Information Scaling Laws in Natural Scenes

In natural scenes, objects and patterns can appear at a wide variety of distances from the viewer. For the same visual pattern viewed at different distances, both the image and our perception of the pattern change over distance. We call the change of the image over distance as image scaling, and the change of our perception over distance as information scaling. While image scaling can be accounted for by the state space theory, information scaling has not been mathematically studied in computer vision. In this paper, we prove two information scaling laws: 1) the entropy rate of the image changes over distance, and 2) the entropy of the posterior distribution of the pattern also changes over distance. These two information scaling laws have deep implications in computer vision: they call for different models of the same visual pattern at different distances, as well as a model transition mechanism for switching models over different distance/scale regimes.