Mapping the Ensemble of Natural Image Patches by Explicit and Implicit Manifolds
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Image patches are fundamental elements for object modeling and recognition. However, there has not been a panoramic study in the literature on the structures of the whole ensemble of natural image patches. In this article, we study the mathematical structures of the ensemble of natural image patches and map 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 these manifolds, different perceptual metrics are used. Then we show a unified framework for learning a probabilistic model on the space of patches by pursuing both types of manifolds under a common information theoretical principle. The connection between the two types of manifolds are realized through 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. In experiments, we cluster the natural image patches and compare the two types of manifolds with a common information theoretical criterion. We also study the transition of the manifolds over scales and show that the complexity peak in a middle entropy regime where most objects and parts reside. 1.