frequencies, then the principal effect of the whitening process is to
emphasize the high frequencies and therefore the edges in the image,
thereby retaining many of the recognizable features. In Figures 4.1
and 4.2 the phase-only filter emphasizes edges more strongly than a
gradient filter for the examples shown.
The advantage of using a phase-only image or high-pass image is
the increase in optical efficiency of the resultant matched filter.
As shown in equation 2.35, the transmission of each hologram element
depends on the magnitude of the reference image Fourier transform. As
the magnitude drops off for high frequencies, so does the transmission
of light through the holographic filter, and hence filter efficiency
is low. If the magnitude is set to unity (phase-only filter) for all
frequencies, the overall efficiency increases dramatically. The image
transform is white and thus the throughput of the absorption hologram
is highest. Horner14 shows that the maximum throughput efficiency of
an ideal autocorrelation of a 2-D rect function is only 44%, while the
autocorrelation using an phase-only filter achieves 100% efficiency.
The phase function, ((u,v) of an image Fourier transform is a
continuous function. To fabricate a phase-only filter for such an
image requires a linear process capable of faithfully reproducing the
whole range of values from 0 to 2 If the phase is quantized so as
to permit only two values, typically 0 and pi, such a filter is known
as a bi-phase filter.
H'(u,v) = sgn [cos 4(u,v)] = +1 if Re [H(u,v)] > 0 (4.4)
= -1 otherwise
where H(u,v) is the Fourier transform of the filter impulse response
h(x,y), the sgn operator gives the sign of the argument, and H'(u,v)