Heat Kernel Smoothing in Irregular Domains

We review the heat kernel smoothing techniques for denoising and regressing data in irregularly shaped domains embedded in Euclidean spaces. This is a problem often encountered in functional data analysis and medical imaging. In this chapter, we present a unified mathematical framework based on the eigenfunctions of the Laplace-Beltrami operators defined on irregular domains. Numerical implementation issues will be addressed as well. Various examples will be presented. We also present a few new theoretical results on the properties of heat kernel smoothing.