Data Dialogue

In this talk, I will present a low dimensional manifold model (LDMM) and apply it to some image reconstruction 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 or weighted graph Laplacian. These two types of discretization enforce the sample point constraints correctly. A semi-local patch distance is used to compute the metric on the patch manifold. Numerical simulations in image processing, hyperspectral imagery reconstruction, and plasma inpainting show that LDMM is a powerful method in reconstruction of noisy and incomplete data.
https://services.math.duke.edu/mcal?abstract-9882