Abstract. A class of scaled gradient projection methods for optimization problems with simple constraints is considered. These iterative algorithms can be useful in variational approaches to image deblurring that lead to minimize convex nonlinear functions subject to nonnegativity constraints and, in some cases, to an additional flux conservation constraint. A special gradient projection method is introduced that exploits effective scaling strategies and steplength updating rules, appropriately designed for improving the convergence rate. We give convergence results for this scheme and we evaluate its effectiveness by means of an extensive computational study on the minimization problems arising from the maximum likelihood approach to image deblurring. Comparisons with the standard expectation maximization algorithm and with other iterative regularization schemes are also reported to show the computational gain provided by the proposed method.

Several methods based on different image models have been proposed and developed for image denoising. Some of them, such as total variation (TV) and wavelet thresholding, are based on the assumption of additive Gaussian noise. Recently the TV approach has been extended to the case of Poisson noise, a model describing the effect of photon counting in applications such as emission tomography, microscopy and astronomy. For the removal of this kind of noise we consider an approach based on a constrained optimization problem, with an objective function describing TV and other edge-preserving regularizations of the Kullback-Leibler divergence. We introduce a new discrepancy principle for the choice of the regularization parameter, which is justified by the statistical properties of the Poisson noise. For solving the optimization problem we propose a particular form of a general scaled gradient projection (SGP) method, recently introduced for image deblurring. We derive the form of the scaling from a decomposition of the gradient of the regularization functional into a positive and a negative part. The beneficial effect of the scaling is proved by means of numerical simulations, showing that the performance of the proposed form of SGP is superior to that of the most efficient gradient projection methods. An extended numerical analysis of the dependence of the solution on the regularization parameter is also performed to test the effectiveness of the proposed discrepancy principle.

In applications of imaging science, such as emission tomography, fluorescence microscopy and optical/infrared astronomy, image intensity is measured via the counting of incident particles (photons, γ-rays, etc). Fluctuations in the emission-counting process can be described by modeling the data as realizations of Poisson random variables (Poisson data). A maximum-likelihood approach for image reconstruction from Poisson data was proposed in the mid-1980s. Since the consequent maximization problem is, in general, ill-conditioned, various kinds of regularizations were introduced in the framework of the so-called Bayesian paradigm. A modification of the well-known Tikhonov regularization strategy results in the data-fidelity function being a generalized Kullback-Leibler divergence. Then a relevant issue is to find rules for selecting a proper value of the regularization parameter. In this paper we propose a criterion, nicknamed discrepancy principle for Poisson data, that applies to both denoising and deblurring problems and fits quite naturally the statistical properties of the data. The main purpose of the paper is to establish conditions, on the data and the imaging matrix, ensuring that the proposed criterion does actually provide a unique value of the regularization parameter for various classes of regularization functions. A few numerical experiments are performed to demonstrate its effectiveness. More extensive numerical analysis and comparison with other proposed criteria will be the object of future work.

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.

Although deconvolution can improve the quality of any type of microscope, the high computational time required has so far limited its massive spreading. Here we demonstrate the ability of the scaled-gradient-projection (SGP) method to provide accelerated versions of the most used algorithms in microscopy. To achieve further increases in efficiency, we also consider implementations on graphic processing units (GPUs). We test the proposed algorithms both on synthetic and real data of confocal and STED microscopy. Combining the SGP method with the GPU implementation we achieve a speed-up factor from about a factor 25 to 690 (with respect the conventional algorithm). The excellent results obtained on STED microscopy images demonstrate the synergy between super-resolution techniques and image-deconvolution. Further, the real-time processing allows conserving one of the most important property of STED microscopy, i.e the ability to provide fast sub-diffraction resolution recordings.