Diffusion-Driven Kernel Design and Algorithms for Dynamic Shape Analysis

Abstract

With the rapidly development of 3D data acquisition, a massive collection of dynamic shapes emerge and become ubiquitous in various real-world applications. It results in the urgent need of techniques for dynamic shape analysis and processing. Accordingly, a large body of literature has been dedicated to this study, in which heat diffusion and related tools are frequently used. This dissertation concentrates on kernels and algorithms derived from the diffusion theory, with the purpose of developing new techniques for dynamic shape analysis. Bivariate kernels represent point-to-point relations on manifolds. We introduce three kernels: geodesic Gaussian, admissible diffusion wavelet, and Mexican hat wavelet. The geodesic Gaussian is a Gaussian with geodesic metric, which is isometric invariant. It is a good approximation to the heat kernel at small scales. For large scales, we propose an efficient computing by a pyramid structure and the semi-group property. The admissible diffusion wavelet comes from diffusion wavelets. It is constructed in a bottom-up fashion by a diffusion operator and its dyadic powers. It can extract details of a function at different scales. The Mexican hat wavelet is defined as the negative first derivative of the heat kernel with respect to time. It is a solution to the heat equation with the Laplace-Beltrami operator as initial condition. We explore applications of these kernels in dynamic shape analysis, including feature detection, multiscale approximation, shape representation, geometry processing, etc. Functions generated by multiple kernels further enrich the kernel family. We present heat kernel coordinates, together with a complete solution for dense registration of partial nonrigid shapes. The coordinates consist of heat kernels from a set of features, and their magnitudes serve as priorities in registration. Based on diffusion wavelets, we propose the probability-density-function distance. It measures probability distributions rather than single points, which makes it resilient to small perturbations. For applications, we apply it to local coordinates and volumetric image registration. Together with some collaborators, we extend our work to anisotropic kernels and more algorithms, which demonstrate the wide application scope of the diffusion theory. At the end, we conclude this dissertation, by comparing the proposed kernel functions, discussing some remaining challenges, and envisioning broader applications.