Image Restoration and Reconstruction using Entropy as a Regularization Functional

Abstract

Entropy has been widely used to solve inverse problems, especially when a complete set of data is not available to obtain a unique solution to the problem. In this paper we show that a Maximum Entropy (ME) approach is appropriate for solving some inverse problems arising at different levels of various image restoration and reconstruction problems: Reconstruction of images in X ray tomography, Reconstruction of images in ultrasound or microwave diffraction tomography (in the case of Born or Rytov approximation) from either Fourier domain data or directly from diffracted field measurements; Restoration of positive images by deconvolution when data is missing. Our contribution is twofold: i) Entropy has been introduced by various approaches: combinatorial considerations, probabilistic and information theory arguments, etc. But in real situations, we also have to take into account the presence of noise on the data. To do this, a c2 statistics is added to the entropy measure. This leads us to a novel interpretation of entropy as a particular choice of regularizing functional. ii) In some of the above-mentioned applications, the object to be restored or reconstructed is a complex valued quantity. Consequently, we extend the definition of the entropy of an image which is considered as a function of ℝ2 to ℂ. The discussion is illustrated with some simulated results in X ray and diffraction tomography which allow a comparison between ME methods and some classical ones to be made. The computational cost and practical implementation of the algorithm are discussed.