Mapping Natural Image Patches by Explicit and Implicit Manifolds

Image patches are fundamental elements for object modeling and recognition. However, there has not been a panoramic study of the structures of the whole ensemble of natural image patches in the literature. In this article, we study the structures of this ensemble by mapping natural image patches into two types of subspaces which we call "explicit manifolds " and "implicit manifolds " respectively. On explicit manifolds, one finds those simple and regular image primitives, such as edges, bars, corners and junctions. On implicit manifolds, one finds those complex and stochastic image patches, such as textures and clutters. On different types of manifolds, different perceptual metrics are used. We propose a method for learning a probabilistic distribution on the space of patches by pursuing both types of manifolds using a common information theoretical criterion. The connection between the two types of manifolds is realized by image scaling, which changes the entropy of the image patches. The explicit manifolds live in low entropy regimes while the implicit manifolds live in high entropy regimes. We study the transition between the two types of manifolds over scale and show that the complexity of the manifolds peaks in a middle entropy regime.