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The choice of the parameter value for regularized inverse problems iscritical to the results and remains a topic of interest. This article exploresa criterion for selecting a good parameter value by maximizing the probabilityof the data, {{with no prior knowledge of the noise variance}}. These conceptsare developed for $\ell_2$ and consequently $\ell_1$ regularization models byway of their Bayesian interpretations. Based on these concepts, an iterativescheme is proposed and demonstrated to converge accurately, and analyticalconvergence results are provided that substantiate these empiricalobservations. For some of the most common inverse problems, including MRI, SAR,denoising, and deconvolution, an extremely efficient algorithm is derived,making the iterative scheme very attractive for real case use. Thecomputational concerns associated with the general case for any inverse problemare also carefully addressed. A robust set of 1D and 2D numerical simulationsconfirm the effectiveness of the proposed approach.

Effective New Methods for Automated Parameter Selection in Regularized Inverse Problems