where F' :is the modified image transform,
F is the original image transform,
and P(u,v) is the frequency emphasis chosen.
Phase-Only Filters
The preceding section describes techniques in which the high
frequencies are emphasized. This emphasis usually improves the
discrimination against false targets and increases hologram
efficiency. Frequency emphasis involves the multiplication of the
image transform by a filter function which attenuates or amplifies the
appropriate frequency components. The filter function adjusts the
spectral magnitude of the image. In the Fourier representation of
images, spectral magnitude and phase tend to play different roles and,
in some situations, many of the important features of a signal are
preserved even when only the phase is retained. Oppenheim15 showed
that when the magnitude portion of an image Fourier Transform is set
to an arbitrary constant and the phase left intact, the reconstructed
image closely resembles the original. Features of an image are
clearly identifiable in a phase-only image but not in a magnitude-only
image. Statistical arguments by Tescher30 and by Pearlman, and Gray31
have been applied to real-part, imaginary-part, and magnitude-phase
encoding of the discrete Fourier transform of random sequences. They
conclude that, for equivalent distortion, the phase angle must be
encoded with 1.37 bits more than the magnitude. Kermisch32 analyzed
image reconstructions from kinoforms, a phase-only hologram. He
developed an expansion of the phase-only reconstructed image I(x,y) in
the form