Modern Methods of Image Reconstruction

Summary

Chapter 1 reviews the image restoration/reconstruction problem in its general setting. We first discuss linear methods for solving the problem of image deconvolution, i.e. the case in which the data is a convolution of a point-spread function and an underlying unblurred image. Next, non-linear methods are introduced in the context of Bayesian estimation, including Maximum-Likelihood and Maximum Entropy methods. Finally, the successes and failures of these methods are discussed along with some of the roots of these problems and the suggestion that these difficulties might be overcome by new (e.g. pixon-based) image reconstruction methods.

Chapter 2 discusses the role of language and information theory concepts for data compression and solving the inverse problem. The concept of Algorithmic Information Content (AIC) is introduced and shown to be crucial to achieving optimal data compression and optimized Bayesian priors for image reconstruction. The dependence of the AIC on the selection of language then suggests how efficient coordinate systems for the inverse problem may be selected. This motivates the selection of a multiresolution language for the reconstruction of generic images.

Chapter 3 introduces pixon-based image restoration/reconstruction methods. The relationship between image Algorithmic Information Content and the Bayesian incarnation of Occam's Razor are discussed as well as the relationship of multiresolution pixon languages and image fractal dimension. Also discussed is the relationship of pixons to the role played by the Heisenberg uncertainty principle in statistical physics and how pixon-based image reconstruction provides a natural extension to the Akaike information criterion for Maximum Likelihood estimation.