The data structure that is frequently used in image processing is based upon the Euclidean Fourier analysis and lacks projectively covariant characteristics. That is, although one can reconstruct a pattern that is translated and rotated in an image plane using only one Fourier transform of the original pattern, when projective distortions are applied this is no longer feasible.
Recently the Principal Investigator has developed a complete projective Fourier analysis consisting of noncompact and compact pictures corresponding to standard and spherical Fourier analyses, respectively. This harmonic analysis is well-suited for developing algorithms for storing and processing visual information that can be used in automated systems for pattern recognition independent of different perspectives between planar objects (i.e., patterns) and the imaging systems. In particular, the analysis allows reconstruction of any perspective distortions of a pattern by using only the projective Fourier transform in the noncompact picture of the original (undistorted) pattern, which was demonstrated in numerical tests. Moreover, the convolutions in both pictures can be used to develop algorithms for projectively-invariant matching of patterns.
This project will develop - a fast algorithm for rendering any perspective distortions of a pattern using efficiently computed projective Fourier transforms in the non-compact picture of the original pattern; and - a fast algorithm for computing projective Fourier transform in the compact picture.
Algorithms for efficient computation of the convolutions in noncompact and compact pictures would follow from the both fast projective transforms and their inverses.
