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Chapter 3:
Review of Signal-Restoration Techniques

3.1 Introduction

Many researchers have proposed different approaches for overcoming the obstacles to solving the inverse problem. Arguably the most popular classical technique is the Wiener-Kolmogorov filter technique. This technique is implemented in the Fourier domain using the power spectra of the respective images and functions. Modern methods of image restoration are most often based on regularization theory. At the heart of these techniques is the idea of conditioning the solution in a certain manner so as to make it stable.

In the rest of this work m, n will denote spatial domain coordinates, while u, v will denote frequency-domain coordinates. Also, small letters represent functions in the spatial domain whereas their capitalized counterparts represent functions in the frequency domain. Letters with a hat on top, such as z, emphasize the fact that the symbol represents an estimate of the value or function.

3.2 Wiener-Kolmogorov Filters

We have used the Fourier inverse filter Fi (u, v) to obtain an estimate for the process x given the observed process y. TheWiener-Kolmogorov filter Fw (u, v) overcomes the stability problems associated with direct inverse filters. This filter was proposed by Bracewell and Heltron based on the work of Wiener and Kolmogorov. A notable feature of the filter is that it is the MMSE linear filter. The filter uses the ratio of the power spectra of the image and noise to prevent noise amplification. It is given by the following equation.

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