Spectral graph based joint vertex-frequency Wiener filtering for image and graph signal denoising

In this thesis, spectral graph based vertex-frequency Wiener filtering scheme is proposed and developed for image and graph signal denoising. Zadeh time-frequency filter concept in classical signal processing is first extended to graph signals and vertex-frequency transfer function of this filter is obtained from its vertex varying impulse response minimizing the mean square error between original and estimated graph signal. A detailed derivation of graph Rihaczek vertex-frequency distribution (GRD) based on a graph shift operator defined by generalized convolution with a delta signal is presented to facilitate the derived Wiener filter. The form of the obtained Wiener filter is different than those of time-frequency Wiener filters prevalent in the classical signal processing. Moreover, the invertibility of the employed GRD is investigated. As another work, a vertex-frequency graph Wiener filter framework in the windowed graph Fourier transform domain is proposed and developed to denoise irregularly structured graph signals only. Under the assumption that the graph signal is deterministic, some algorithms are proposed to implement these proposed vertex-frequency Wiener filters, and they are applied for denoising of a set of images and irregular graph signals, and their performances are compared with recent denoising methods.