We consider regions of images that exhibit smooth statistics, and pose the question of characterizing the "essence" of these regions that matters for recognition. Ideally, this would be a statistic (a function of the image) that does not depend on viewpoint and illumination, and yet is sufficient for the task. In this manuscript, we show that such statistics exist. That is, one can compute deterministic functions of the image that contain all the "information" present in the original image, except for the effects of viewpoint and illumination. We also show that such statistics are supported on a "thin" (zero-measure) subset of the image domain, and thus the "information" in an image that is relevant for recognition is sparse. Yet, from this thin set one can reconstruct an image that is equivalent to the original up to a change of viewpoint and local illumination (contrast). Finally, we formalize the notion of "information" an image contains for the purpose of viewpoint- and illumination- invariant tasks, which we call "actionable information" following ideas of J. J. Gibson.