Laplace-Beltrami Operator for Spectral Point Clouds Processing

1. Laplace-Beltrami Operator forSpectral Point Clouds Processing Yusu Wang Computer Science and Engineering Dept The Ohio State University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA

2. Dagstuhl 2009 This Talk • Many geometric networks • Concern points from Euclidean space • This talk: • Points sampled from an underlying Riemannian manifold • Approximating Laplace operator faithfully • In non-statistical setting • Converges to ground truth for denser sampling • [Joint work with Belkin and Sun] Goal: Form a framework for spectral point clouds analysis in geometric setting

3. Dagstuhl 2009 • It is a fundamental geometric object • Two manifolds M and N isometric • ∆Mand∆N share same eigenvalues & eigenfunctions Laplace-Beltrami Operator • Given a manifold M, Laplace-Beltrami operator ∆ • Operates on functions ∆ f = g, withf , g: M -> R • ∆ f = div (grad f) • If M = R2, • then ∆ f = ∂ 2f / ∂x2 + ∂ 2f / ∂y2 Contain all intrinsic geometry info. !

4. Eigenfunctions of Laplacian: • Minimizing Dirichlet’s energy: • Model dimensionality reduction (Laplace Eigenmap[Belkin&Niyogi 2003]) • Relation to heat diffusion operator 1 -1 Example I • Clustering: • ∆ f = 0 if f is constant on each component • Take Eigenfunction corresponding to 0 Eigenvalue • Segmentation etc Dagstuhl 2009

5. Example II: • Matching: Let Φ1, Φ2, …. be solutions to ∆ Φ = λΦ Then, use eigen-coordinate: Tsinghua U. 2009

6. Example III: • Smoothing • Eigenfunctions form a basis for functions on manifold • Relation to Heat diffusion operator • Relation to curvature flow Tsinghua U. 2009

7. In summary: • Intrinsic, isometry invariant • Eigenfunctions form a basis • Low eigenvalue corresponds to low-frequency mode • Relation to heat diffusion and other geometric quantities A spectral processing framework based on the Laplace operator Dagstuhl 2009

8. Dagstuhl 2009 Point Clouds (PCD) Setting • In practice, underlying manifold is seldom available • Only available through certain discrete approximation • Piecewise-linear approximation, points cloud. • First step in a spectral point clouds processing framework: • Approximate Laplace operator from discrete setting • As sampling becomes denser, converges to ground truth

9. Discrete Setting • Laplace operator is linear operator • ∆ f = g • In discrete setting, with vertices • V = { v1, v2, …., vn} • A function f:M -> R • n-vector f = [f(v1), f(v2), …, f(vn)] = [f1, f2, …, fn] • Discrete Laplace operator • n x n matrix L such that L f = g Goal: Construct matrix L

10. Previous Work • Point clouds: • Neighborhood graph => graph Laplace [Chung97] : • L f(v) = ∑u neighbor of v (f(u)-f(v)) / dv • If points are sampled from a Riemannian manifold: • Weighted graph Laplace [Belkin & Niyogi, 03] • Converge for points sampled from probabilistic distribution v

11. Previous Work • Surface meshes: • Cotangent scheme [Pinkall and Polthier93] [Desbrun et al. 99] • L f(x) = • Weak Convergence [Dziuk88] [Hildebrandt et al. 06] • No convergence result for general mesh [Xu06] • Cannot be extended to PCD Goal: General point clouds, with point-wise convergence: lime->0 || D f – Le f ||∞ = 0 for any fixed f

12. Dagstuhl 2009 Framework • Approximation via Heat operator [BN05] • Ht f : the heat distribution after time t • Ht = e-∆t=> Ht ≈ I - ∆t • ∆ = limt->0 (I – Ht) / t Lt≈ ( I – Ht ) / t [BN05] Next step: Approximate Lt in discrete setting

13. Dagstuhl 2009 y Discrete Case • Goal: Approximate from discrete samples • Statistical setting • w(y) = 1 / n • Mesh K: (e, r)-approximates M • w(y) = one-third of the 1-ring area

14. Point Clouds Data • Reconstruct Mesh? • Too expensive in high dimensions (1) Needs area weight w(y)around each point y (2) For every point x, a small neighborhood suffices Dagstuhl 2009

15. A Local Algorithm Approximate the area weight of each point using local information • Algorithm: for each p from P • Take neighboring points Q of p • Approx. tangent space Tp from Q • Project Q onto Tp, denoted by Q’ • Construct Vor(Q’), the Voronoi diagram of Q’ • Return Aq , area of Voronoi cell of q from Vor(Q’) Dagstuhl 2009

16. Dagstuhl 2009 Remarks • New PCD-Laplace operator: • Time complexity exponential only in intrinsic dimension • Error depends on sampling condition e • | L f(x) - ∆ f(x) | = O(ε) • More accurate even for uniformly sampled points • Need to know intrinsic dimension a priori • [SoCG08,SoDA09, Joint work with Belkin and Sun]

17. Dagstuhl 2009 Concluding Remarks • Spectral point clouds processing framework ! • Gradients / integral of a function • Segmentation / clustering based on heat kernel • Geometric network under Riemannian metric? • The use of diffusion distance metric • Bound error when using few embedding coordinates? • Complementing statistical methods? • How geometric and statistical methods can augment each other?

18. Dagstuhl 2009 Diffusion distance in Eigenspace • Map M to Eigen-space • f1, f2, …. : Eigenfunctions of ∆M • l1, l2, …. : Eigenvalues of ∆M • Euclidean distance in Eigen-space: Diffusion distance [Lafon]

19. y x Properties of Eigen-mapping • Two isometric objects • Have the same image in the eigenspace • Diffusion distance • Intuitive physical meaning • Robust, stable Dagstuhl 2009

20. Gradient in Eigenspace • In Eigenspace, gradient estimation becomes easier. • Metric in Eigenspace isometric upto a constant • Approximation becomes easier