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