Patterns with different phases but same statistics.

Many successful methods in various vision tasks rely on statistical analysis of visual patterns. However, we are interested in covering the gap between the underlying mathematical representation of the visual patterns and their statistics. With this general trend, in this paper a relationship between phase structure of a class of patterns and their moments after and before filtering have been considered. First, a general formula between the phase structure and moments of the images is obtained. Second, a theorem is developed that states under which conditions two visual patterns with the same frequencies but different phases have the same moments up to a certain moment. Finally, a theorem is developed that explains, given a set of filters, under which conditions two visual patterns with both different frequencies and different phases have the same subband statistics.