Harmonic Shape Analysis: From Fourier to Wavelets

Harmonic analysis studies the representation of functions as the linear combination of basic wave-like functions. It plays a fundamental role in the processing of time-series signals and images. Recent years have witnessed many efforts to adapt the Fourier and wavelet analysis to the domain of 3D shapes. The manifold Fourier analysis relies on the eigenfunctions of the Laplace-Beltrami operator to analyze geometric shapes. Analogous to classic Fourier basis, the eigenfunctions of the Laplace-Beltrami operator form an orthonormal basis of a Hilbert space on the manifold, and can be used to decompose functions defined on the manifold as generalized Fourier series. The manifold Fourier analysis is essential in solving the heat equation and heat kernel on manifold. Since the eigenvalues and eigenfunctions of the Beltrami-Laplace operator are intrinsic and globally shape-aware, they are well suited for constructing isometry-invariant shape descriptors and distance metrics, facilitating higherlevel analysis tasks such as matching, registration and retrieval. Wavelet transform allow signals to be decomposed into elementary forms at different positions and scales. It is advantageous to Fourier transform in that wavelet functions can be simultaneously localized in both time/space and frequency domain. Various types of manifold wavelets have been proposed, based on subdivision, diffusion operator or spectral decomposition. In this report, we first explain the theoretical background of the manifold Fourier analysis, focusing on its connection with signal processing and heat diffusion. We also give an overview of shape analysis applications based on spectral methods. We then provide a survey of various types of manifold wavelet transform, particularly the spectral manifold wavelet transform. Finally we present some of our on-going research on spectral shape analysis, including isometric shape matching, registration and retrieval.