Coordinate transformations, symmetries, and GHT

A conjugate harmonic function pair is a suitable curvi-linear coordinate pair with which image iso-gray curves can be described. In this representation, images or parts of images which are symmetric with respect to such a function pair, have iso-gray curves which are lines or parallel line patterns. Detecting these lines, or generalized linear symmetry fitting as it will be called, corresponds to finding invariants of Lie groups of transformations. The associated Lie infinitesimal operators help constructing detection algorithms which are simple. The procedure, which is shown to be an extension of the generalized Hough transform, performs detection by voting and accumulating evidence for the searched pattern. Experimental results illustrating the theory are presented.