On the Order of Regularization Methods

Natterer, Frank

doi:10.1007/978-3-0348-5460-3_14

Cited by 11 publications

(1 citation statement)

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“…What is known is that such solution, in the case of experimental data, although non negative, is not acceptable from a physical viewpoint, since the intrinsic ill-posedness of the inverse problem induces noise amplification. There are two ways to regularize a statistical inverse problem: first, with an add of information on the solution realized by a prior probability in a Bayesian framework, or, second, without adding information on the solution but simply stopping the iterative approximation process before getting the limit solution [17]. This paper focuses on this second approach.…”

Section: Introductionmentioning

confidence: 99%

Regularization of constrained maximum likelihood iterative algorithms by means of statistical stopping rule

Benvenuto,

Piana

2012

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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.

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Section: Introductionmentioning

confidence: 99%