Regularization of inverse problems with adaptive discrepancy terms: application to multispectral data

n this paper, a general framework for the inversion of a linear operator in the case where one seeks several components from several observations is presented. The estimation is done by minimizing a functional balancing discrepancy terms by regularization terms. The regularization terms are adapted norms that enforce the desired properties of each component. The main focus of this paper is the definition of the discrepancy terms. Classically, these are quadratic. We present novel discrepancy terms adapt to the observations. They rely on adaptive projections that emphasize important information in the observations. Iterative algorithms to minimize the functionals with adaptive discrepancy terms are derived and their convergence and stability is studied. The methods obtained are compared for the problem of reconstruction of astrophysical maps from multifrequency observations of the Cosmic Microwave Background. We show the added flexibility provided by the adaptive discrepancy terms.