In this article, various notions of edges encountered in digital image process- ing are reviewed in terms of compact representation (or completion). We show that critical exponents defined in Statistical Physics lead to a much more coherent definition of edges, consistent across the scales in acquisitions of natural phenomena, such as high resolution natural images or turbulent acquisitions. Edges belong to the multiscale hierarchy of an underlying dy- namics, they are understood from a statistical perspective well adapted to fit the case of natural images. Numerical computation methods for the eval- uation of critical exponents in the non-ergodic case are recalled, which apply for the vast majority of natural images. We study the framework of re- constructible systems in a microcanonical formulation, show how it redefines edge completion, and how it can be used to evaluate and assess quantitatively the adequation of edges as candidates for compact representations. We study with particular attention the case of turbulent data, in which edges in the classical sense are particularly challenged. Tests are conducted and evalu- ated on a standard database for natural images. We test the newly intro- duced compact representation as an ideal candidate for evaluating turbulent cascading properties of complex images, and we show better reconstruction performance than the classical tested methods.