Spectral Graph Based Vertex-Frequency Wiener Filtering for Image and Graph Signal Denoising

In this article, we propose and develop a spectral graph based vertex varying Wiener filtering framework in the joint vertex-frequency domain for denoising of graph signals defined on weighted, undirected and connected graphs. To this end, we first extend the Zadeh time-frequency filter concept to graph signals and obtain vertex-frequency transfer function of the proposed Wiener filter by transforming its vertex varying impulse response that minimizes the mean square error between original and recovered signals. To facilitate the derived Wiener filter, we present a detailed derivation of a recently proposed graph Rihaczek vertex-frequency signal distribution (GRD) so as to match the structure of the proposed graph Zadeh filter, based on a graph translation operator defined by generalized convolution with a delta signal. We express the filter transfer function in terms of this graph transform of the original, noiseless signal. The form of the obtained Wiener filter is, interestingly, different than those of time-frequency Wiener filters prevalent in the classical signal processing. We also investigate the invertibility of the employed GRD. Assuming that the original graph signal of interest can be viewed as deterministic, we propose two algorithms to implement the proposed vertex-frequency Wiener filter from a single realization of a noisy, input signal. We derive mean and variance of the GRD of the noisy signal, since they are required in one of these algorithms. We apply proposed Wiener filter algorithms to denoise a standard set of images and three irregularly structured graph signals, and demonstrate their competitiveness with compared high performance denoising methods.