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 Visualization of f=0 RBF fitting
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 Post-processing approach.
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. Basic Optimization Algorithm
Triangle centroids are projected onto the implicit surface.
Mesh vertices are optimized according to tangent planes.
estimated
numerically

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 weighted resampling +
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. + Dual/Primal mesh optimization
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. (ε : Threshold of adaptive subdivision)
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. 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.
Edge flipping

36. Why It Works Well without Edge Flipping?