Statistics of natural time-varying images

Abstract

Natural time-varying images possess substantial spatiotemporal correlations. We measure these correlations-or equivalently the power spectrum-for an ensemble of more than a thousand segments of motion pictures and we find significant regularities. More precisely, our measurements show that the dependence of the power spectrum on the spatial frequency, f, and temporal frequency, w, is in general non-separable and is given by f−m−1F(w/f), where F(w/f) is a non-trivial function of the ratio w/f. We give a theoretical derivation of this scaling behaviour and show that it emerges from objects with a static power spectrum ∼f−m, appearing at a wide range of depths and moving with a distribution of velocities relative to the observer. We show that in the regime of relatively high temporal and low spatial frequencies the power spectrum becomes independent of the details of the velocity distribution and that it is separable into the product of spatial and temporal power spectra with the temporal part given by the universal power law ∼w−2. Making some reasonable assumptions about the form of the velocity distribution we derive an analytical expression for the spatiotemporal power spectrum which is in excellent agreement with the data for the entire range of spatial and temporal frequencies of our measurements. The results in this paper have direct implications for neural processing of time-varying images in the visual pathway.