Wavelet‐Based Image Deconvolution and Reconstruction

Pustelnik, Nelly; Benazza-Benhayia, Amel; Zheng, Yuling; Pesquet, Jean-Christophe

doi:10.1002/047134608x.w8294

Cited by 50 publications

(52 citation statements)

References 179 publications

(214 reference statements)

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“…The proposed DA approach can also be applied to the problem of drawing random variables from a high-dimensional Gaussian distribution with parameters m and G as defined in (5) and (6). The introduction of auxiliary variables can be especially useful in facilitating the sampling process in a number of problems that we discuss below.…”

Section: High-dimensional Gaussian Distributionmentioning

confidence: 99%

“…Other common choices can be found for instance in [3,4]. Moreover, Ψ(V·) is related to some prior knowledge one can have about x, and V ∈ R M×N is a linear transform that can describe, for example, a frame analysis [5] or a discrete gradient operator [6]. Within a Bayesian framework, it is related to a prior distribution of density p(x) whose logarithm is given by log p(x) = −Ψ(Vx).…”

Section: Introductionmentioning

confidence: 99%

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An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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Abstract:In this paper, we are interested in Bayesian inverse problems where either the data fidelity term or the prior distribution is Gaussian or driven from a hierarchical Gaussian model. Generally, Markov chain Monte Carlo (MCMC) algorithms allow us to generate sets of samples that are employed to infer some relevant parameters of the underlying distributions. However, when the parameter space is high-dimensional, the performance of stochastic sampling algorithms is very sensitive to existing dependencies between parameters. In particular, this problem arises when one aims to sample from a high-dimensional Gaussian distribution whose covariance matrix does not present a simple structure. Another challenge is the design of Metropolis-Hastings proposals that make use of information about the local geometry of the target density in order to speed up the convergence and improve mixing properties in the parameter space, while not being too computationally expensive. These two contexts are mainly related to the presence of two heterogeneous sources of dependencies stemming either from the prior or the likelihood in the sense that the related covariance matrices cannot be diagonalized in the same basis. In this work, we address these two issues. Our contribution consists of adding auxiliary variables to the model in order to dissociate the two sources of dependencies. In the new augmented space, only one source of correlation remains directly related to the target parameters, the other sources of correlations being captured by the auxiliary variables. Experiments are conducted on two practical image restoration problems-namely the recovery of multichannel blurred images embedded in Gaussian noise and the recovery of signal corrupted by a mixed Gaussian noise. Experimental results indicate that adding the proposed auxiliary variables makes the sampling problem simpler since the new conditional distribution no longer contains highly heterogeneous correlations. Thus, the computational cost of each iteration of the Gibbs sampler is significantly reduced while ensuring good mixing properties.

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Section: High-dimensional Gaussian Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

Self Cite

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“…Their Lipschitz constants are denoted β T C and β K respectively [24]. The other three functions however are not differentiable and f T V involves a linear transformation H such as:…”

Section: Algorithmmentioning

confidence: 99%

A Primal-Dual Algorithm for Link Dependent Origin Destination Matrix Estimation

Michau

Pustelnik

Borgnat

et al. 2017

IEEE Trans. on Signal and Inf. Process. over Networks

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Origin-Destination Matrix (ODM) estimation is a classical problem in transport engineering aiming to recover flows from every Origin to every Destination from measured traffic counts and a priori model information. In addition to traffic counts, the present contribution takes advantage of probe trajectories, whose capture is made possible by new measurement technologies. It extends the concept of ODM to that of Link dependent ODM (LODM), keeping the information about the flow distribution on links and containing inherently the ODM assignment. Further, an original formulation of LODM estimation, from traffic counts and probe trajectories is presented as an optimisation problem, where the functional to be minimized consists of five convex functions, each modelling a constraint or property of the transport problem: consistency with traffic counts, consistency with sampled probe trajectories, consistency with traffic conservation (Kirchhoff's law), similarity of flows having close origins and destinations, positivity of traffic flows. A primal-dual algorithm is devised to minimize the designed functional, as the corresponding objective functions are not necessarily differentiable. A case study, on a simulated network and traffic, validates the feasibility of the procedure and details its benefits for the estimation of an LODM matching real-network constraints and observations. * Preliminary versions of this work were presented in [1,2].

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“…In order to find an appropriate solution to an ill-posed inverse problem like (1), variational methods incorporate prior information on the sought variable x, through constraints or regularization functions, such as the total variation and its various extensions [3] or sparsity-promoting functions [4]. This leads to the following minimization problem, min x∈C f (Hx, y) + λR(x)…”

Section: Introductionmentioning

confidence: 99%

Deep unfolding of a proximal interior point method for image restoration

Bertocchi¹,

Chouzenoux²,

Corbineau³

et al. 2020

Inverse Problems

107

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Variational methods are widely applied to ill-posed inverse problems for they have the ability to embed prior knowledge about the solution. However, the level of performance of these methods significantly depends on a set of parameters, which can be estimated through computationally expensive and timeconsuming methods. In contrast, deep learning offers very generic and efficient architectures, at the expense of explainability, since it is often used as a black-box, without any fine control over its output. Deep unfolding provides a convenient approach to combine variational-based and deep learning approaches. Starting from a variational formulation for image restoration, we develop iRestNet, a neural network architecture obtained by unfolding a proximal interior point algorithm. Hard constraints, encoding desirable properties for the restored image, are incorporated into the network thanks to a logarithmic barrier, while the barrier parameter, the stepsize, and the penalization weight are learned by the network. We derive explicit expressions for the gradient of the proximity operator for various choices of constraints, which allows training iRestNet with gradient descent and backpropagation. In addition, we provide theoretical results regarding the stability of the network for a common inverse problem example. Numerical experiments on image deblurring problems show that the proposed approach compares favorably with both state-of-the-art variational and machine learning methods in terms of image quality.

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Wavelet‐Based Image Deconvolution and Reconstruction

Cited by 50 publications

References 179 publications

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

A Primal-Dual Algorithm for Link Dependent Origin Destination Matrix Estimation

Deep unfolding of a proximal interior point method for image restoration

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