The recovery of a signal as it would be observed by a hypothetical, perfectly resolving, apparatus is an interesting and challenging problem. This engineer's "dream" fits in the general framework of inverse problems. Over the years, various methods were invented to deal with the generally ill-conditioned nature of inverse problems. In physics, engineering, and statistics the ultimate goal was and still is that of converting the real problem into one which is "well posed" in the sense that the statement of the problem gives just enough information to determine one unique solution. This means that, besides the data, some prior information about the system has to be included in the general formulation of the problem. Then the question of how much confidence one has in the data and prior information arises. The consideration of a fidelity criterion measure becomes a logical direction to pursue. This way of treating the problem is known as the regularization of the ill-posed problem.;Relying on the notion of mutual information as a distance, new restoration methods are presented and a general framework that encompasses and unifies these new methods with classical restoration techniques is established. Signal restoration methods based on generalized entropic distances (also based in information theory) are shown to lead to very elegant signal recovery algorithms.;In this thesis we employ the information theoretic mutual information functional and generalized entropic distances as fidelity measures of prior knowledge. These functionals are regarded as distances between the sought signal and an available prior. Assuming that some knowledge about the contaminating noise power is available (or estimable), the general problem is then formulated as an extremum constrained problem and solved using Lagrange method. The resulting solution is iterative and the issues of proving convergence and uniqueness of solution are a key element to this thesis.
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