Rongjie Lai University of Southern California Joint work with: Jian Liang, Alvin Wong, Hongkai Zhao 1 Geometric Understanding of Point Clouds using Laplace-Beltrami operator

 Motivation and problems  Solving PDEs on point clouds  Moving least square method  Local mesh method  Geometric understanding of point clouds  Conclusion Outlines

 Point Cloud is a set of points with no ordering and connection. It is the most basic, natural and intrinsic representation of geometric object and data with nonlinear structures in 3D and high dimension.  Point clouds can be ideally treated as data sampled from n-dimensional manifold in R^d. Motivation and Problems 3D modeling high dimension data global recognition intrinsic distance

Challenges  Highly unstructured geometric objects. Global mesh based approaches or implicit approaches can not be applied.  No natural basis for representation.  Non-uniform sampling  Solve geometric PDEs and variation problems on point clouds. Need to approximate differential operators and integrals on point clouds.  Obtain global structure and geometric understanding of point clouds then underlying nonlinear structures. Questions ?

Previous methods  PDEs on point clouds PCD Laplacian: M Blkin, J. Sun, Y. Wang, 2009 o Use heat diffusion in the ambient space R^d o Lower order approximation o Only restricted on Laplacian  Integral approximation on point clouds o Statistical approach, Monte carlo integration o Cauchy- Crofton formula. X. Li et. al o Voronoi weight scheme. C. Liu et. al 2009

Solving PDEs on point clouds Two systematic methods are developed to solve PDEs on point clouds. Key features of our methods:  No global mesh or global parameterization is needed.  Only local information such as K nearest neighbors are needs.  Our methods works for manifolds with point clouds sampled from manifold with any dimension and co-dimension.  The complexity scales well with the total number of points and the true dimension of the underlying manifold. Moreover, we would like to have geometric understanding of point clouds using the results of PDEs on point clouds.

Differential operators on manifolds  Surface gradient operator  Laplace-Beltrami operator

Moving Least square method  K nearest neighbor (KNN)  Locally principle component analysis (PCA) Local Coordinate system construction

Moving Least square method  KNN of have local coordinate  Find a local degree two polynomial by minimizing the following weighted sum: Local manifold approximation is a smooth representation of the surface near

Moving Least square method Local function approximation  Similarly, locally approximate function by minimizing the following weighted sum:  In the local coordinate system, manifolds and functions are smoothly defined.  Then standard definition of differential operators in differential geometry can be applied. Key points:

Local mesh method  K nearest neighbor (KNN)  Locally principle component analysis (PCA)  Global triangle Mesh structures are hard to have, in particular in high dimension case.  local mesh structures are relatively easy to have in the flat tangent space.  Local Delaunay triangulation can be applied on the tangent space, then local connectivity are adapted to the neighbor of each point. Local Mesh construction

Local mesh method Differential operators approximation

Local mesh method Integral approximation Mass Matrix : Stiffness Matrix :

Point clouds from n-dimension manifolds in R^d  Using local PCA on KNN can find local dimension of the manifold as well as the tangent and normal spaces.  Local approximation of manifolds and functions can be parameterized on tangent spaces.  Local mesh structures on tangent space can be also obtained. The complexity of our methods scales well with the true dimension of the manifold

Integrals on point clouds C. Luo, J. Sun, Y. Wang. Integral estimation from point cloud in d-dimensional space: a geometric view. SCG. 2009

The heat diffusion using both methods

The Ekinoal equation

Geodesic tracking

The Helmholtz equation M Blkin, J. Sun, Y. Wang, Constructing Laplace operator from point clouds in R^d. ACM-SIAM SDA, 2009

The Helmholtz equation

Relation between LB Eigen-system and surface geometry  Sturm-Liouville: Heat kernel:  Heat trace asymptotic expansion:  LB eigenfunctions are Morse functions.  LB spectral embedding. where B is the boundary of M, K is the Gauss curvature of M and J is the mean curvature of B in M. Moreover, if M is a closed surface with euler number (M), then c 2 = 2 (M)/3

From local to global: Laplace-Beltrami operator Reeb graphs and Skeleton structures Generically, LB eigenfunctions are a Morse function for most of surfaces

Point clouds comparison/registration

Point clouds clustering/segmentation/

Global information: conformal structures extraction

Conclusions: ◦ Two systematic methods to solve PDEs on point clouds ◦ Solve problems on any dimension and co-dimension ◦ Computational complexity scales well with the number of points and the true dimension of the underlying manifolds. ◦ Has potential to extract geometric information for nonlinear structures. ◦ Explore more applications on 3D modeling and data science.

Thanks for your attention! For more computation about the LB eigenproblem on manifolds and applications, please check Rongjie Lai University of Southern California Our papers: Rongjie Lai, Jian Liang and Hongkai Zhao, A Local Mesh Method for Solving PDEs on Point Clouds, UCLA CAM Report, Jian Liang, Rongjie Lai, Tsz Wai Wong and Hongkai Zhao, Geometric Understanding of Point Clouds Using Laplace-Beltrami operator, CVPR, 2012 Jian Liang and Hongkai Zhao, Solving Partial Differential Equations on Point Clouds, UCLA CAM Report, 12-25