Factorization of Natural 4×4 Patch Distributions

Abstract

The lack of sufficient machine readable images makes impossible the direct computation of natural image 4 × 4 block statistics and one has to resort to indirect approximated methods to reduce their domain space. A natural approach to this is to collect statistics over compressed images; if the reconstruction quality is good enough, these statistics will be sufficiently representative. However, a requirement for easier statistics collection is that the method used provides a uniform representation of the compression information across all patches, something for which codebook techniques are well suited. We shall follow this approach here, using a fractal compression–inspired quantization scheme to approximate a given patch B by a triplet ( D _ B , μ _ B , σ _ B ) with σ _ B the patch’s contrast, μ _ B its brightness and D _ B a codebook approximation to the mean–variance normalization ( B – μ _ B )/ σ _ B of B . The resulting reduction of the domain space makes feasible the computation of entropy and mutual information estimates that, in turn, suggest a factorization of the approximation of p ( B ) ≃ p ( D _ B , μ _ B , σ _ B ) as p ( D _ B , μ _ B , σ _ B ) ≃ p ( D _ B ) p ( μ ) p ( σ )Φ(|| ∇ ||), with Φ being a high contrast correction.