Читать книгу Graph Spectral Image Processing - Gene Cheung - Страница 14

Closely related to the combinatorial graph, Laplacian L, are other variants of Laplacian operators, each with their own unique spectral properties. A normalized graph Laplacian L_n = D^−1/2LD^−1/2 is a symmetric normalized variant of L. In contrast, a random walk graph Laplacian L_r = D⁻¹L is an asymmetric normalized variant of L.A generalized graph Laplacian L_g = L + diag(D) is a graph Laplacian with self-loops d_i,i at nodes i – called the loopy graph Laplacian in Dörfler and Bullo (2013) – resulting in a general symmetric matrix with non-positive off-diagonal entries for a positive graph (Biyikoglu et al. 2005). Eigen-decomposition can also be performed on these operators to acquire a set of graph frequencies and graph Fourier modes. For example, normalized variants L_n and L_r (which are similarity transforms of each other) share the same eigenvalues between 0 and 2. While L and L_n are both symmetric, L_n does not have the constant vector as an eigenvector. Asymmetric L_r can be symmetrized via left and right diagonal matrix multiplications (Milanfar 2013a). Different variation operators will be used throughout the book for different applications.