Higher-Order Statistics and the Multivariate Probability Density Function of Natural Images

The statistical structure of natural images is analysed by two complementary approaches: by the measurement of higher-order spectra and by our method of 'wavelet statistics', which explores interesting projections of the multivariate probability density function (pdf). We show that the probability mass is concentrated in low-dimensional subspaces which represent intrinsically one-dimensional image features, and that the multivariate pdf thus resembles a 'hyper star'. The polyspectra show a corresponding concentration of polyspectral energy for aligned frequency components. This basic statistical structure provides the explanation for the generation of a sparse representation by oriented filters: the kurtosis is maximised if the filter decomposition is adapted to the shape of the tri-spectrum, which is closely related to the fact that only oriented filters can cause the projection of a maximum part of the probability mass to the zero level. Since the hyper star is not separable in Cartesian axes, linear filters cannot yield independent components of natural images, and there remain substantial statistical dependences across orientation, phase, scale, and position. We show how these dependences are exploited by basic cortical nonlinearities, like the phase-invariance of complex cells, cortical gain control, and end-stopping. The visual cortex thus seems to pursue a higher-order whitening strategy