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This paper presents a multilevel algorithm specifically designed for radio-interferometric imaging in astronomy. The proposed algorithm is used to solve the uSARA (unconstrained Sparsity Averaging Reweighting Analysis) formulation of this image restoration problem. Multilevel algorithms rely on a hierarchy of approximations of the objective function to accelerate its optimization. In contrast to the usual multilevel approaches where this hierarchy is derived in the parameter space, here we construct the hierarchy of approximations in the observation space. The proposed approach is compared to a reweighted forward-backward procedure, which is the backbone iteration scheme for solving the uSARA problem.

Selecting the fastest algorithm for a specific signal/image processing task is a challenging question. We propose an approach based on the Performance Estimation Problem framework that numerically and automatically computes the worst-case performance of a given optimization method on a class of functions. We first propose a computer-assisted analysis and comparison of several first-order primal optimization methods, namely, the gradient method, the forward-backward, Peaceman-Rachford, and Douglas-Rachford splittings. We tighten the existing convergence results of these algorithms and extend them to new classes of functions. Our analysis is then extended and evaluated in the context of the primal-dual Chambolle-Pock and Condat-V\~u methods., Comment: 5 pages, 7 figures

Plug-and-play algorithms constitute a popular framework for solving inverse imaging problems that rely on the implicit definition of an image prior via a denoiser. These algorithms can leverage powerful pre-trained denoisers to solve a wide range of imaging tasks, circumventing the necessity to train models on a per-task basis. Unfortunately, plug-and-play methods often show unstable behaviors, hampering their promise of versatility and leading to suboptimal quality of reconstructed images. In this work, we show that enforcing equivariance to certain groups of transformations (rotations, reflections, and/or translations) on the denoiser strongly improves the stability of the algorithm as well as its reconstruction quality. We provide a theoretical analysis that illustrates the role of equivariance on better performance and stability. We present a simple algorithm that enforces equivariance on any existing denoiser by simply applying a random transformation to the input of the denoiser and the inverse transformation to the output at each iteration of the algorithm. Experiments on multiple imaging modalities and denoising networks show that the equivariant plug-and-play algorithm improves both the reconstruction performance and the stability compared to their non-equivariant counterparts.

A common approach to solve inverse imaging problems relies on finding a maximum a posteriori (MAP) estimate of the original unknown image, by solving a minimization problem. In thiscontext, iterative proximal algorithms are widely used, enabling to handle non-smooth functions and linear operators. Recently, these algorithms have been paired with deep learning strategies, to further improve the estimate quality. In particular, proximal neural networks (PNNs) have been introduced, obtained by unrolling a proximal algorithm as for finding a MAP estimate, but over a fixed number of iterations, with learned linear operators and parameters. As PNNs are based on optimization theory, they are very flexible, and can be adapted to any image restoration task, as soon as a proximal algorithm can solve it. They further have much lighter architectures than traditional networks. In this article we propose a unified framework to build PNNs for the Gaussian denoising task, based on both the dual-FB and the primal-dual Chambolle-Pock algorithms. We further show that accelerated inertial versions of these algorithms enable skip connections in the associated NN layers. We propose different learning strategies for our PNN framework, and investigate their robustness (Lipschitz property) and denoising efficiency. Finally, we assess the robustness of our PNNs when plugged in a forward-backward algorithm for an image deblurring problem.

This paper presents a multilevel framework for inertial and inexact proximal algorithms, that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems and we show that it considerably accelerates the convergence of the corresponding one-level (i.e. standard) version of the methods, for large-scale images.

This paper presents a multilevel FISTA algorithm, based on the use of the Moreau envelope to build the correction brought by the coarse models, which is easy to compute when the explicit form of the proximal operator of the considered functions is known. This approach is supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We evaluate our approach on image restoration problems and we show that it outperforms classical FISTA for large-scale images.

