Harmonic Analysis on SL(2,C) and Projectively Adapted Pattern Representation

Among all image transforms the classical (Euclidean) Fourier trans-form has had the widest range of applications in image processing. Here its projective analogue, given by the double cover group SL(2,C) of the projective group PSL(2,C) for patterns, is developed. First, a projectively invariant classification of patterns is constructed in terms of orbits of the group PSL(2,C) acting on the image plane (with com-plex coordinates) by linear-fractional transformations. Then, SL(2,C)-harmonic analysis, in the noncompact picture of induced representa-tions, is used to decompose patterns into the components invariant under irreducible representations of the principal series of SL(2,C). Usefulness in digital image processing problems is studied by pro-viding a camera model in which the action of SL(2,C) on the com-plex image plane corresponds to, and exhaust, planar central projec-tions as produced when aerial images of the same scene are taken from different vantage points. The projectively adapted properties of the SL(2,C)-harmonic analysis, as applied to the problems in im-age processing, are confirmed by computational tests. Therefore, it should be an important step in developing a system for automated perspective-independent object recognition.