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Mesh Segmentation via Spectral Embedding and Contour Analysis

Mesh Segmentation via Spectral Embedding and Contour Analysis. Speaker: Min Meng 2007.11.22. Background knowledge. Spectrum of matrix. Given an nxn matrix M Eigenvalues Eigenvectors By definition The spectrum of matrix M. The Spectral Theorem.

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Mesh Segmentation via Spectral Embedding and Contour Analysis

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  1. Mesh Segmentationvia Spectral Embeddingand 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. Mesh laplacians • 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. Geometric laplacian M • 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

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

  26. Iterations of spectral cut

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

  28. 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

  29. Salience-guided spectral cut

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

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

  32. Spectral 1D embedding -faces • Given sample faces

  33. 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

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

  35. Experimental results

  36. L-embedding

  37. Pro.

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

  39. Robustness of sampling • Two samples reside on different parts

  40. Cor. • Segmentation measure • Salience measure Manually searched automatic

  41. Thanks!

  42. Q&A

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