Random Fields and Wiener Filtering

A central problem in the application of random fields is the estimation of various statistical parameters from real data. The Wiener filter that will be discussed later requires knowledge of the spectral content of the image signal and the background noise. In practice these are, in general, not known and have to be estimated from the image.

3.1 Autocorrelation and Power Spectrum

A collection of an infinite number of random variables defined on an «-dimensional space (x e R") is called a random field. The autocorrelation function of a random field f (x) is defined as the expected value of the product of two samples of the random field,

where E denotes the statistical expectation operator. A random field is said to be stationary if the expectation value and the autocorrelation function are shift-invariant, i.e., the expectation value is independent of the spatial position vector x, and the autocorrelation is a function only of x = x — x'.

The power spectrum of a stationary random process is defined by the Fourier transform of the autocorrelation function.

Since the autocorrelation function is always symmetric, the power spectrum is always a real function.

A random process n(x) is called a white noise process if

From Eq. (16), the power spectrum of a white process is a constant:

The Wiener-Khinchin theorem [6] states that

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