Continuous and discrete Mexican hat wavelet transforms on manifolds
- 作者：Tingbo Hou ; thou@cs.stonybrook.edu ; Hong Qin qin@cs.stonybrook.edu
- 关键词：Mexican hat wavelet ; Heat diffusion ; Biharmonic wavelet ; Feature detection ; Geometry processing
- 刊名：Graphical Models
- 出版年：2012
- 期刊代码：130_15240703
- 类别：cp
- 出版时间：July, 2012
- 卷：74
- 期：4
- 页码：221-232
- 文件大小：1690 K
摘要
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.