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Sensory systems exploit the statistical regularities of natural signals, and thus, a fundamental goal for understanding biological sensory systems, and creating artificial sensory systems, is to characterize the statistical structure of natural signals. Here, we use a simple conditional moment method to measure natural image statistics relevant for three fundamental visual tasks: (i) estimation of missing or occluded image points, (ii) estimation of a high-resolution image from a low-resolution image ("super resolution"), and (iii) estimation of a missing color channel. We use the conditional moment approach because it makes minimal invariance assumptions, can be applied to arbitrarily large sets of training data, and provides (given sufficient training data) the Bayes optimal estimators. The measurements reveal complex but systematic statistical regularities that can be exploited to substantially improve performance in the three tasks over what is possible with some standard image processing methods. Thus, it is likely that these statistics are exploited by the human visual system.

Statistics for optimal point prediction in natural images