The chains studied in this paper generalize Chern–Moser chains for CR structures. They form a distinguished family of one dimensional submanifolds in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be equivalently encoded as Cartan geometries (of different types). The aim of this paper is to study the relation between
these two Cartan geometries for Lagrangean contact structures and partially integrable almost CR structures.
We develop a general method for extending Cartan geometries which generalizes the Cartan geometry interpretation of Fefferman’s
construction of a conformal structure associated to a CR structure. For the two structures in question, we show that the Cartan geometry associated to the family of chains can be obtained in that way if and only if the original parabolic contact structure is torsion free. In particular, the procedure works exactly on the subclass of (integrable) CR structures.
This tight relation between the two Cartan geometries leads to an explicit description of the Cartan curvature associated to the
family of chains. On the one hand, this shows that the homogeneous models for the two parabolic contact structures give rise to examples of non–flat path geometries with large automorphism groups. On the other hand, we show that one may (almost) reconstruct
the underlying torsion free parabolic contact structure from the Cartan curvature associated to the chains. In particular, this leads to a very conceptual proof of the fact that chain preserving contact diffeomorphisms are either isomorphisms or antiisomorphisms of parabolic contact structures.
Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear
dimensionality reduction. Graph Laplacian is one widely used discrete laplacian on point cloud. It was
previously proved by Belkin and Niyogi  that the eigenvectors and eigenvalues of the graph Laplacian
converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in
the limit of innitely many data points sampled independently from the uniform distribution over the
manifold. Recently, we introduced Point Integral method (PIM)  to solve elliptic equations and
corresponding eigenvalue problem on point clouds. In this paper, we prove that the eigenvectors and
eigenvalues obtained by PIM converge in the limit of innitely many random samples. Moreover, one
estimate of the rate of the convergence is given.
Poisson equation on point cloud with Dirichlet boundary condition plays important
role in many problems. In this paper, we use the volume constraint proposed by Du et.al to handle
the Dirichlet boundary condition in the point integral method for Poisson equation on point cloud. We
prove that the solution given by volume constraint converges to the true solution as the point cloud
converges to the underlying smooth manifold.
We present the application of a low dimensional manifold model (LDMM) on hyperspectral
image (HSI) reconstruction. An important property of hyperspectral images is that the
patch manifold, which is sampled by the three-dimensional blocks in the data cube, is generally of
a low dimensional nature. This is a generalization of low-rank models in that hyperspectral images
with nonlinear mixing terms can also fit in this framework. The point integral method (PIM) is used
to solve a Laplace-Beltrami equation over a point cloud sampling the patch manifold in LDMM.
Both numerical simulations and theoretical analysis show that the sample points constraint is correctly
enforced by PIM. The framework is demonstrated by experiments on the reconstruction of
both linear and nonlinear mixed hyperspectral images with a significant number of missing voxels
and several entirely missing spectral bands.
In this paper, we propose a novel low dimensional manifold model (LDMM) and
apply it to some image processing problems. LDMM is based on the fact that the patch manifolds
of many natural images have low dimensional structure. Based on this fact, the dimension of the
patch manifold is used as a regularization to recover the image. The key step in LDMM is to solve
a Laplace-Beltrami equation over a point cloud which is solved by the point integral method. The
point integral method enforces the sample point constraints correctly and gives better results than the
standard graph Laplacian. Numerical simulations in image denoising, inpainting and super-resolution
problems show that LDMM is a powerful method in image processing.