L 1-Regularized inverse problems for image deblurring via bound- and equality-constrained optimization

Johnathan M. Bardsley, Marylesa Howard

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


Image deblurring is typically modeled as an ill-posed, linear inverse problem. By adding an L1-penalty to the negative-log likelihood function, the resulting minimization problem becomes well-posed. Moreover, the penalty enforces sparsity. The difficulty with L1-penalties, however, is that they are non-differentiable. Here we replace the L1-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.

Original languageEnglish
Title of host publicationAssociation for Women in Mathematics Series
Number of pages17
StatePublished - 2018

Publication series

NameAssociation for Women in Mathematics Series
ISSN (Print)2364-5733
ISSN (Electronic)2364-5741


Dive into the research topics of 'L 1-Regularized inverse problems for image deblurring via bound- and equality-constrained optimization'. Together they form a unique fingerprint.

Cite this