shift in one direction and the normal shift can be left in the other
dimension. This would provide scale invariance, but the resultant
correlation would only yield the location of the target in only one
dimension (i.e. the x coordinate).
In another example, the scale can be converted to shift in one
dimension and rotation converted to shift in another dimension. Such
a two-dimensional optical correlator could provide correlations on
rotated and scaled objects but would no longer predict the location of
the object. The two-dimensional nature of the optical processor
allows the correlator to be invariant to both deformations. In order
to provide invariance to other deformations two at at time, a
coordinate transformation is needed to convert that deformation to a
coordinate shift. The Mellin Transform is an excellent example of
such a transformation used to provide scale and rotation invariance.
The Fourier transform is invariant to translation shift in two
dimensions. To provide invariance to other deformations, a coordinate
transformation is needed to convert each deformation to a shift. To
provide scale invariance a logarithmic transformation is used. The
logarithmic transformation converts a multiplicative scale change to
an additive shift. This shifted version will correlate with the
logarithmically transformed reference pattern. To provide rotation
invariance, a transformation is performed to map the angle to each
point in the image to a theta coordinate. If an object rotates in
the test image, it is translated along the theta coordinate. Usually
the two transformation are combined into the log r, theta
transformation. The test image as well as the reference image is
converted to polar form to provide r and theta values for each pixel.