3 - Utilisation de l'entropie dans les problèmes de restauration et de reconstruction d'images

In a great number of image reconstruction and restoration problems we have to solve an integral equation of the first kind which is an ill posed inverse problem . Therefore one cannot obtain a unique and stable solution without introducing an a priori information on the solution, The Bayesian approach is a coherent one for solving inverse problems because it lets us to take into account and to process in the same way the a priori information on the solution and the data . This approach can be resumed as the following : (i) Assign an a priori probability distribution to the parameters to translate our knowledge on these parameters . (ii) Assign a probability distribution to the measured data to translate the errors and the noise on the data . (iii) Use the Bayes' rule to transmit the information contained in the data to the parameters, i . e. calculate the a posteriori probability distribution of these parameters . (iv) Define a decision rule to determine the parameters values . One must note that : (i) This approach can be used only in problems which can be described by a finite number of parameters (for example when the integral equation is discretized) . (ii) The notion of probability in this approach is not always connected to the frequency of the realization of a random variable . (iii) While it is easy to assign a probability distribution to the measured data to translate the existence of noise on these data, it is more difficult to assign an a priori probability distribution on the unknown parameters of the problem . The maximum entropy principle permits us to choose a probability distribution which is coherent with our a priori knowledge on these parameters and which is less compromising in the sense that it does not introduce any supplementary information . In this paper we use this approach to establish a method for solving the integral equations of the f rst kind in which the entropy of the solution is used as a regularization functional . This method is then used for solving many inverse problems : image restoration by deconvolution in the situation of missing data, image reconstruction in X ray tomography and diffraction tomography, and the multivariable Fourier syrtthesis problem . A great number of simulation results are showed and a comparison is made between these results and those obtained by other usual linear methods .