We explore the relationships between two classical means of mathematically representing visual patterns that are invariant under geometric transformations. One, based on integral transforms, in particular, Fourier transforms, produces transform magnitudes invariant to pairwise combinations of translations, rotations, and size changes. The second, based on the degree to which the pattern remains invariant to differential operators (which are the infinitesimal generators of the geometric transformations), results in algebraic relations between pattern structures and group theoretical properties of the transforms. Formal relationships are established between these representations, relating the kernel properties of the integral transforms to the associated Lie transformation groups.
© 1988 Optical Society of America
Original Manuscript: June 11, 1987
Manuscript Accepted: December 8, 1987
Published: May 1, 1988
Mario Ferraro and Terry M. Caelli, "Relationship between integral transform invariances and Lie group theory," J. Opt. Soc. Am. A 5, 738-742 (1988)