skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


  • NSF Public Access
  • Search Results
  • A new nonstationary preconditioned iterative method for linear discrete ill‐posed problems with application to image deblurring
Title: A new nonstationary preconditioned iterative method for linear discrete ill‐posed problems with application to image deblurring
Abstract Discrete ill‐posed inverse problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods aim at reducing this sensitivity. This article considers an iterative regularization method, based on iterated Tikhonov regularization, that was proposed in M. Donatelli and M. Hanke, Fast nonstationary preconditioned iterative methods for ill‐posed problems, with application to image deblurring,Inverse Problems, 29 (2013), Art. 095008, 16 pages. In this method, the exact operator is approximated by an operator that is easier to work with. However, the convergence theory requires the approximating operator to be spectrally equivalent to the original operator. This condition is rarely satisfied in practice. Nevertheless, this iterative method determines accurate image restorations in many situations. We propose a modification of the iterative method, that relaxes the demand of spectral equivalence to a requirement that is easier to satisfy. We show that, although the modified method is not an iterative regularization method, it maintains one of the most important theoretical properties for this kind of methods, namely monotonic decrease of the reconstruction error. Several computed experiments show the good performances of the proposed method.  more » « less
Award ID(s):
1720259
PAR ID:
10453751
Author(s) / Creator(s):
 ;  ;  ;  
Publisher / Repository:
Wiley Blackwell (John Wiley & Sons)
Date Published:
Journal Name:
Numerical Linear Algebra with Applications
Volume:
28
Issue:
2
ISSN:
1070-5325
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Sampling theory in signal and image processing)
    null (Ed.)
    Many inverse problems involve two or more sets of variables that represent different physical quantities but are tightly coupled with each other. For example, image super-resolution requires joint estimation of the image and motion parameters from noisy measurements. Exploiting this structure is key for efficiently solving these large-scale optimization problems, which are often ill-conditioned. In this paper, we present a new method called Linearize And Project (LAP) that offers a flexible framework for solving inverse problems with coupled variables. LAP is most promising for cases when the subproblem corresponding to one of the variables is considerably easier to solve than the other. LAP is based on a Gauss–Newton method, and thus after linearizing the residual, it eliminates one block of variables through projection. Due to the linearization, this block can be chosen freely. Further, LAP supports direct, iterative, and hybrid regularization as well as constraints. Therefore LAP is attractive, e.g., for ill-posed imaging problems. These traits differentiate LAP from common alternatives for this type of problem such as variable projection (VarPro) and block coordinate descent (BCD). Our numerical experiments compare the performance of LAP to BCD and VarPro using three coupled problems whose forward operators are linear with respect to one block and nonlinear for the other set of variables. 
    more » « less
  2. ; ; ; (, Conference on Neural Information Processing Systems (NeurIPS))
    Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using the Bregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance. 
    more » « less
  3. ; ; (, BIT Numerical Mathematics)
    null (Ed.)
    Abstract Randomized methods can be competitive for the solution of problems with a large matrix of low rank. They also have been applied successfully to the solution of large-scale linear discrete ill-posed problems by Tikhonov regularization (Xiang and Zou in Inverse Probl 29:085008, 2013). This entails the computation of an approximation of a partial singular value decomposition of a large matrix A that is of numerical low rank. The present paper compares a randomized method to a Krylov subspace method based on Golub–Kahan bidiagonalization with respect to accuracy and computing time and discusses characteristics of linear discrete ill-posed problems that make them well suited for solution by a randomized method. 
    more » « less
  4. ; (, Numerical Algorithms)
    null (Ed.)
    Abstract Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p -norm, and a regularization term, that is defined in terms of a q -norm. We allow 0 < p , q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed. 
    more » « less
  5. ; ; (, Computational Methods for Inverse Problems in Imaging)
    Non-stationary regularizing preconditioners have recently been proposed for the acceleration of classical iterative methods for the solution of linear discrete ill-posed problems. This paper explores how these preconditioners can be combined with the flexible GMRES iterative method. A new structure-respecting strategy to construct a sequence of regularizing preconditioners is proposed. We show that flexible GMRES applied with these preconditioners is able to restore images that have been contaminated by strongly non-symmetric blur, while several other iterative methods fail to do this. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email