Statistics of gradient directions in natural images

Interest in finding statistical regularities in natural images has been growing since the advent of information theory and the advancem ent o f the efficient coding hypothesis that the human visual system is optimised to encode natural visual stimuli. In this thesis, a statistical analy­ sis o f gradient directions in an ensem ble o f natural images is reported. Inform ation-theoretic measures have been used to com pute the am ount o f dependency which exists betw een triples o f gradient directions at separate im age locations. Control experim ents are perform ed on other image classes: phase randomized natural im ages, whitened natural images, and G aussian noise images. The main results show that for an ensem ble o f natural images the average am ount o f de­ pendency between two and three gradient directions is the same as for an ensem ble o f phase randomized natural images. This result does not extend to i) the amount dependency between gradient m agnitudes, ii) gradient directions at high gradient magnitude locations, or iii) indi­ vidual natural images. Furtherm ore, no significant synergetic dependencies are found between triples o f gradient directions in an ensem ble natural images; a synergetic dependency is an increase in dependency between a pair o f gradient directions given the interaction o f a third gradient direction. Additional experim ents are perform ed to establish both the generality and specificity o f the main results by studying the gradient direction dependencies o f ensembles o f noise (random phases) images with varying power law power spectra. The results o f the additional experim ents indicate that, for ensembles of images with varying power law power spectra, the am ount o f dependency between two and three gradient directions is determ ined by the ensem ble’s mean power spectrum rather than the phase spectrum. A framework is also presented for future work and preliminary results are provided for the dependency between second order derivative m easurem ents (shape index) for up to 9-point configurations.