Approximating Gradients for Meshes and Point Clouds via Diffusion Metric

1. Approximating Gradients for Meshes and Point Clouds via Diffusion Metric Chuanjiang Luo, Issam Safa and Yusu Wang Department of Computer Science and Engineering The Ohio State University

2. Motivation • Gradient is one of the most differential objects. • Input: • Points P sampling a Riemannian manifold M • Function F: M -> R given as values at points in P • Goal: • Approximating gradient field, critical points of F SGP 2009

3. Related Work • Given a mesh K approximating M • Compute a piecewise linear approximation of F on K • Point cloud data • Local optimization [CGAL] • Statistical setting [Mukherjee et al 08] • Issue: • Sensitive to noise : • Smooth gradient field ? • How to generate gradient field at multi-scales ? SGP 2009

4. Our Contribution • Initiate the study of gradients under a different metric on the underlying manifold M • Method of computing gradients from the eigenspace • Provide a natural way of smoothing gradient • New discrete Laplace operator for point clouds data SGP 2009

5. 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. ! SGP 2009

6. Laplace-Beltrami Operator • Applications: • Smoothing • Matching • Clustering • In summary: • Intrinsic, isometry invariant • Eigenfunctions form a basis • Low eigenvalue corresponds to low-frequency mode SGP 2009

7. Laplace-Beltrami Operator • In summary: • Intrinsic, isometry invariant • Eigenfunctions form a basis • Low eigenvalue corresponds to low-frequency mode SGP 2009

8. Cotangent Scheme • For mesh only • Discrete PCD Laplace Operator [Belkin et al 09] • The only one for point clouds setting with convergence guarantee • New Discrete Laplace Operator for Point Cloud • Real eigenvalue and eigenvectors • Eigenvectors form an orthonormal basis with respect to area weight. SGP 2009

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

10. y x Diffusion Distance • Stable, robust • Intuitive physical meaning SGP 2009

11. Gradient in Eigenspace • Given a square-integrable function F: M -> R • Corresponds to G: Ft(M) -> R in Eigenspace as G(Ft(x)) = F(x) • and • Set • Given point cloud P • Construct Laplace operator, compute fi’s and li’s • Given a function F: P -> R • Compute ai = < F, fi> and set up V • For each x, approximate TF(x) in Eigen-space • Project V onto TF(x) SGP 2009

12. When using Gaussian kernel to approximate heat kernel For fixed t, Summary • Coarser gradient by taking only top k Eigenfunctions • Eigenfunctions with higher eigenvalues  noise • Smooth noises in functions and in manifold simultaneously! • Time complexity remains same ! • Compute gradient using a different metric SGP 2009

13. Experimental Results (1) 150 50 20 10 5 SGP 2009

14. Experimental Results (2) ground truth our method quadratic fitting SGP 2009

15. Experimental Results (3) • Gradient field for a molecular surface • Input function: Connolly function 1000 Eigenfunctions 100 Eigenfunctions SGP 2009

16. Experimental Results (4) • Jacobi sets of two functions • Compute correlation of two functions • Jacobi sets between two coordinates functions • Produce silhouette of input object • Previously, • no algorithm to compute / simplify jacobi sets from point clouds data SGP 2009

17. Jacobi Sets combinatorial 1000 eigenfunctions 50 eigenfunctions SGP 2009

18. Jacobi Sets Laplace Eigen-mapping computed Only from one model ! SGP 2009

19. Critical Points SGP 2009

20. From mesh 200 eigenfunctions Critical Points cont. SGP 2009

21. Timing Download: http://www.cse.ohio-state.edu/~luoc/eigen-gradient.htm SGP 2009

22. Conclusion and Future Work • New discrete point-cloud Laplace operator • Novel way to compute gradient in eigenspace • Accurate, robust to noise, easy to simplify Future Work • Spectral point clouds processing framework • Efficient way for constructing Laplace and eigen-decomposition Thank you! SGP 2009