formed on the film rather than a sharp point. The impulse response is
a line, and the corresponding frequency response of the distorting
system is a sine function (sin u /u). To retrieve the European
photo collection, merely multiply the Fourier transform of the
pictures by u/sin u and re-image.
In the physical implementation of this process, there are several
practical problems. To multiply the image transform by the inverse
function, a transparency with the appropriate response is produced.
In general, a transparency can only attenuate the light striking it.
That is, the transparency can only represent non-negative real values
less than one. Herein lies the problem. The inverse response
required to correct a specific distortion may, in fact, be complex.
In some cases, a combination of two transparencies can be combined to
provide complex values. One transparency is used for amplitude or
attenuation, and another phase transparency or phase plate is used to
provide the appropriate phase shift at each point. A phase
transparency can be produced by bleaching film with an appropriate
latent image induced in the emulsion. Chu, Fienup, and Goodman18
demonstrated a technique in color film which consists of three
emulsions. One emulsion was used as an amplitude transparency and
another emulsion was used as a phase plate. The appropriate patterns
were determined by a computer and the film was given the proper
exposure using colored filters.
Even with a two-transparency system, not all distortions are
possible to remove. Note that in the linear blur case, the inverse
response is u/sin u. The denominator goes to zero for specific values
of u, and the response has a pole at those values. The filter cannot