Higher-order statistics of natural images and their exploitation by operators selective to intrinsic dimensionality

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Abstract
Natural images contain considerable statistical redundancies beyond the level of second-order correlations. To identify the nature of these higher-order dependencies, we analyze the bispectra and trispectra of natural images. Our investigations reveal substantial statistical dependencies between those frequency components which are aligned to each other with respect to orientation. We argue that operators which are selective to local intrinsic dimensionality can optimally exploit such redundancies. We also show that the polyspectral structure we find for natural images helps to understand the hitherto unexplained superiority of orientation-selective filter decompositions over isotropic schemes like the Laplacian pyramid. However any essentially linear scheme can only partially exploit this higher-order redundancy. We therefore propose nonlinear i2D-selective operators which exhibit close resemblance to hypercomplex and end-stopped cells in the visual cortex. The function of these operators can be interpreted as a higher-order whitening of the input signal.

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