Fourier-based and Rational Graph Filters for Spectral Processing.

Abstract

Data are represented as graphs in a wide range of applications, such as Computer Vision (e.g., images) and Graphics (e.g., 3D meshes), network analysis (e.g., social networks), and bio-informatics (e.g., molecules). In this context, our overall goal is the definition of novel Fourier-based and graph filters induced by rational polynomials for graph processing, which generalise polynomial filters and the Fourier transform to non-Euclidean domains. For the efficient evaluation of discrete spectral Fourier-based and wavelet operators, we introduce a spectrum-free approach, which requires the solution of a small set of sparse, symmetric, and well-conditioned linear systems and is oblivious of the evaluation of the Laplacian or kernel spectrum. Approximating arbitrary graph filters with rational polynomials provides a more accurate and numerically stable alternative with respect to polynomials. To achieve these goals, we also study the link between spectral operators, wavelets, and filtered convolution with integral operators induced by spectral kernels.