An inner–outer iterative method for edge preservation in image restoration and reconstruction ^*

Abstract

We present a new inner–outer iterative algorithm for edge enhancement in imaging problems. At each outer iteration, we formulate a Tikhonov-regularized problem where the penalization is expressed in the two-norm and involves a regularization operator designed to improve edge resolution as the outer iterations progress, through an adaptive process. An efficient hybrid regularization method is used to project the Tikhonov-regularized problem onto approximation subspaces of increasing dimensions (inner iterations), while conveniently choosing the regularization parameter (by applying well-known techniques, such as the discrepancy principle or the L -curve criterion, to the projected problem). This procedure results in an automated algorithm for edge recovery that does not involve regularization parameter tuning by the user, nor repeated calls to sophisticated optimization algorithms, and is therefore particularly attractive from a computational point of view. A key to the success of the new algorithm is the design of the regularization operator through the use of an adaptive diagonal weighting matrix that effectively enforces smoothness only where needed. We demonstrate the value of our approach on applications in x-ray CT image reconstruction and in image deblurring, and show that it can be computationally much more attractive than other well-known strategies for edge preservation, while providing solutions of greater or equal quality.

A computational framework for edge-preserving regularization in dynamic inverse problems

“…We select the best (i.e., the one that produces the smallest RRE) regularization parameter out of 15 candidate values. (d) Inner-outer reweighting scheme: We follow the inner-outer approach introduced in [29], where the authors present an IRLS approach that uses an adaptive diagonal weighting matrix that shares some common features with spatial anisotropic TV involving the discrete spatial gradient operator L s (2.3) and a projection-based iterative method developed in [45] to solve the corresponding sequence of generalform Tikhonov problems. We extended this approach to spatio-temporal TV by considering the spatio-temporal first-derivative operator D 1 (3.1) rather than L s .…”

Computational methods for large-scale inverse problems: a survey on hybrid projection methods

“…An IRN method was proposed in [213] that employs CGLS during the inner iterations, while [4] uses preconditioned LSQR applied to a problem that is effectively transformed into standard form. The authors of [78] propose an inner-outer iterative method that enhances edges through multiplicative updates of the weights, and which exploits the hybrid projection method based on joint bidiagonalization method (4.4). The methods based on flexible Krylov subspaces described in 4.3.3 can be as well extended to handle problem (4.28), but the strategies adopted to achieve this are regularizer-dependent: in other words, no universal approach is possible, even for the special case R(x) = Lx p p .…”

Learning regularization parameters of inverse problems via deep neural networks

“…Sparsity-promoting regularizers based on p regularization and inner-outer schemes for edge and or discontinuity preservation have gained popularity in recent years [3,77], but selecting regularization parameters for these settings is not trivial. Various extensions of hybrid projection methods to more general settings have been developed [12,27,28]. Such methods exploit iteratively reweighted approaches and flexible preconditioning of Krylov subspace methods in order to avoid expensive parameter tuning, but can still be costly if many problems must be solved.…”