Abstract
This paper presents a novel approach to the regularization of linear problems involving total variation (TV) penalization, with a particular emphasis on image deblurring applications. The starting point of the new strategy is an approximation of the non-differentiable TV regularization term by a sequence of quadratic terms, expressed as iteratively reweighted 2-norms of the gradient of the solution. The resulting problem is then reformulated as a Tikhonov regularization problem in standard form, and solved by an efficient Krylov subspace method. Namely, flexible GMRES is considered in order to incorporate new weights into the solution subspace as soon as a new approximate solution is computed. The new method is dubbed TV-FGMRES. Theoretical insight is given, and computational details are carefully unfolded. Numerical experiments and comparisons with other algorithms for TV image deblurring, as well as other algorithms based on Krylov subspace methods, are provided to validate TV-FGMRES.
Highlights
This paper considers large-scale discrete ill-posed problems of the form b = Ax + e, (1.1)where the matrix A ∈ RN×N is ill-conditioned with ill-determined rank
The best reconstructions computed by the GMRES(D), the total variation (TV)-flexible GMRES (FGMRES), and the FBTV methods are displayed in Fig. 10
More details are visible in the image restored by TV-FGMRES with respect to the one restored by the FBTV method, which is more blocky
Summary
Where the matrix A ∈ RN×N is ill-conditioned with ill-determined rank
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