The restoration problem is essentially one of optimal filtering with respect to some error criterion. The mean-squared error (MSE) criterion has formed the basis for most published work in this area [7-9].

In the case of linear stationary estimation we wish to estimate the process /with a linear combination of values of the data g. This can be expressed by a convolution operation f=h* g,

where h is a linear filter. A very general statement of the estimation problem is: given a set of data g, find the estimate f of an image fthat minimizes some distance ||f — f\\.

By using the mean-squared error, it is possible to derive the estimate with the principle o/orthogonality [6]:

Inserting the convolution expression in Eq. (21) thus gives

When the filter operates in its optimal condition, the estimation error (f — f) is orthogonal to the data g. In terms of the correlation functions Rg and Rgg,Eq. (23) can be expressed as [2]

By Fourier transforming both sides of the equation, we get S/g = HSgg, resulting in the familiar Wiener filter

When the data are the sum of the image and stationary white noise of zero mean and variance ai, then g = f + n

Rgg = Rif + Rnn, and the stationary Wiener filter is given by where n is the dimension of the signal (the theorem follows directly from Eq. 16, by taking the inverse Fourier transform of S(u) for x = 0). This means that the integral of the power spectrum of any process is positive. It can also be shown that the power spectrum is always nonnegative [6].

The cross-correlation function of two random processes is defined as

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