Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

Harizanov, Stanislav; Pesquet, Jean-Christophe; Steidl, Gabriele

doi:10.1007/978-3-642-38267-3_11

Cited by 25 publications

(28 citation statements)

References 22 publications

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“…Inequality constraints are frequently used in image processing. For instance, they can be derived from the underlying geometry of the problem, like in [34], in the context of Poisson-noise denoising. We can also mention the work in [35], where inequality constraints are used in a problem of deformable image matching to ensure that the estimated image deformation is injective and preserves the topology.…”

Section: Variational Formulation and Notationmentioning

confidence: 99%

Deep unfolding of a proximal interior point method for image restoration

Bertocchi¹,

Chouzenoux²,

Corbineau³

et al. 2020

Inverse Problems

105

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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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Section: Variational Formulation and Notationmentioning

confidence: 99%

Deep unfolding of a proximal interior point method for image restoration

Bertocchi¹,

Chouzenoux²,

Corbineau³

et al. 2020

Inverse Problems

105

View full text Add to dashboard Cite

show abstract

“…The resulting optimization approach can be solved through an alternating projection technique [15], where both the data fidelity term and the regularization term are based on the KL divergence. The problem was formulated in a similar manner by Richardson and Lucy [16,17], whereas more general forms of the regularization functions were considered by others [18][19][20][21][22][23][24][25]. In particular, some of these works are grounded on proximal splitting methods [19,22,23].…”

Section: Kullback-leibler Divergencementioning

confidence: 99%

“…This difficulty can be circumvented when the constraint can be expressed as the lower-level set of some separable function, by making use of epigraphical projection techniques. Such approaches have attracted interest in the last years [25,62,[92][93][94][95]. The idea consists of decomposing the constraint of interest into the intersection of a half-space and a number of epigraphs of simple functions.…”

Section: Connection With Epigraphical Projectionsmentioning

confidence: 99%

Proximity Operators of Discrete Information Divergences

Gheche

Chierchia²,

Pesquet

2018

IEEE Trans. Inform. Theory

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Information divergences allow one to assess how close two distributions are from each other. Among the large panel of available measures, a special attention has been paid to convex ϕ-divergences, such as Kullback-Leibler, Jeffreys, Hellinger, Chi-Square, Renyi, and I α divergences. While ϕ-divergences have been extensively studied in convex analysis, their use in optimization problems often remains challenging. In this regard, one of the main shortcomings of existing methods is that the minimization of ϕ-divergences is usually performed with respect to one of their arguments, possibly within alternating optimization techniques. In this paper, we overcome this limitation by deriving new closed-form expressions for the proximity operator of such two-variable functions. This makes it possible to employ standard proximal methods for efficiently solving a wide range of convex optimization problems involving ϕ-divergences. In addition, we show that these proximity operators are useful to compute the epigraphical projection of several functions. The proposed proximal tools are numerically validated in the context of optimal query execution within database management systems, where the problem of selectivity estimation plays a central role. Experiments are carried out on small to large scale scenarios.

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“…Furthermore, in applications such as astronomy, medicine, and fluorescence microscopy where signals are acquired via photon counting devices, like CMOS and CCD cameras, the number of collected photons is related to some non-additive counting errors resulting in a shot noise [1-3, 17, 18]. The latter is non-additive, signal-dependent and it can be modeled by a Poisson distribution [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. In this case, when the noise is assumed to be Poisson distributed, the implicit assumption is that Poisson noise dominates over all other noise kinds.…”

Section: Introductionmentioning

confidence: 99%

A Variational Bayesian Approach for Image Restoration—Application to Image Deblurring With Poisson–Gaussian Noise

Marnissi

Zheng

Chouzenoux

et al. 2017

IEEE Trans. Comput. Imaging

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In this paper, a methodology is investigated for signal recovery in the presence of non-Gaussian noise. In contrast with regularized minimization approaches often adopted in the literature, in our algorithm the regularization parameter is reliably estimated from the observations. As the posterior density of the unknown parameters is analytically intractable, the estimation problem is derived in a variational Bayesian framework where the goal is to provide a good approximation to the posterior distribution in order to compute posterior mean estimates. Moreover, a majorization technique is employed to circumvent the difficulties raised by the intricate forms of the non-Gaussian likelihood and of the prior density. We demonstrate the potential of the proposed approach through comparisons with state-of-the-art techniques that are specifically tailored to signal recovery in the presence of mixed Poisson-Gaussian noise. Results show that the proposed approach is efficient and achieves performance comparable with other methods where the regularization parameter is manually tuned from the ground truth.

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Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

Cited by 25 publications

References 22 publications

Deep unfolding of a proximal interior point method for image restoration

Deep unfolding of a proximal interior point method for image restoration

Proximity Operators of Discrete Information Divergences

A Variational Bayesian Approach for Image Restoration—Application to Image Deblurring With Poisson–Gaussian Noise

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