Moving Least Squares Multiresolution Surface Approximation

1. Moving Least Squares Multiresolution Surface Approximation Boris Mederos Luiz Velho Luiz Henrique De Figueirdo

2. Overview • About Authors • Introduction • Related Works • This work • Results • Conclusion

3. About Authors Boris Mederos PhD of Instituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro, Brasil, Advisor: Luiz Velho and Luiz Henrique de Figueiredo research interests lie in surface reconstruction and CG

4. Luiz Velho • Full Researcher at IMPA and Leading Scientist of the VISGRAF Laboratory. • various aspects of computer graphics and related areas. the central focus of his work is the investigation of multiscale models and hierarchical computational methods associated with them

5. Luiz Henrique De Figueirdo • Associate Researcher at IMPA and a member of its Visgraf laboratory. • Research interests include computational geometry, geometric modeling, and interval methods in computer graphics, specially applications of affine arithmetic.

6. Introduction The problem of surface reconstruction and refinement from scatted points without normals has received a growing amount of attention in computer graphics. and there are several algorithms known for this problem

7. Related Works • 3D Delaunay triangulation a new Voronoi based surface algorithm .SIGGRAPH’98 • Greedy approach The ball-pivoting algorithm for surface reconstruction 99 • Incremental algorithms computes a set of representative points and triangulates these representative points. Curve and surface reconstruction from regular and non regular point sets 101-126 2001

8. This Work • Clustering • Reduction • Triangulation • Refinement

9. Clustering This step is to partition the original set of points Q into a finite set of clusters. This method is based on a BSP tree,Each node contains a sub set First define subdivision criteria for BSP tree:

10. Clustering • Since C is a symmetric positive semi-definite 3×3 matrix,its three eigenvalues are real and order them as

11. Clustering Hence the ratio can be used for the curvature of S around p Now the subdivision criteria is 1 the ratio is larger than a tolerance 2 the number n is larger than

12. Clusters

13. Reduction This algorithm uses a new method based on moving least squares theory (MLS) to find a representative point for each cluster.

14. Reduction Assuming the centroid of the set of points in the cluster is c,now we abtain the point where the weighted covariance matrix M is a 3×3 matrix whose entries are

15. Reduction • To compute the vector nc, we can start with t = 0,and the minimization problem can be rewritten in the form While B is the matrix of weight covariance

16. Reduction Using the direction as above ,we can compute

17. Trianulation • The algorithm computes a sequence of triangulated surface with border. At each step, it choose a border edge, finds a new triangle associated to this edge, and updates the current surface. • it maintains a half-edge data structure H and a list L of half-edge.

18. Pseudo-code

19. Pseudo-code

20. Triangulation

21. Refinement Refine the initial coarse triangulation ,first refine an edge uv: For each edge uv,its mid-point is m, And its normal

22. Refinement • Minimize the following functional with respect to t:

23. Refinement

24. Results

25. Coclusion • The new method computes representative points • Triangulation algorithm does not need to compute 3D Delaunay triangulation • Refinement method is fast and gives a fine triangular mesh.

26. The end THANK YOU!

27. MLS • The introduction First find a local reference domain (plane) for r, then use the domain to compute a local bivariate polynomial approximation to the surface in a neighborhood of r

28. Reference domain • The local plane: is computed so as to minimize a local weighted sum of squared distances of the points pi to the plane

29. Local map To compute local bivariate polynomial approximation first Let