Image Restoration and Reconstruction using Entropy as a Regularization Functional

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.

[1]  Stephen J. Wernecke,et al.  Maximum Entropy Image Reconstruction , 1977, IEEE Transactions on Computers.

[2]  G. Demoment,et al.  Tomographie de diffraction et synthse de Fourier maximum d'entropie , 1987 .

[3]  S. Gull,et al.  Maximum entropy algorithm applied to image enhancement , 1980 .

[4]  G. Demoment,et al.  Maximum entropy diffraction tomography , 1986, ICASSP '86. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[5]  John Skilling,et al.  Maximum entropy method in image processing , 1984 .

[6]  J. Justice.  Maximum entropy and bayesian methods in applied statistics , 1986 .

[7]  B. F. Logan,et al.  The Fourier reconstruction of a head section , 1974 .

[8]  E. T. Jaynes,et al.  Maximum Entropy and Bayesian Methods in Applied Statistics: Monkeys, Kangaroos, and N , 1986 .

[9]  S. F. Gull,et al.  The Entropy of an Image , 1985 .

[10]  John Skilling,et al.  Image restoration by a powerful maximum entropy method , 1982, Comput. Vis. Graph. Image Process..

[11]  Bobby R. Hunt,et al.  Bayesian Methods in Nonlinear Digital Image Restoration , 1977, IEEE Transactions on Computers.

[12]  E. T. Jaynes,et al.  BAYESIAN METHODS: GENERAL BACKGROUND ? An Introductory Tutorial , 1986 .

[13]  B. Frieden.  Statistical models for the image restoration problem , 1980 .

[14]  E. Jaynes.  On the rationale of maximum-entropy methods , 1982, Proceedings of the IEEE.

[15]  G. Demoment,et al.  Maximum entropy Fourier synthesis with application to diffraction tomography. , 1987, Applied optics.

[16]  C. R. Smith,et al.  Maximum-Entropy and Bayesian Methods in Inverse Problems , 1985 .