A new look at entropy for solving linear inverse problems

Entropy-based methods are widely used for solving inverse problems, particularly when the solution is known to be positive. Here, we address linear ill-posed and noisy inverse problems of the form z = Ax+ n with a general convex constraint x 2 X , where X is a convex set. Although projective methods are well adapted to this context, we study alternative methods which rely highly on some “information-based” criteria. Our goal is to clarify the role played by entropy in this field, and to present a new point of view on entropy, using general tools and results coming from convex analysis. We present then a new and broad scheme for entropic-based inversion of linear-noisy inverse problems. This scheme was introduced by Navaza in 1985 in connection with a physical modeling for crystallographic applications, and further studied by DacunhaCastelle and Gamboa. Important features of this paper are: i) a unified presentation of many well-known reconstruction criteria, ii) proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics, iii) a description of practical inversion of data using the aforementioned criteria, and iv) a presentation of some reconstruction results.