### Regularization of multiplicative iterative algorithms with nonnegative constraint

"... Abstract. This paper studies the regularization of constrained Maximum Likelihood iterative algorithms applied to incompatible ill-posed linear inverse problems. Specifically we introduce a novel stopping rule which defines a regularization algorithm for the Iterative Space Reconstruction Algorithm ..."

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Abstract. This paper studies the regularization of constrained Maximum Likelihood iterative algorithms applied to incompatible ill-posed linear inverse problems. Specifically we introduce a novel stopping rule which defines a regularization algorithm for the Iterative Space Reconstruction Algorithm in the case of Least-Squares minimization. Further we show that the same rule regularizes the Expectation Maximization algorithm in the case of Kullback-Leibler minimization provided a welljustified modification of the definition of Tikhonov regularization is introduced. The performances of this stopping rule are illustrated in the case of an image reconstruction problem in X-ray solar astronomy.

### Corrigendum: The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise

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### International Journal for Uncertainty Quantification, 1(1):xxx–xxx, 2015 A METROPOLIS-HASTINGS METHOD FOR LINEAR INVERSE PROBLEMS WITH POISSON LIKELIHOOD AND GAUSSIAN PRIOR

"... Poisson noise models arise in a wide range of linear inverse problems in imaging. In the Bayesian setting, the Poisson likelihood function together with a Gaussian prior yields a posterior density function that is not of a well known form and is thus difficult to sample from, especially for large-sc ..."

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Poisson noise models arise in a wide range of linear inverse problems in imaging. In the Bayesian setting, the Poisson likelihood function together with a Gaussian prior yields a posterior density function that is not of a well known form and is thus difficult to sample from, especially for large-scale problems. In this work, we present a method for computing samples from posterior density functions with Poisson likelihood and Gaussian prior, using a Gaussian approximation of the posterior as an independence proposal within a Metropolis-Hastings framework. To define our priors, we use Gaussian and Laplace-distributedMarkov random fields, which are the Bayesian analogues of smoothness and total variation regularization, respectively. For the Laplace prior, a Gaussian approximation is used, whereas for the Gaussian prior, the scaling (or regularization) parameter is sampled using a hierarchical Gibbs sampler, eliminating the need to choose a regularization parameter a priori. The results are demonstrated on synthetic data–including a synthetic X-ray radiograph generated from a radiation transport code–and on real images used to calibrate a pulsed power high-energy X-ray source at a U.S. Department of Energy X-ray radiography facility.

### Inverse Problems and Imaging doi:10.3934/ipi.xx.xx.xx Volume X, No. 0X, 20xx, X–XX A REWEIGHTED `2 METHOD FOR IMAGE RESTORATION WITH POISSON AND MIXED POISSON-GAUSSIAN NOISE

"... (Communicated by the associate editor name) Abstract. We study weighted `2 fidelity in variational models for Poisson noise related image restoration problems. Gaussian approximation to Poisson noise statistic is adopted to deduce weighted `2 fidelity. Different from the tra-ditional weighted `2 app ..."

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(Communicated by the associate editor name) Abstract. We study weighted `2 fidelity in variational models for Poisson noise related image restoration problems. Gaussian approximation to Poisson noise statistic is adopted to deduce weighted `2 fidelity. Different from the tra-ditional weighted `2 approximation, we propose a reweighted `2 fidelity with sparse regularization by wavelet frame. Based on the split Bregman algorithm introduced in [22], the proposed numerical scheme is composed of three easy subproblems that involve quadratic minimization, soft shrinkage and matrix vector multiplications. Unlike usual least square approximation of Poisson noise, we dynamically update the underlying noise variance from previous es-timate. The solution of the proposed algorithm is shown to be the same as the one obtained by minimizing Kullback-Leibler divergence fidelity with the same regularization. This reweighted `2 formulation can be easily extended to mixed Poisson-Gaussian noise case. Finally, the efficiency and quality of the proposed algorithm compared to other Poisson noise removal methods are demonstrated through denoising and deblurring examples. Moreover, mixed Poisson-Gaussian noise tests are performed on both simulated and real digital images for further illustration of the performance of the proposed method.

### J Math Imaging Vis DOI 10.1007/s10851-014-0553-9 Numerical Methods for Parameter Estimation in Poisson Data Inversion

, 2014

"... Abstract In a regularized approach to Poisson data inver-sion, the problem is reduced to the minimization of an objec-tive function which consists of two terms: a data-fidelity func-tion, related to a generalized Kullback–Leibler divergence, and a regularization function expressing a priori informat ..."

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Abstract In a regularized approach to Poisson data inver-sion, the problem is reduced to the minimization of an objec-tive function which consists of two terms: a data-fidelity func-tion, related to a generalized Kullback–Leibler divergence, and a regularization function expressing a priori information on the unknown image. This second function is multiplied by a parameter β, sometimes called regularization parame-ter, which must be suitably estimated for obtaining a sensible solution. In order to estimate this parameter, a discrepancy principle has been recently proposed, that implies the min-imization of the objective function for several values of β. Since this approach can be computationally expensive, it has also been proposed to replace it with a constrained minimiza-tion, the constraint being derived from the discrepancy prin-ciple. In this paper we intend to compare the two approaches from the computational point of view. In particular, we pro-pose a secant-based method for solving the discrepancy equa-tion arising in the first approach; when this root-finding algo-rithm can be combined with an efficient solver of the inner

### Image restoration with spatially variable PSF

"... We present a method for the restoration of astronomical images obtained with Adaptive Optics (AO) systems. In order to maximize the scientific return from AO data and, in general, from the data of the next generation telescopes, we developed a restoration method based on deconvolution for the de-blu ..."

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We present a method for the restoration of astronomical images obtained with Adaptive Optics (AO) systems. In order to maximize the scientific return from AO data and, in general, from the data of the next generation telescopes, we developed a restoration method based on deconvolution for the de-blurring of images degraded by a spatially variable PSF. The deconvolution method is based on a partition of the image domain in partially overlapping sub-domains where the PSF can be assumed to be space invariant. The software, called Patch, is written in IDL language and is freely distributed to the community. Here we report a general description of the method and of its graphical interface. The potentiality of the Software Patch have been tested on two completely different astrophysical scenarios: a crowded stellar field and an extended galaxy. Despite the very conservative assumptions made on the Point Spread Function (assumed to be strongly variable across the field of view) , we obtained good results in terms of image reconstruction both for the stellar (point-like) case and for the extended galaxy.