Spectral Mesh Processing

1. Spectral Mesh Processing SIGGRAPH Asia 2011 Course Elements of Geometry Processing Hao (Richard) Zhang, Simon Fraser University

2. What do you see?

8. Solved in a new view …

9. ! ? Different view or function space The same problem, phenomenon or data set, when viewed from a different angle, or in a new function space,may better reveal its underlying structure to facilitate the solution.

10. Spectral: a solution paradigm Solving problems in a different function space using a transform  spectral transform

11. Spectral: a solution paradigm Solving problems in a different function space using a transform  spectral transform y x x y

12. Shape correspondence A first example

13. Shape correspondence Rigid alignment easy, but how about shape pose? A first example

14. Shape correspondence A first example • Spectral transform normalizes shape pose

15. Shape correspondence A first example • Spectral transform normalizes shape pose Rigid alignment

16. Spectral mesh processing Use eigenstructures of appropriately defined linear mesh operators for geometry analysis and processing Spectral = use of eigenstructures

17. Eigenstructures Eigenvalues and eigenvectors: Diagonalization or eigen-decomposition: Projection into eigensubspace: DFT-like spectral transform: Au = u, u  0 A = UUT y’ = U(k)U(k)Ty ŷ = UTy

18. Function space • Global basis provided by eigenvectors • Eigenvectors are solutions of global optimization problem • Eigenstructures reflect global (non-local) info in the shape • Spectral is HIP

19. Overall picture Eigendecomposition: A = UUT Input mesh y  A = U UT e.g. y’ = U(3)U(3)Ty = Some mesh operator A

20. Classification of applications What eigenstructures are used Eigenvalues: signature for shape characterization Eigenvectors: form spectral embedding (a transform) Eigenprojection: also a transform  DFT-like

21. Classification of applications What eigenstructures are used Eigenvalues: signature for shape characterization Eigenvectors: form spectral embedding (a transform) Eigenprojection: also a transform  DFT-like Dimensionality of spectral embeddings 1D: mesh sequencing 2D or 3D: graph drawing or mesh parameterization Higher D: clustering, segmentation, correspondence

22. Classification of applications What eigenstructures are used Eigenvalues: signature for shape characterization Eigenvectors: form spectral embedding (a transform) Eigenprojection: also a transform  DFT-like Dimensionality of spectral embeddings 1D: mesh sequencing 2D or 3D: graph drawing or mesh parameterization Higher D: clustering, segmentation, correspondence Mesh operator used Combinatorial vs. geometric, 1st-order vs. higher order

23. More details in survey H. Zhang, O. van Kaick, and R. Dyer, “Spectral Mesh Processing”, Computer Graphics Forum (Eurographics STAR 2007), 2011. Eigenvector (of a combinatorial operator) plots on a sphere

24. This talk • What is the spectral approach exactly? • Three perspectives or views • Motivations • Sampler of applications • Difficulties and challenges

25. The three perspectives • A signal processing perspective • A very gentle introduction  boredom alert  • Signal compression and filtering • Relation to discrete Fourier transform (DFT) • Geometric perspective: global and intrinsic • Dimensionality reduction

26. The first perspective • A signal processing perspective • A very gentle introduction  boredom alert  • Signal compression and filtering • Relation to discrete Fourier transform (DFT) • Geometric perspective: global and intrinsic • Dimensionality reduction

27. The smoothing problem • Smooth out rough features of a contour (2D shape)

28. Laplacian smoothing • Move each vertex towards the centroid of its neighbours • Here: • Centroid = midpoint • Move half way

29. Laplacian smoothing and Laplacian • Local averaging • 1D discrete Laplacian

30. Smoothing result • Obtained by 10 steps of Laplacian smoothing

31. Signal representation • Represent a contour using a discrete periodic 2D signal x-coordinates of the seahorse contour

32. Smoothing operator In matrix form x component only y treated same way

33. 1D discrete Laplacian operator • Smoothing and Laplacian operator

34. Spectral analysis of a signal • Signal as linear combination of Laplacian eigenvectors

35. Spectral analysis of a signal Signal as linear combination of Laplacian eigenvectors DFT-like spectral transform X Spatial domain Spectral domain Project X along eigenvector

36. Plot of eigenvectors • First 8 eigenvectors of the 1D Laplacian More oscillation as eigenvalues (frequencies) increase

37. Relation to DFT • Smallest eigenvalue of L is zero • Each remaining eigenvalue has multiplicity 2 • The plotted real eigenvectors are not unique to L • One particular set of eigenvectors of L are the DFT basis • Both sets exhibit similar oscillatory behaviours wrt frequencies

38. Reconstruction and compression • Reconstruction using k leading coefficients • A form of spectral compression with info loss given by L2

39. Plot of transform coefficients • Fairly fast decay as eigenvalue increases

40. Reconstruction examples n = 401 n = 75

41. A filter applied to spectral coefficients Laplacian smoothing as filtering • Recall the Laplacian smoothing operator • Repeated application of S

42. Examples: low-pass filtering m = 1 m = 50 m = 5 m = 10 Filter:

43. Computational issues • No need to compute spectral coefficients for filtering • Polynomial (e.g., Laplacian): matrix-vector multiplication • Spectral compression needs explicit spectral transform

44. (x, y, z) Spectral mesh transform • Signal representation • Vectors of x, y, z vertex coordinates • Laplacian operator for meshes • Encodes connectivity and geometry • Combinatorial: graph Laplacians and variants • Discretization of the continuous Laplace-Beltrami operator  by Bruno • The same kind of spectral transform and analysis

45. Spectral mesh compression

46. Main references

47. A geometric perspective • Classical Euclidean geometry • Primitives not represented in coordinates • Geometric relationships deduced in a pure and self-contained manner • Use of axioms

48. A geometric perspective • Descartes’ analytic geometry • Algebraic analysis tools introduced • Primitives referenced in global frame  extrinsicapproach

49. Intrinsic approach • Riemann’s intrinsic view of geometry • Geometry viewed purely from the surface perspective • Metric: “distance” between points on surface • Many spaces (shapes) can be treated simultaneously: isometry

50. Spectral: intrinsic view • Spectral approach takes the intrinsic view • Intrinsic mesh informationcaptured via a linear mesh operator • Eigenstructures of the operator present the intrinsic geometric information in an organized manner • Rarely need all eigenstructures, dominant ones often suffice