The multifractal structure of contrast changes in natural images: From sharp edges to textures

We present a formalism that leads naturally to a hierarchical description o f the different contrast structures in images, providing precise definition s of sharp edges and other texture components. Within this formalism, we ac hieve a decomposition of pixels of the image in sets, the fractal component s of the image, such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborh ood. A crucial role in this description of images is played by the behavior of contrast differences under changes in scale. Contrary to naive scaling ideas where the image is thought to have uniform transformation properties (Field, 1987), each of these fractal components has its own transformation law and scaling exponents. A conjecture on their biological relevance is al so given.