Monitoring the Covid19 pandemic constitutes a critical societal stake that received considerable research efforts. The intensity of the pandemic on a given territory is efficiently measured by the reproduction number, quantifying the rate of growth of daily new infections. Recently, estimates for the time evolution of the reproduction number were produced using an inverse problem formulation with a nonsmooth functional minimization. While it was designed to be robust to the limited quality of the Covid19 data (outliers, missing counts), the procedure lacks the ability to output credibility interval based estimates. This remains a severe limitation for practical use in actual pandemic monitoring by epidemiologists that the present work aims to overcome by use of Monte Carlo sampling. After interpretation of the nonsmooth functional into a Bayesian framework, several sampling schemes are tailored to adjust the nonsmooth nature of the resulting posterior distribution. The originality of the devised algorithms stems from combining a Langevin Monte Carlo sampling scheme with Proximal operators. Performance of the new algorithms in producing relevant credibility intervals for the reproduction number estimates and denoised counts are compared. Assessment is conducted on real daily new infection counts made available by the Johns Hopkins University. The interest of the devised monitoring tools are illustrated on Covid19 data from several different countries.

This work designs an image restoration deep network relying on unfolded Chambolle-Pock primal-dual iterations. Each layer of our network is built from Chambolle-Pock iterations when specified for minimizing a sum of a $\ell_2$-norm data-term and an analysis sparse prior. The parameters of our network are the step-sizes of the Chambolle-Pock scheme and the linear operator involved in sparsity-based penalization, including implicitly the regularization parameter. A backpropagation procedure is fully described. Preliminary experiments illustrate the good behavior of such a deep primal-dual network in the context of image restoration on BSD68 database.

Monitoring the evolution of the Covid19 pandemic constitutes a critical step in sanitary policy design. Yet, the assessment of the pandemic intensity within the pandemic period remains a challenging task because of the limited quality of data made available by public health authorities (missing data, outliers and pseudoseasonalities, notably), that calls for cumbersome and ad-hoc preprocessing (denoising) prior to estimation. Recently, the estimation of the reproduction number, a measure of the pandemic intensity, was formulated as an inverse problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that formulation lacks robustness against the limited quality of the Covid19 data and confidence assessment. The present work aims to address both limitations: First, it discusses solutions to produce a robust assessment of the pandemic intensity by accounting for the low quality of the data directly within the inverse problem formulation. Second, exploiting a Bayesian interpretation of the inverse problem formulation, it devises a Monte Carlo sampling strategy, tailored to a nonsmooth log-concave a posteriori distribution, to produce relevant credibility intervalbased estimates for the Covid19 reproduction number. Clinical relevance Applied to daily counts of new infections made publicly available by the Health Authorities for around 200 countries, the proposed procedures permit robust assessments of the time evolution of the Covid19 pandemic intensity, updated automatically and on a daily basis.

This work focuses on a parameter-free joint piecewise smooth image denoising and contour detection. Formulated as the minimization of a discrete Mumford-Shah functional and estimated via a theoretically grounded alternating minimization scheme, the bottleneck of such a variational approach lies in the need to fine-tune their hyperparameters, while not having access to ground truth data. To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk. Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters. Its practical implementation is performed using a forward differentiation of the alternating scheme minimizing the Mumford-Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometry and noise levels, assessing the accuracy and the robustness of the proposed procedure. The resulting parameter-free piecewise-smooth estimation and contour detection procedure, not requiring prior image processing expertise nor annotated data, can then be applied to real-world images.

