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