Nonlinear image operators, higher-order statistics, and the AND-like combinations of frequency components

The frequency domain plays a key role in the description of signals and systems. In the classical approaches, the individual frequency components are treated as independent. In linear systems, the superposition principle restricts the filtering to an OR-like processing of independent complex exponentials. Likewise, the classical second-order statistic (the power spectrum) measures only the occurrence of each individual frequency component, independent of whether it occurs in a systematic combination with other components or not. This basic limitation can be overcome by the extension of the classical approaches to nonlinear systems and higher-order statistics, which makes it possible to selectively address AND-like combinations of frequency components. We measure which AND combinations are statistically most relevant in natural images, and investigate how this statistical structure can be exploited by nonlinear Volterra filters