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Harmonic analysis on SL(2,ℂ) and projectively adapted pattern representation) and projectively adapted pattern representation

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Abstract

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

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  1. Department of Computer and Mathematical Sciences, University of Houston-Downtown, USA

    Jacek Turski

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  1. Jacek Turski
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Communicated by Todd Quinto

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Turski, J. Harmonic analysis on SL(2,ℂ) and projectively adapted pattern representation) and projectively adapted pattern representation. The Journal of Fourier Analysis and Applications 4, 67–91 (1998). https://doi.org/10.1007/BF02475928

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  • DOI: https://doi.org/10.1007/BF02475928

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