Daily pandemic surveillance, often achieved through the estimation of the reproduction number, constitutes a critical challenge for national health authorities to design countermeasures. In an earlier work, we proposed to formulate the estimation of the reproduction number as an optimization problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that first formulation significantly lacks robustness against the Covid-19 data low quality (irrelevant or missing counts, pseudo-seasonalities,.. .) stemming from the emergency and crisis context, which significantly impairs accurate pandemic evolution assessments. The present work aims to overcome these limitations by carefully crafting a functional permitting to estimate jointly, in a single step, the reproduction number and outliers defined to model low quality data. This functional also enforces epidemiology-driven regularity properties for the reproduction number estimates, while preserving convexity, thus permitting the design of efficient minimization algorithms, based on proximity operators that are derived analytically. The explicit convergence of the proposed algorithm is proven theoretically. Its relevance is quantified on real Covid-19 data, consisting of daily new infection counts for 200+ countries and for the 96 metropolitan France counties, publicly available at Johns Hopkins University and Sant{\'e}-Publique-France. The procedure permits automated daily updates of these estimates, reported via animated and interactive maps. Open-source estimation procedures will be made publicly available.

Polarimetric imaging is one of the most effective techniques for high-contrast imaging and characterization of circumstellar environments. These environments can be characterized through direct-imaging polarimetry at near-infrared wavelengths. The SPHERE/IRDIS instrument installed on the Very Large Telescope in its dual-beam polarimetric imaging (DPI) mode, offers the capability to acquire polarimetric images at high contrast and high angular resolution. However dedicated image processing is needed to get rid of the contamination by the stellar light, of instrumental polarization effects, and of the blurring by the instrumental point spread function. We aim to reconstruct and deconvolve the near-infrared polarization signal from circumstellar environments. We use observations of these environments obtained with the high-contrast imaging infrared polarimeter SPHERE-IRDIS at the VLT. We developed a new method to extract the polarimetric signal using an inverse approach method that benefits from the added knowledge of the detected signal formation process. The method includes weighted data fidelity term, smooth penalization, and takes into account instrumental polarization. The method enables to accurately measure the polarized intensity and angle of linear polarization of circumstellar disks by taking into account the noise statistics and the convolution of the observed objects by the instrumental point spread function. It has the capability to use incomplete polarimetry cycles which enhance the sensitivity of the observations. The method improves the overall performances in particular for low SNR/small polarized flux compared to standard methods.

This paper provides a theoretical and numerical comparison of classical first-order splitting methods for solving smooth convex optimization problems and cocoercive equations. From a theoretical point of view, we compare convergence rates of gradient descent, forward-backward, Peaceman-Rachford, and Douglas-Rachford algorithms for minimizing the sum of two smooth convex functions when one of them is strongly convex. A similar comparison is given in the more general cocoercive setting under the presence of strong monotonicity and we observe that the convergence rates in optimization are strictly better than the corresponding rates for cocoercive equations for some algorithms. We obtain improved rates with respect to the literature in several instances by exploiting the structure of our problems. Moreover, we indicate which algorithm has the lowest convergence rate depending on strong convexity and cocoercive parameters. From a numerical point of view, we verify our theoretical results by implementing and comparing previous algorithms in well-established signal and image inverse problems involving sparsity. We replace the widely used $\ell_1$ norm with the Huber loss and we observe that fully proximal-based strategies have numerical and theoretical advantages with respect to methods using gradient steps., Comment: 20 pages, 18 figures

The shear resulting from internal wave reflections can play a crucial role in the transport and resuspension of sediments in oceanic conditions. In particular, when these waves undergo a \textit{critical reflection} phenomenon, the reflected wave can produce a very large shear. Separating the reflected wave from the incident wave is a technical challenge since the two waves share the same temporal frequency. In our study, we present a series of experimental measurements of internal waves in \textit{critical reflection} configuration and we analyze them using the 2D-VMD-prox decomposition method. This decomposition method was adapted to specifically decompose waves in an internal wave critical reflection, showing an improvement in its performance with respect to preexisting internal wave decomposition methods. Being able to confidently isolate the reflected wave allowed us to compare our results to a viscous and non-linear model for critical reflection, that correctly describes the dependence of the shear rate produced in the boundary as a function of the experimental parameters., Comment: 14 pages, 20 figures

