Geometric Fourier Analysis of the Conformal Camera for Active Vision

Suppose one intends to design an active vision system that should perform some artificial intelligence functions. For instance, it should recognize a planar object (or a three-dimensional object containing a piece of a planar surface) in a dynamic scene. Ideally, such a system should be built upon some data model representing visual inputs and algorithms storing, processing, and analyzing visual information that are well adapted to image transformations produced by different perspectives between planar objects and the imaging system. In spite of its importance, this problem remained unsolved until recently. In this article, building on the author's work, projective Fourier analysis for patterns is constructed in the framework of geometric Fourier analysis on groups and homogeneous spaces. It is done by identifying in the conformal camera the group $\mathbf{SL(}2,\mathbb{C)}$, which gives image projective transformations by acting through linear-fractional mappings on the image plane---homogeneous under th...

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