This paper systematically studies the well-known Mexican hat wavelet (MHW) on manifold geometry, including its derivation, properties, transforms, and applications. The MHW 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 MHW 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 MHW is localized in both space and frequency, which enables space-frequency analysis of input functions. We defined its continuous and discrete transforms as convolutions of bivariate kernels, and propose a fast method to compute convolutions by Fourier transform. To broaden its application scope, we apply the MHW to graphics problems of feature detection and geometry processing. © 2012 Elsevier Inc. All rights reserved.
CITATION STYLE
Hou, T., & Qin, H. (2012). Continuous and discrete Mexican hat wavelet transforms on manifolds. In Graphical Models (Vol. 74, pp. 221–232). https://doi.org/10.1016/j.gmod.2012.04.010
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