The image is transformed into a coordinate system where one axis is

log r and the other axis is theta. In this system, scale changes

shift the object along the log r axis and rotation shifts the object

along the theta axis. Because this transform, known as the Mellin-

Fourier transform, is itself not shift invariant, it is normally

applied to the Fourier transform of the test image. This provides the

shift invariance but loses the location information in the test scene.

The cross correlation between the transformed test and reference

images no longer can provide the location of the object but does

determine the presence of the object, its size, and its rotation

relative to the reference pattern.

 To perform the Mellin-Fourier transform for shift, scale, and

rotation invariance, the input image is first Fourier transformed and

the resultant power spectral density recorded. This magnitude array

is converted to polar coordinates and the linear radial frequency is

converted to a logarithmic radial coordinate. The new coordinate

space (log r,theta) is used for cross-correlation of the input image

with similarly transformed reference images. A high speed technique

is required to convert the image into log r, theta coordinates at a

speed compatible with the optical processor. This has been

demonstrated using holograms to perform geometrical

transformations.46-50 To do this, the coordinate transforming

hologram must influence the light from each point and introduce a

specific deflection to the light incorporating such modifications as

local stretch, rotation, and translation.

 A practical correlator system might incorporate such an optical

transforming system or a sensor which collects data in the appropriate