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Data are represented as graphs in a wide range of applications, such asComputer Vision (e.g., images) and Graphics (e.g., 3D meshes), network analysis(e.g., social networks), and bio-informatics (e.g., molecules). In thiscontext, our overall goal is the definition of novel Fourier-based and graphfilters induced by rational polynomials for graph processing, which generalisepolynomial filters and the Fourier transform to non-Euclidean domains. For theefficient evaluation of discrete spectral Fourier-based and wavelet operators,we introduce a spectrum-free approach, which requires the solution of a smallset of sparse, symmetric, well-conditioned linear systems and is oblivious ofthe evaluation of the Laplacian or kernel spectrum. Approximating arbitrarygraph filters with rational polynomials provides a more accurate andnumerically stable alternative with respect to polynomials. To achieve thesegoals, we also study the link between spectral operators, wavelets, andfiltered convolution with integral operators induced by spectral kernels.According to our tests, main advantages of the proposed approach are (i) itsgenerality with respect to the input data (e.g., graphs, 3D shapes),applications (e.g., signal reconstruction and smoothing, shape correspondence),and filters (e.g., polynomial, rational polynomial), and (ii) a spectrum-freecomputation with a generally low computational cost and storage overhead.

Fourier-based and Rational Graph Filters for Spectral Processing