ABSTRACT
This chapter presents the Mexican hat wavelet defined on manifold data. This continuous wavelet is rigorously derived from the heat kernel by taking the negative first-order derivative with respect to time. As a solution to the heat equation, it has a clear initial condition: the Laplace-Beltrami operator. Following a popular methodology in mathematics, we analyze the wavelet and its transforms from a Fourier perspective. By formulating Fourier transforms of bivariate kernels and convolutions, we obtain its explicit expression in the Fourier domain, which is a scaled differential operator continuously dilated via heat diffusion. The wavelet has localization in both space and frequency, which enables space-frequency analysis of input functions.
