Factorization and structure of natural 4×4 patch densities

Kostadin Koroutchev; Jose R. Dorronsoro

doi:10.1117/1.2402520

1 December 2006 Factorization and structure of natural 4×4 patch densities

Kostadin Koroutchev, Jose R. Dorronsoro

Author Affiliations +

Optical Engineering, Vol. 45, Issue 12, 127003 (December 2006). https://doi.org/10.1117/1.2402520

Abstract

The objective of this work is to give an approximate statistical model of the distribution of 4×4 pixel natural image patches B. Given the huge size of the sample space, we collect the required statistics not directly over B, but, instead, over a fractal compression inspired representation of B, namely by a triplet (D^B,μ_B,σ_B), with σ_B being the patch's contrast, μ_B its brightness, and D^B a codebook representation of the mean-variance normalization of B:(B-μ_B)/σ_B. While not coinciding exactly with the true natural patch density p(B), the density _p^{^}(B)=p(D^B,μ_B, σ_B) should give an adequate approximation of p(B), because B^~₌σ_BD^B+μ_B. Our first main result is a factorization of the probability density p(D,μσ,) as p(D,μσ,)^~₌p(D)p(μ)p(σ)Φ(||ΔB||), with Φ being a high-contrast correction. Here, the brightness term p(μ) is largely irrelevant, and our second main result deals with the structure of the other two factors, showing that p(σ) follows an exponential distribution, and that p(D) is uniformly distributed with respect to volume in image space. These results are largely independent of the codebook used.

©(2006) Society of Photo-Optical Instrumentation Engineers (SPIE)

Citation Download Citation

Kostadin Koroutchev and Jose R. Dorronsoro "Factorization and structure of natural 4×4 patch densities," Optical Engineering 45(12), 127003 (1 December 2006). https://doi.org/10.1117/1.2402520

Published: 1 December 2006

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