Local probability distribution of natural signals in sparse domains

In this paper we investigate the local probability density function (pdf) of natural signals in sparse domains. The statistical properties of natural signals are characterized more accurately in the sparse domains because the sparse domain coefficients (SDCs) have heavy-tailed distribution and have reduced correlation with adjacent coefficients. Our experiments show that a conditionally (given locally estimated variance and shape) independent Bessel K-form (BKF) pdf locally fits the sparse domain´s coefficients of natural signals, accurately. To justify this observation, we also investigate the pdf of the locally estimated variance and suggest a Gamma pdf for the locally estimated variance. Since commonly used sparse transformations are orthonormal, the pdf of the sparse domain coefficients must converge to Gaussian distribution by virtue of central limit theorem assuming that natural signals are locally wide sense stationary for small window sizes. Interestingly, we observe that the pdf of the normalized data (on the locally estimated variance) exhibit a Gaussian pdf, which justifies why the BKF pdf is an appropriate fit.