1. Mesh Segmentation via Spectral Embedding and Contour Analysis Speaker: Min Meng 2007.11.22

2. Background knowledge

3. Spectrum of matrix Given an nxn matrix M Eigenvalues Eigenvectors By definition The spectrum of matrix M

4. The Spectral Theorem Let S be a real symmetric matrix of dimension n, the eigendecomposition of S Where are diagonal matrix of eigenvalues are eigenvectors are real, V are orthogonal

5. Spectral method Solve the problem by manipulating Eigenvalues Eigenvectors Eigenspace projections Combination of these quantities Which derived from an appropriately defined linear operator

6. Use of spectral method Use of eigenvalues Global shape descriptors Graph and shape matching

7. Use of spectral method Use of eigenvectors Spectral embedding K-D embedding

8. Use of spectral method Use of eigenprojections Project the signal into a different domain Mesh compression Remove high-frequency Spectral watermark Remove low-frequency

9. Mesh laplacians Mesh laplacian operators Linear operators Act on functions defined on a mesh Mesh laplacians

10. Combinatorial mesh laplacians Defined by the graph associated with mesh Adjacency matrix W Graph : Normalized graph: Geometric mesh laplacians

11. Overview

12. Outline 2D Spectral embedding - vertices 2D Contour analysis 1D Spectral embedding - faces line search with salience

13. 2D Spectral projections-point Graph laplacian L Structural segmentability Geometric laplacian M Geometrical segmentability

14. Graph laplacian L Adjacency matrix W, graph laplacian L L is positive semi-definite and symmetric Its smallest eigenvalue Corresponding eigenvector v is constant vector Choose k=3 to spectral 2D embedding

15. Graph laplacian L Spectral projection Branch is retained Capture structural segmentability

16. Geometric laplacian M Geometric matrix W For edge e=(i, j) Others Geometric laplacian M

17. If an edge e=(i, j) Takes a large weight Mesh vertices from concave region Pulled close Geometric segmentability

18. Contour analysis Segmentability analysis Sampling points (faces)

19. Contour extract

20. Contour Convexity Area-based Struggle with boundary defects perimeter-based Sensitive to noise Combinational measure

21. Contour Convexity

22. Convexity and Segmentability Not exactly the same concept

23. Inner distance Consider two points Inner distance defined as the length of the shortest path connecting them within O Insensitive to shape bending

24. Multidimensional scaling (MDS) Provide a visual representation of the pattern of proximities

26. Segmentability analysis Segmentability score Four steps : If return Compute embedding of via MDS if return If return Compute embedding of via MDS if return

27. Iterations of spectral cut

28. Sampling points (faces) Integrated bending score (IBS) I is inner distance E is Euclidean distance

29. Sampling points (faces) Two samples The first sample s1, maximizes IBS The second s2, has largest distance from s1 Sample points reside on different parts

30. Salience-guided spectral cut

31. Spectral 1D embedding -faces Compute matrix A Adjacent faces Construct the dual graph of mesh is the shortest path between their dual vertices

32. Spectral 1D embedding -faces Nystrom approximation Let If Approximate eigenvector of A

33. Spectral 1D embedding -faces Given sample faces

34. salient cut: line search Part salience Sub-mesh M, the part Q Vs : part size Vc : cut strength Vp : part protrusiveness Require an appropriate weighting between three factors

35. salient cut: line search Part salience When L used, When M used,

36. Experimental results

38. L-embedding

40. Pro.

41. Segmentability analysis : automatic Graph laplacian - L Geometric laplacian - M MDS based on inner distance

42. Robustness of sampling Two samples reside on different parts

43. Cor. Segmentation measure Salience measure Manually searched automatic

44. Thanks!

45. Q&A