A discrepancy principle for Poisson data

Abstract

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.

Deconvolution of post-adaptive optics images of faint circumstellar environments by means of the inexact Bregman procedure

“…We denote this error ρ w (for sake of completeness, we point out that we enumerate the pixels from 1 to 1024 in both directions). As previously mentioned, in practical applications, only an estimate of β opt can be obtained (see Bertero et al 2010;Bardsley & Goldes 2009). Moreover, the numerical experience (see Benfenati & Ruggiero 2015) has shown that these methods cannot provide satisfactory results for some classes of images (such as the image treated in this work) because of the hypothesis underlying these techniques.…”

“…The parameter β is named regularization parameter and it measures the trade-off between f 0 and f 1 . In practical application, one needs the value that gives the best reconstruction, but actually this value is very difficult to estimate (see Bertero et al 2010;Bardsley & Goldes 2009, for the Poisson case).…”

“…A practical strategy consists in estimating β with one of the methods mentioned previously in the paper (e.g. Bertero et al 2010;Bardsley & Goldes 2009), and then multiply the obtained value (namely β) by a factor γ > 1. Four scenarios may arise by adopting this strategy:…”

Space Variant Blind Image Restoration

“…The two first terms corresponds to the data fidelity component related to the Poisson statistic , Bertero et al [2010]). The third term is the total variation function (Rudin et al [1992]) which allows to smooth homogeneous areas of the recovered images while preserving sharp edges.…”

Some proximal methods for Poisson intensity CBCT and PET

“…FISTA is an acceleration procedure for non constraint algorithms [5], and in the special case of the TV functional and ℓ 1 , additional constraints such as the positivity of the solution can be addressed [4]. The choice of the parameter λ in the regularization term was studied in [6] for data fidelity terms issued from a Poisson noise. In [13] an algorithm which belongs to primal-dual methods is described since primal and dual variables are successively computed in the algorithm.…”