Research in Shape Analysis pp 143-159 | Cite as

*L*^{1}-Regularized Inverse Problems for Image Deblurring via Bound- and Equality-Constrained Optimization

## Abstract

Image deblurring is typically modeled as an ill-posed, linear inverse problem. By adding an *L*^{1}-penalty to the negative-log likelihood function, the resulting minimization problem becomes well-posed. Moreover, the penalty enforces sparsity. The difficulty with *L*^{1}-penalties, however, is that they are non-differentiable. Here we replace the *L*^{1}-penalty by a linear penalty together with bound and equality constraints. We consider two statistical models for measurement error: Gaussian and Poisson. In either case, we obtain a bound- and equality-constrained minimization problem, which we solve using an iterative augmented Lagrangian (AL) method. Each iteration of the AL method requires the solution of a bound-constrained minimization problem, which is convex-quadratic in the Gaussian case and convex in the Poisson case. We recommend two highly efficient methods for the solution of these subproblems that allows us to apply the AL method to large-scale imaging examples. Results are shown on synthetic data in one and two dimensions, as well as on a radiograph used to calibrate the transmission curve of a pulsed-power X-ray source at a US Department of Energy radiography facility.

## Notes

### Acknowledgements

This work was done in part by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946 with the US Department of Energy and supported by the Site Directed Research and Development program. The US government retains, and the publisher, by accepting the article for publication, acknowledges that the US government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. The US Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). DOE/NV/25946--2476.

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