Symmetry Interpretation of Complex Moments and the Local Power Spectrum

Abstract

Complex moments yield estimates of the local N-folded image symmetry because they give optimal solutions to the generalized line-fitting problem. Instead of computing them in the spatial domain, they can be computed in the local frequency domain because the local (Gabor) power spectrum is translation invariant (or varying slowly in practice) inside homogeneously textured regions. In other words, they make it possible to detect and discriminate linear, rectangular, hexagonal/triangular, etc., structures. When computed for different frequency bands they make it possible to distinguish between textures having different structures at different scales. Experimental results on texture segmentation confirm this. Furthermore, complex moments of the local power spectrum are shown to be related to Lie operators as well as prolate spheroidal functions, which allow for implementations that bypass the Gabor filtering.