represent those values, and the practical filter is merely an
approximation to the ideal filter. It is worth noting that when the
distorting response reduces a frequency component to zero or below
some noise threshold, that component cannot be recovered. That is,
information is usually lost during the distorting process and inverse
filtering cannot recover it.
It is desirable to remove noise from a corrupted image. Although
it is not always possible to remove all of the noise, the
relationships between the input and output of a.linear system are
known. A linear system is optimized when most of the noise is
removed. To optimize a system the input must be specified, the system
design restrictions known, and a criterion of optimization accepted.
The input may be a combination of known and random signals and noises.
The characteristics of the input such as the noise spectrum or
statistics must be available. The classes of systems are restricted
to those which are linear, space-invariant, and physically realizable.
The criterion of the optimization is dependent on the application.
The optimum filters include the least mean-square-error (Wiener)
filter and the matched filter. The Wiener filter minimizes the mean-
squared-error between the output of the filter and actual signal
input. The Wiener filter predicts the least mean-squared-error
estimate of the noise-corrupted input signal. Thus, the output of the
Wiener filter is an approximation to the input signal. The output of
the matched filter is not an approximation to the input signal but
rather a prediction of whether a specific input signal is present.
The matched filter does not preserve the input image. This is not the
objective. The objective is to distort the input image and filter the