optical and digital, tend to suffer from two types of difficulties.

They tend to be too sensitive to differences likely to occur in the

desired pattern. These differences are termed "within-class

variations." Second, they tend to be too insensitive to differences

between real and false targets. These are "between-class variations."

While other deformations in the object condition are possible in

specific applications, translation, rotation, and scale are the most

common in pattern recognition whether it is accomplished optically or

digitally.

 Deformation Invariant Optical Pattern Recognition

 The basic operation performed in an optical processor is a two-

dimensional Fourier transform. Matched spatial filters are used to

perform correlations between an input image and a reference pattern.

While the reference pattern may exist in the input image, it may be

deformed by scale, rotation or geometrical distortion. The Fourier

transform is invariant to shift in two dimensions (see equation 2.6).

It is not however invariant to scale or rotation, and a dramatic loss

in signal-to-noise ratio (3 to 30 dB) occurs for small scale changes

(2%) or rotation (3.50).44

 In some applications it is desirable to give up translation or

shift invariance in substitution for some other deformation

invariance. The technique described by Casasent and Psaltis45

involves a space variant coordinate transformation to convert the

deformation under consideration to a shift in the new coordinate

system. Because the optical processor performs two-dimensional

transforms, it is insensitive to shifts in two dimensions. Thus, two

separate invariances can be accommodated. Scale can be converted to a