Geometric Fourier Analysis of the Conformal Camera for Active Vision
Jacek Turski · SIAM Review · 2004
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 the group action. This analysis is being implemented in perspectively adapted digital image processing, and its basic components are tested for binary images in computer simulations. It is recognized that the data model of digital image representation developed in the article is explicitly designed for foveated sensors, the use of which in active vision systems is presently limited due to the lack of such a data model.