Abstract
In this paper, we investigate the usefulness of adding a box-constraint to the minimization of functionals consisting of a data-fidelity term and a total variation regularization term. In particular, we show that in certain applications an additional box-constraint does not effect the solution at all, i.e., the solution is the same whether a box-constraint is used or not. On the contrary, i.e., for applications where a box-constraint may have influence on the solution, we investigate how much it effects the quality of the restoration, especially when the regularization parameter, which weights the importance of the data term and the regularizer, is chosen suitable. In particular, for such applications, we consider the case of a squared L 2 data-fidelity term. For computing a minimizer of the respective box-constrained optimization problems a primal-dual semi-smooth Newton method is presented, which guarantees superlinear convergence.
Highlights
An observed image g, which contains additive Gaussian noise with zero mean and standard deviation σ, may be modeled as g = K û + n where û is the original image, K is a linear bounded operator and n represents the noise
Newton methods have not been presented for box-constrained total variation minimization
The contribution of the paper is three-sided: (i) We present a semi-smooth Newton method for the box-constrained total variation minimization problems (3) and (6). (ii) We investigate the influence of the box-constraint on the solution of the total variation minimization models with respect to the regularization parameter. (iii) In case the noise-level is not at hand, we propose a new automatic regularization parameter selection algorithm based on the box-constraint information
Summary
Newton methods have not been presented for box-constrained total variation minimization In this setting, differently to the before mentioned approaches, the box-constraint adds some additional difficulties in deriving the dual problems, which have to be calculated to obtain the desired method; see Section 4 for more details. That our approach differs significantly from the Newton-like scheme presented in [29], where a smooth objective functional with a box-constraint is considered This allows in [29] to derive a Newton method without dualization. The contribution of the paper is three-sided: (i) We present a semi-smooth Newton method for the box-constrained total variation minimization problems (3) and (6). (ii) We investigate the influence of the box-constraint on the solution of the total variation minimization models with respect to the regularization parameter.
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