Circular harmonic wavelets: a tool for optimum scale-orientation independent pattern recognition

The classical problem of recognition of patterns irrespective of their actual size, displacement and orientation is approached in a context of estimation theory. To simplify mathematical derivations, the image and the reference pattern are represented on a complex support, which converts the four positional parameters into two complex numbers: complex displacement and complex scale factor. The latter one represents isotropic dilations with its magnitude, and rotations with its phase. In this context, evaluation of the Likelihood function under additive Gaussian noise assumption allows to relate basic template matching strategy to wavelet theory. In particular, it is shown that using Circular Harmonic wavelets (CHW) drastically simplifies the problem from a computational viewpoint. A general purpose pattern detection/estimation scheme is further introduced by decomposition of the images on a orthogonal basis formed by complex Laguerre-Gauss Harmonic Wavelets. Based on this decomposition a solution of ambiguity problems with a progressive coarse to fine parameter estimation is finally presented.