Volume 38, pp. 44-68, 2011.
Error estimates for general fidelities
Martin Benning and Martin Burger
Abstract
Appropriate error estimation for regularization methods in imaging and inverse problems is of enormous importance
for controlling approximation properties and understanding types of solutions that are particularly favoured. In
the case of linear problems, i.e., variational methods with quadratic fidelity and quadratic regularization, the
error estimation is well-understood under so-called source conditions. Significant progress for nonquadratic
regularization functionals has been made recently after the introduction of the Bregman distance as an appropriate
error measure. The other important generalization, namely for nonquadratic fidelities, has not been analyzed so far.
In this paper we develop a framework for the derivation of error estimates in the case of rather general fidelities
and highlight the importance of duality for the shape of the estimates. We then specialize the approach for several
important fidelities in imaging (
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Key words
error estimation, Bregman distance, discrepancy principle, imaging, image processing, sparsity
AMS subject classifications
47A52, 65J20, 49M30