Local symmetry features in image processing

A model for circular symmetry is used to describe a loeal neighbourhood. A definition of circular symmetry is given which implies detection of one-dimensionality of a 2-D image arter a coordinate transformation. The coordinate transformation is such that Archimecles' spirals map to straight lines. The Fourier transform of a circularly symmetric image, in these coordinates provides an energy concentration to a line in a certain direction. Locsl neighbourhoods consisting of one circie or sev· era1 concentric circ!es showa concentration of energy to a line. This is alsa the case for lines with a common intersection point. These two types of circularly symmetric images map to two orthogonal lines in this special Fourier dornain. Archimedes' spirals map continuously to lines with directions between these two orthogonallines incorporating circ!es, hal f lines and spirals into the same model. Fitting a line in the least square sense in this special Fourier transform domain is shown to be possible to accomplish in the spatial domain as a convolution carried out on the partiai derivative image. The necessary filters are derived. Two algorithms based on interpretation of the error of the fitted optimal line and its orientation are implemented. ODe is dependent on the energy of the variation of the local image, the other is not. Both use the same optimal estimate of the orientation of the fitted line. Experiments are carried out utilizing the implemented algorithrns showing very good detection properties for spirals, circles, concentric circ!es, line end! and intersection point of a set of lines.