signal-to-noise ratio is large, the energy reaching the correlation
plane may be too small to measure. The efficiency of the hologram,
the ratio of the power in the correlation to the power in the input
test image, is an important criterion in evaluating a practical CGH-
matched filter. Mathematically, it is given as
ff Ig(x,y) f*(x,y) 2dx dy
= _____________________ (4.1)
H
ff Ig(x,y)I2dxdy
where H has been coined the Horner efficiency,28 f is the reference
scene, g is the test scene, and denotes an ideal correlation. The
correlation derived from a Vander Lugt-matched filter is not ideal.
To determine the Horner efficiency for a CGH-matched filter, equation
4.1 must include an accurate model of the encoding scheme. This
efficiency can be measured experimentally using a known input source
and calibrated detectors. Caulfield28 estimated that efficiencies for
certain matched filters could be as low as 10-6. Butler and Riggins29
used models of CGH filters to verify Caulfield's prediction and went
on to recommend techniques for improving the efficiency.
The matched filter is used to determine the presence of a target
in a large scene. A test scene is correlated with a reference, and
the correlation plane is thresholded to indicate the target location.
Occasionally, the Vander Lugt filter will generate correlation values
above the threshold in areas where no target exists. Accordingly, the
correlation of an actual target corrupted by noise may be lower than
the threshold. Due to the presence of noise, random and otherwise,
the performance of the filter must be measured in terms of the
probability of detection and the probability of false alarm. The
probability of detection, Pd, is defined as the probability that a