The Statistical Properties of Local Log-Contrast in Natural Images

The study of natural image statistics considers the statistical properties of large collections of images from natural scenes, and has applications in image processing, computer vision, and visual computational neuroscience. In the past, a major focus in the field of natural image statistics have been the statistics of outputs of linear filters. Recently, attention has been turning to nonlinear models. The contribution of this paper is the empirical analysis of the statistical properties of a central nonlinear property of natural scenes: the local log-contrast. To this end, we have studied both second-order and higher-order statistics of local log-contrast. Second-order statistics can be observed from the average amplitude spectrum. To examine higher-order statistics, we applied a higher-order-statistics-based model called independent component analysis to images of local log-contrast. Our results on second-order statistics show that the local log-contrast has a power-law-like average amplitude spectrum, similarly as the original luminance data. As for the higherorder statistics, we show that they can be utilized to learn intuitively meaningful spatial local-contrast patterns, such as contrast edges and bars. In addition to shedding light on the fundamental statistical properties of natural images, our results have important consequences for the analysis and design of multilayer statistical models of natural image data. In particular, our results show that in the case of local log-contrast, oriented and localized second-layer linear operators can be learned from the higher-order statistics of the nonlinearly mapped output of the first layer.