Pattern Recognition in Images by Symmetries and Coordinate Transformations

A theory for detecting general curve families by means of symmetry measurements is presented. Symmetries are modeled by iso-gray curves of conjugate harmonic function pairs which define coordinate transformations. Harmonic function pair coordinates render the target curve patterns as parallel lines, which is defined here as linear symmetry. Detecting these lines, or generalized linear symmetry fitting, corresponds to finding invariants of Lie groups of transformations. A technique based on total least square error minimization for estimating the invariance parameters is presented. It uses the Lie infinitesimal operators to construct feature extraction methods that are efficient and simple to implement. The technique, which is shown to be an extension of the generalized Hough Transform, enables detection by voting and accumulating evidence for the searched pattern. In this approach complex valued votes are permitted, where the phase of the vote identifies the member of the family of patterns that are detectable. Experimental results illustrating the theory and its application to real as well as synthetic images are presented.