Factorization and structure of natural 4×4 patch densities
Kostadin Koroutchev · Optical Engineering · 2006
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 (DB,µB,B), with B being the patch's contrast, µB its brightness, and DB a codebook representation of the mean-variance normalization of B:(B?µB)/B. While not coinciding exactly with the true natural patch density p(B), the density (B)=p(DB,µB,B) should give an adequate approximation of p(B), because BBDB+µB. Our first main result is a factorization of the probability density p(D,µ,) as p(D,µ,)p(D)p(µ)p()(||B||), with 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() 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.