh distributions. Averaging by Projection

22. Curvature Weighted Resampling • Sampling should be dense near high curvature regions. Uniform resampling causes a skip here. Small bump Uniform weight Curvature weight

23. Effectiveness • Small bumps are well reconstructed. Uniform resampling + Primal/dual mesh optimization Curvature weightedresampling + Primal/dual mesh optimization

24. Gathering All Together Curvature weighted resampling Input Adaptive subdivision Dual/Primal mesh optimization else If user is satisfied output

25. Adaptive subdivision • Linear 1-to-4 split rule is applied on highly curved triangles. + Dual/Primal mesh optimization “Cat” model provided by HyperFun project.

26. Decimation • Garland-Heckbert method using • Tolerance: 90% reduction

27. Gathering All Together Curvature weighted resampling Input Adaptive subdivision Dual/Primal mesh optimization else If user is satisfied Mesh Decimation output

28. The number of triangles

29. Large adaptive ε 3 subdivision steps Small threshold ε 5 subdivision steps (ε: Threshold of adaptive subdivision)

30. Contents • Basic Optimization Method • Combining with Adaptive Remeshing and Subdivision • Discussion

31. Comparison with Mesh Evolution Approach • Faster and more accurate than mesh evolution approach. Mesh evolution 20 sec. (stabilized) Primal/Dual mesh optimization 1 sec.

32. Stanford bunny represented by RBF with 10,000 centers. (FastRBF developed by FarField Technology) Optimization takes several hours (Direct evaluation)

33. Dual Contouring of Hermite Data SIG’02 • Also good for reconstruction of sharp features • Tao Ju, Frank Losasso, Scott Schaefer, Joe Warren,“Dual Contouring of Hermite Data”. • Dual mesh to marching cubes mesh.

34. Speed: they(sig’02) > we(sm’02) • Their method is not post-processing. • Control of sampling rate: we(sm’02) > they(sig’02) • Octtree based adaptive sampling. Our Their

35. Edge flipping Conclusion and Problems • A mesh optimization method is developed. • Primal/Dual mesh optimization. • Not so fast if the implicit function is complex. • Adaptive voxelization. • Requirement of correct topology in the input mesh. • Can not optimize this pattern.