or
h(x,y) = s(-x,-y). (2.34)
Equation 2.34 shows that the impulse response of the matched filter
(with white noise) is simply the signal image in reverse order
(inverted and perverted). Thus, the filter is said to be matched to
the signal. Filtering with a matched filter is equivalent to cross-
correlating with the expected signal or pattern. That is,
O(x,y) = Rhs(x,y)
= ff s(xo,yo)h(xo-x,yo-y) dxo dy (2.35)
Also, it can be seen that the frequency response of the matched filter
is equivalent to that of the signal but with the phase negated so that
the output of the filter is real. That is, the matched filter removes
the phase variations and provides a real valued output.19
Matched filters are used extensively in radar signal
processing, seismic data processing, and communications. These
filters are implemented using electronic circuitry and digital
computers. For image processing, the need to process large two-
dimensional arrays places a large burden on conventional filtering
techniques. For these applications, optical processing techniques
provide the highest throughput speeds for matched filtering. One such
optical processing technique was proposed by Vander Lugt7 in 1969.
Vander Lugt Filtering
If an image is placed on a transparent sheet and illuminated by a
plane wave of coherent light, its Fourier transform is formed using a
simple lens.19 Once the Fourier transform is formed, specific
frequency components in the image can be removed or attenuated. The