Nonnegative least-squares image deblurring: improved gradient projection approaches

Abstract

The least-squares approach to image deblurring leads to an ill-posed problem. The addition of the nonnegativity constraint, when appropriate, does not provide regularization, even if, as far as we know, a thorough investigation of the illposedness of the resulting constrained least-squares problem has still to be done. Iterative methods, converging to nonnegative least-squares solutions, have been proposed. Some of them have the 'semi-convergence' property, i.e. early stopping of the iteration provides 'regularized' solutions. In this paper we consider two of these methods: the projected Landweber (PL) method and the iterative image space reconstruction algorithm (ISRA). Even if they work well in many instances, they are not frequently used in practice because, in general, they require a large number of iterations before providing a sensible solution. Therefore, the main purpose of this paper is to refresh these methods by increasing their efficiency. Starting from the remark that PL and ISRA require only the computation of the gradient of the functional, we propose the application to these algorithms of special acceleration techniques that have been recently developed in the area of the gradient methods. In particular, we propose the application of efficient step-length selection rules and line-search strategies. Moreover, remarking that ISRA is a scaled gradient algorithm, we evaluate its behaviour in comparison with a recent scaled gradient projection (SGP) method for image deblurring. Numerical experiments demonstrate that the accelerated methods still exhibit the semi-convergence property, with a considerable gain both in the number of iterations and in the computational time; in particular, SGP appears definitely the most efficient one.

Semiconvergence and Relaxation Parameters for Projected SIRT Algorithms

“…Such constraints incorporate prior physical knowledge about the solution, and therefore they typically lead to smaller reconstruction errors (see Figure 2.1 for an example). Some applications of projected iterative methods in seismology, image restoration, nonnegative matrix factorization, matrix completion, and supervised learning can be found in [1], [2], [3], [7], [23], [28], [31], [32]. We focus on regularizing iterations with semiconvergence, where the iteration number plays the role of the regularizing parameter.…”

“…[1], [19], [20], [22], [27]. The semiconvergence of projected iterative methods has been noted in several papers [2], [3], [7]. It is also discussed in [19] and analyzed in an infinite dimensional setting by Eicke [16].…”

Limited-memory scaled gradient projection methods for real-time image deconvolution in microscopy

“…As can be observed from equation (6), the RL algorithm is a special scaled gradient method where the diagonal scaling matrix has the current iteration on the main diagonal and the variables' nonnegativity is ensured by the assumptions on A, b, y and x (0) . Convergence properties of the algorithm have been proved in various situations in [20,21,22,23,24,25,26].…”

“…In [3] numerical evidence has been also provided indicating remarkable gain in the convergence rate over the classical BarzilaiBorwein (BB) step-length rule [4]. Since in the last years promising image reconstruction algorithms have been designed by exploiting BB-based rules within gradient methods [5,6,7,8,9,10,11], it is worthwhile to investigate if useful acceleration can be achieved with the new step-length selection idea. In particular, we focus on the algorithm for image deconvolution in microscopy provided by the Scaled Gradient Projection (SGP) method recently developed in [12], that can be appropriately modified for managing the step-length rule proposed in [3].…”

A modified non-negative LMS algorithm and its stochastic behavior analysis

“…It is often imposed on the parameters to estimate in order to avoid physically absurd and uninterpretable results. Non-negative least-square problems (NNLS) have been addressed in applications ranging from image deblurring [1] to impulse response estimation [2]. Non-negative matrix factorization (NMF) [3], which is closely related to blind deconvolution problems, have also found direct application in hyperspectral imaging [4].…”