Читать книгу Graph Spectral Image Processing - Gene Cheung - Страница 25

Filtering in the graph frequency domain seems to be an intuitive extension of Fourier domain filtering into the graph setting. In fact, it coincides with time-domain filtering in a special case, beyond the intuition.

Suppose that the underlying graph is a cycle graph with length N, and its graph Laplacian L_cycle is assumed as follows:

[1.17]

where its blank elements are zero. It is well known that the eigenvector matrix of L_cycle is the DFT (Strang 1999), i.e.

[1.18]

in which

[1.19]

In other words, when we consider a cycle graph and assume its associated graph Laplacian is L_cycle, its GFT is the DFT. Therefore, graph spectral filtering in equation [1.13] is identical to the time-domain filtering. Note that, while U is the DFT, the interval of its eigenvalues is not equal to 2πk/N. Specificallly, the kth eigenvalue of L_cycle is λ_k = 2 − 2cos(2πk/N).