Pattern Recognition in Images by Symmetries and Coordinate Transformations

Abbreviations CT: Coordinate transformation FIR: Finite impulse response HFP: Harmonic function pair ILS: Innnitesimal linear symmetry GHT: Generalized Hough transform GLS: Generalized linear symmetry SNR: Signal to noise ratio 1 Abstract A theory for detecting general curve families by means of symmetry measurements in the coordinate transformed originals is presented. Symmetries are modeled by iso-gray curves of conjugate harmonic function pairs which also deene the coordinate transformations. Harmonic function pair coordinates render the target curve patterns as parallel lines, which is deened here as linear symmetry. Detecting these lines, or generalized linear symmetry tting as it will be called, corresponds to nding invariants of Lie groups of transformations. A technique based on least square error minimization for estimating the invariance parameters is presented. It uses the Lie innnitesimal operators to construct feature extraction methods that are eecient 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 identiies 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.