I(x,y) = A [Io'(x,y) + 1/8 Io'(x,y/@Ro'(X,Y)
+ 3/64 Io'(x,y)Ro'(x,y)@Ro'(x,y) + .] (43)
where Io'(x,y) is the normalized irradiance of the original object,
Ro'(x,y) is the two-dimensional autocorrelation function of Io'(x,y)
and denotes convolution. The first term represents the desired
image, and the higher terms represent the degradation. Kermisch
showed that the first term contributed 78% to the total radiance in
the image, giving a ratio of 1.8 bits.
The phase-only image typically emphasizes the edges as in the case
of the high-pass filtering as shown in Figure 4.1. This phase-only
filtering is closely related to the high-pass filter. Most images
have spectra where the magnitude tends to drop off with frequency. In
the phase-only image, the magnitude of each frequency component is set
to unity. This implies multiplying each pixel magnitude by its
reciprocal. The Fourier transform tends to fall off at high
frequencies for most images, giving a mound-shaped transform. Thus,
the phase-only process applied to a mound-shaped Fourier Transform is
high-pass filtering. The phase-only image has a high-frequency
emphasis which accentuates edges. The processing to obtain the phase-
only image is highly non-linear. Although the response 1/IF(u,v)l
generally emphasizes high frequencies over low frequencies, it will
have spectral details associated with it which could affect or
obliterate important features in the original. Oppenheim15 proposed
that if the Fourier transform is sufficiently smooth, then
intelligibility will be retained in the phase-only reconstruction.
That is, if the transform magnitude is smooth and falls off at high