A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations

Dupé, François-Xavier; Fadili, Jalal M.; Starck, Jean-Luc

doi:10.1109/tip.2008.2008223

Cited by 175 publications

(166 citation statements)

References 48 publications

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“…A way to generalize our method to Poisson noise would be to include a VST (Anscombe 1948;Dupé et al 2009;Shearer et al 2012) in the acquisition model in (4), both on the observations Y and on the convolution result U(h) X (see Sect. 7 for more details).…”

Section: Noise Estimationmentioning

confidence: 99%

“…In the presence of Poisson noise, the problem can be formulated using the Kullback-Leibler (KL) divergence as the data fidelity term (Fish et al 1995;Prato et al 2012Prato et al , 2013, which represents a more complex function to minimize. To avoid dealing with the KL divergence, the Poisson corrupted data can also be handled through a Variance Stabilization Transform (VST) (Dupé et al 2009;Shearer et al 2012;Shearer 2013). The VST provides an approximated Gaussian noise distributed data and thus allows working with a regularized LS formulation.…”

Section: Introductionmentioning

confidence: 99%

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Non-parametric PSF estimation from celestial transit solar images using blind deconvolution

Gonzalez

Delouille

Jacques

2016

J. Space Weather Space Clim.

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Context: Characterization of instrumental effects in astronomical imaging is important in order to extract accurate physical information from the observations. The measured image in a real optical instrument is usually represented by the convolution of an ideal image with a Point Spread Function (PSF). Additionally, the image acquisition process is also contaminated by other sources of noise (read-out, photon-counting). The problem of estimating both the PSF and a denoised image is called blind deconvolution and is ill-posed. Aims: We propose a blind deconvolution scheme that relies on image regularization. Contrarily to most methods presented in the literature, our method does not assume a parametric model of the PSF and can thus be applied to any telescope. Methods: Our scheme uses a wavelet analysis prior model on the image and weak assumptions on the PSF. We use observations from a celestial transit, where the occulting body can be assumed to be a black disk. These constraints allow us to retain meaningful solutions for the filter and the image, eliminating trivial, translated, and interchanged solutions. Under an additive Gaussian noise assumption, they also enforce noise canceling and avoid reconstruction artifacts by promoting the whiteness of the residual between the blurred observations and the cleaned data. Results: Our method is applied to synthetic and experimental data. The PSF is estimated for the SECCHI/EUVI instrument using the 2007 Lunar transit, and for SDO/AIA using the 2012 Venus transit. Results show that the proposed non-parametric blind deconvolution method is able to estimate the core of the PSF with a similar quality to parametric methods proposed in the literature. We also show that, if these parametric estimations are incorporated in the acquisition model, the resulting PSF outperforms both the parametric and non-parametric methods.

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Section: Noise Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-parametric PSF estimation from celestial transit solar images using blind deconvolution

Gonzalez

Delouille

Jacques

2016

J. Space Weather Space Clim.

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“…Some reconstruction algorithms start by applying a variance-stabilizing transform such that the Anscombe's transform [3] or the Haar-Fisz transform [24] to go closer to the hypothesis of gaussian noise [19] and [25]. Since for small values of the parameter in the Poisson noise, these transforms do not yield exactly gaussian noise, in the following, we will keep the hypothesis of Poisson noise and we will deal directly with the data fidelity terms (11) and (17) derived from the physic of the problem.…”

Section: Incorporating Noisementioning

confidence: 99%

“…The forward-backward splitting algorithm [14] focus on the minimization of the sum of a convex differentiable function f • A + g with f satisfying a L-Lipschitz continuous gradient, A a linear operator and a convex eventually non smooth function g. This algorithm was used in the context of image deconvolution under a Poisson noise when a Anscombe's transform is applied in [19]. In [23] the authors study the Alternative Direction Method of Multipliers (ADMM) described in the setting of Poisson noise, a method which doesn't require any differentiability of f or g. The algorithm PPXA which deals with the sum of possibly more than 2 functions (the functions may even be non differentiable) was applied in the setting of dynamical PET [46], [45].…”

Section: Incorporating Noisementioning

confidence: 99%

Some proximal methods for Poisson intensity CBCT and PET

Anthoine¹,

Aujol²,

Boursier³

et al. 2012

Inverse Problems &Amp; Imaging

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Cone-Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) are two complementary medical imaging modalities providing respectively anatomic and metabolic information of a patient. In the context of public health, one must address the problem of dose reduction of the potentially harmful quantities related to each exam protocol : X-rays for CBCT and radiotracer for PET. Two demonstrators based on a technological breakthrough (acquisition devices work in photon-counting mode) have been developed and we investigate in this paper the two related tomographic reconstruction problems. We formulate separately the CBCT and the PET problems in two general frameworks that encompass the physics of the acquisition devices and the specific discretization of the object to reconstruct. These objects may be observed from a limited number of angles of views and we take into account the specificity of the Poisson noise. We propose various fast numerical schemes based on proximal methods to compute the solution of each problem. In particular, we show that primal-dual approaches are well suited in the PET case when considering non differentiable regularizations such as Total Variation. Experiments on numerical simulations and real data are in favor of the proposed algorithms when compared with the well-established methods.

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“…Note that this method appears mainly to be well-founded for denoising problems. When a linear degradation operator H is present, a better approach consists of adopting a variational framework [8,13] where one minimizes a data fidelity term…”

Section: Introductionmentioning

confidence: 99%

Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

Harizanov

Pesquet

Steidl

2013

Lecture Notes in Computer Science

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Abstract. This papers deals with the restoration of images corrupted by a non-invertible or ill-conditioned linear transform and Poisson noise. Poisson data typically occur in imaging processes where the images are obtained by counting particles, e.g., photons, that hit the image support. By using the Anscombe transform, the Poisson noise can be approximated by an additive Gaussian noise with zero mean and unit variance. Then, the least squares difference between the Anscombe transformed corrupted image and the original image can be estimated by the number of observations. We use this information by considering an Anscombe transformed constrained model to restore the image. The advantage with respect to corresponding penalized approaches lies in the existence of a simple model for parameter estimation. We solve the constrained minimization problem by applying a primal-dual algorithm together with a projection onto the epigraph of a convex function related to the Anscombe transform. We show that this epigraphical projection can be efficiently computed by Newton's methods with an appropriate initialization. Numerical examples demonstrate the good performance of our approach, in particular, its close behaviour with respect to the I-divergence constrained model.

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A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations

Cited by 175 publications

References 48 publications

Non-parametric PSF estimation from celestial transit solar images using blind deconvolution

Non-parametric PSF estimation from celestial transit solar images using blind deconvolution

Some proximal methods for Poisson intensity CBCT and PET

Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

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