Self-Similarity Properties of Natural Images

Abstract

Scale invariance is a fundamental property of ensembles of nat(cid:173) ural images [1]. Their non Gaussian properties [15, 16] are less well understood, but they indicate the existence of a rich statis(cid:173) tical structure. In this work we present a detailed study of the marginal statistics of a variable related to the edges in the images. A numerical analysis shows that it exhibits extended self-similarity [3, 4, 5]. This is a scaling property stronger than self-similarity: all its moments can be expressed as a power of any given moment. More interesting, all the exponents can be predicted in terms of a multiplicative log-Poisson process. This is the very same model that was used very recently to predict the correct exponents of the structure functions of turbulent flows [6]. These results allow us to study the underlying multifractal singularities. In particular we find that the most singular structures are one-dimensional: the most singular manifold consists of sharp edges.