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Graph Heat Kernel Based Image Smoothing
Abstract
This chapter presents a new method for smoothing both gray-scale and color images, which relies on the heat diffusion equation on a graph. The image pixel lattice using a weighted undirected graph is presented. The edge weights of the graph are determined by the Gaussian weighted distances between local neighboring windows. The associated Laplacian matrix (the degree matrix minus the adjacency matrix) is computed then. The authors capture anisotropic diffusion across this weighted graph-structure with time by the heat equation, and find the solution, i.e. the heat kernel, 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. The relationship is also demonstrated between the authors’ method, standard diffusion-based PDEs, Fourier domain signal processing, and spectral clustering. The effectiveness of the method is illustrated by experiments and comparisons on standard images.
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