Sparse modelling of natural images and compressive sensing

Abstract

This thesis concerns the study of the statistics of natural images and compressive sensing for two main objectives: 1) to extend our understanding of the regularities exhibited by natural images of the visual world we regularly view around us, and 2) to incorporate this knowledge into image processing applications. Previous work on image statistics has uncovered remarkable behavior of the dis tributions obtained from filtering natural images. Typically we observe high kurtosis, non-Gaussian distributions with sharp central cusps, which are called sparse in the literature. These results have become an accepted fact through empirical findings us ing zero mean filters on many different databases of natural scenes. The observations have played an important role in computational and biological applications, where re searchers have sought to understand visual processes through studying the statistical properties of the objects that are being observed. Interestingly, such results on sparse distributions also share elements with the emerging field of compressive sensing. This is a novel sampling protocol where one seeks to measure a signal in already com pressed format through randomised projections, while the recovery algorithm consists of searching for a constrained solution with the sparsest transformed coefficients. In view of prior art, we extend our knowledge of image statistics from the monochrome domain into the colour domain. We study sparse response distributions of filters constructed on colour channels and observe the regularity of the distributions across diverse datasets of natural images. Several solutions to image processing problems emerge from the incorporation of colour statistics as prior information. We give a Bayesian treatment to the problem of colorizing natural gray images, and formulate image compression schemes using elements of compressive sensing and sparsity. We also propose a denoising algorithm that utilises the sparse filter responses as a regular- isation function for the effective attenuation of Gaussian and impulse noise in images. The results emanating from this body of work illustrate how the statistics of natural images, when incorporated with Bayesian inference and sparse recovery, can have deep implications for image processing applications