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Graph spectral image smoothing using the heat kernel

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JournalPattern recognition
DatePublished - Nov 2008
Issue number11
Volume41
Number of pages15
Pages (from-to)3328-3342
Original languageEnglish

Abstract

A new method for smoothing both gray-scale and color images is presented that relies on the heat diffusion equation on a graph. We represent the image pixel lattice using a weighted undirected graph. The edge weights of the graph are determined by the Gaussian weighted distances between local neighboring windows. We then compute the associated Laplacian matrix (the degree matrix minus the adjacency matrix). Anisotropic diffusion across this weighted graph-structure with time is captured by the heat equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigensystem with time. Image smoothing is accomplished by convolving the heat kernel with the image, and its numerical implementation is realized by using the Krylov subspace technique. The method has the effect of smoothing within regions, but does not blur region boundaries. We also demonstrate the relationship between Our method, standard diffusion-based PDEs, Fourier domain signal processing and spectral Clustering. Experiments and comparisons on standard images illustrate the effectiveness of the method. (C) 2008 Elsevier Ltd. All rights reserved.

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