Iterative Nonlinear Gaussianization Algorithm for Image Simulation and Synthesis

Abstract

In this study, a statistical algorithm ---- Iterative Nonlinear Gaussianization Algorithm (INGA) is proposed, which seeks a nonlinear map from a set of dependent random variables to independent Gaussian random variables. A direct motivation of the INGA is to extend the principal component analysis (PCA) which transforms a set of correlated random variables into uncorrelated (independent up to second order) random variables. The present study was motivated by extension of the Principal Component Analysis (PCA) for dimension reduction and statistical image synthesis. There exists currently one such extension--called Independent Component Analysis (ICA)--which proves to be successful for samples that come from a linear combination of statistically independent sources. Nevertheless, the ICA is approximate in the sense of minimizing a differential entropy, and there are cases of interest which do not meet the condition of linear superposition. The INGA which enables transforming a random vector into statistically independent components. In order to quantify the performance of each algorithm, we have chosen the Edgeworth Kullback-Leibler Distance (E.KLD) which serves to measure the ``distance" between two distributions in multi-dimensions. Equipped with the measure of E.KLD, we have demonstrated the superiority of the INGA by several examples which are in general poorly synthesized by the PCA, or even not so well synthesized by the ICA. However, there are cases for which the both ICA and INGA do not produce satisfactory synthesis. Several numerical examples including synthetic and real-life image databases show the capabilities and limitations of INGA.