  In this paper we propose a new statistical stopping rule for constrained
maximum likelihood iterative algorithms applied to ill-posed inverse problems.
To this aim we extend the definition of Tikhonov regularization in a
statistical framework and prove that the application of the proposed stopping
rule to the Iterative Space Reconstruction Algorithm (ISRA) in the Gaussian
case and Expectation Maximization (EM) in the Poisson case leads to well
defined regularization methods according to the given definition. We also prove
that, if an inverse problem is genuinely ill-posed in the sense of Tikhonov,
the same definition is not satisfied when ISRA and EM are optimized by
classical stopping rule like Morozov's discrepancy principle, Pearson's test
and Poisson discrepancy principle. The stopping rule is illustrated in the case
of image reconstruction from data recorded by the Reuven Ramaty High Energy
Solar Spectroscopic Imager (RHESSI). First, by using a simulated image
consisting of structures analogous to those of a real solar flare we validate
the fidelity and accuracy with which the proposed stopping rule recovers the
input image. Second, the robustness of the method is compared with the other
classical stopping rules and its advantages are shown in the case of real data
recorded by RHESSI during two different flaring events.
