Lower bounds on the redundancy of natural images

The light intensities of natural images exhibit a high degree of redundancy. Knowing the exact amount of their statistical dependencies is important for biological vision as well as compression and coding applications but estimating the total amount of redundancy, the multi-information, is intrinsically hard. The common approach is to estimate the multi-information for patches of increasing sizes and divide by the number of pixels. Here, we show that the limiting value of this sequence--the multi-information rate--can be better estimated by using another limiting process based on measuring the mutual information between a pixel and a causal neighborhood of increasing size around it. Although in principle this method has been known for decades, its superiority for estimating the multi-information rate of natural images has not been fully exploited yet. Either method provides a lower bound on the multi-information rate, but the mutual information based sequence converges much faster to the multi-information rate than the conventional method does. Using this fact, we provide improved estimates of the multi-information rate of natural images and a better understanding of its underlying spatial structure.