Optimal Wiener and homomorphic filtration: Review

Abstract

In this paper we analyze the methods of optimal restoration of signals and images from the degradations that are defined by the additive noise and blurring effects. The Wiener filters which are the best linear filters are described, as well as the geometric-mean filters that compose a family of filters indexed by a parameter k varying between 0 and 1. These filters are used to provide frequency-based filtering that mitigates the noise suppression of the optimal-linear Wiener filter in the blurred-signal-plus-noise model. For k=1 and k=0, the geometric-mean filter gives the inverse filter and the Wiener filter for the model, respectively. The geometric mean filter for k=1/2, which is called the homomorphic filter, is the optimal linear filter for the model under power-spectral-density (PSD) equalization. This constraint requires the PSD of the filtered signal to be equal to the PSD of the uncorrupted signal that it estimates. Robustness of the optimal filters is also described and examples of image restoration are given. In the 2-D case, the Wiener and homomorphic filters have very effective realization when images are described in tensor representation as an unique set of 1-D image-signals and the 2-D discrete Fourier transform (DFT) is split into a set of 1-D DFTs of image-signals.