The convergence of many proximal algorithms involving a gradient descent relies on its Lipschitz constant. To avoid computing it, backtracking rules can be used. While such a rule has already been designed for the forward-backward algorithm (FBwB), this scheme is not flexible enough when a non-differentiable penalization with a linear operator is added to a constraint. In this work we propose a backtracking rule for the primal-dual scheme (PDwB), and evaluate its performance for the epigraphical constrained high dynamical reconstruction in high contrast polarimetric imaging, under TV penalization., Comment: in Proceedings of iTWIST'20, Paper-ID: 12, Nantes, France, December, 2-4, 2020

Image restoration remains a challenging task in image processing. Numerous methods tackle this problem, often solved by minimizing a non-smooth penalized co-log-likelihood function. Although the solution is easily interpretable with theoretic guarantees, its estimation relies on an optimization process that can take time. Considering the research effort in deep learning for image classification and segmentation, this class of methods offers a serious alternative to perform image restoration but stays challenging to solve inverse problems. In this work, we design a deep network, named DeepPDNet, built from primal-dual proximal iterations associated with the minimization of a standard penalized likelihood with an analysis prior, allowing us to take advantage of both worlds. We reformulate a specific instance of the Condat-Vu primal-dual hybrid gradient (PDHG) algorithm as a deep network with fixed layers. The learned parameters are both the PDHG algorithm step-sizes and the analysis linear operator involved in the penalization (including the regularization parameter). These parameters are allowed to vary from a layer to another one. Two different learning strategies: "Full learning" and "Partial learning" are proposed, the first one is the most efficient numerically while the second one relies on standard constraints ensuring convergence in the standard PDHG iterations. Moreover, global and local sparse analysis prior are studied to seek a better feature representation. We apply the proposed methods to image restoration on the MNIST and BSD68 datasets and to single image super-resolution on the BSD100 and SET14 datasets. Extensive results show that the proposed DeepPDNet demonstrates excellent performance on the MNIST and the more complex BSD68, BSD100, and SET14 datasets for image restoration and single image super-resolution task.

Numerous fields of nonlinear physics, very different in nature, produce signals and images, that share the common feature of being essentially constituted of piecewise homogeneous phases. Analyzing signals and images from corresponding experiments to construct relevant physical interpretations thus often requires detecting such phases and estimating accurately their characteristics (borders, feature differences, ...). However, situations of physical relevance often comes with low to very low signal to noise ratio precluding the standard use of classical linear filtering for analysis and denoising and thus calling for the design of advanced nonlinear signal/image filtering techniques. Additionally, when dealing with experimental physics signals/images, a second limitation is the large amount of data that need to be analyzed to yield accurate and relevant conclusions requiring the design of fast algorithms. The present work proposes a unified signal/image nonlinear filtering procedure, with fast algorithms and a data-driven automated hyperparameter tuning, based on proximal algorithms and Stein unbiased estimator principles. The interest and potential of these tools are illustrated at work on low-confinement solid friction signals and porous media multiphase flows.

Penalized Least Squares are widely used in signal and image processing. Yet, it suffers from a major limitation since it requires fine-tuning of the regularization parameters. Under assumptions on the noise probability distribution, Stein-based approaches provide unbiased estimator of the quadratic risk. The Generalized Stein Unbiased Risk Estimator is revisited to handle correlated Gaussian noise without requiring to invert the covariance matrix. Then, in order to avoid expansive grid search, it is necessary to design algorithmic scheme minimizing the quadratic risk with respect to regularization parameters. This work extends the Stein's Unbiased GrAdient estimator of the Risk of Deledalle et al. to the case of correlated Gaussian noise, deriving a general automatic tuning of regularization parameters. First, the theoretical asymptotic unbiasedness of the gradient estimator is demonstrated in the case of general correlated Gaussian noise. Then, the proposed parameter selection strategy is particularized to fractal texture segmentation, where problem formulation naturally entails inter-scale and spatially correlated noise. Numerical assessment is provided, as well as discussion of the practical issues., Comment: Submitted to SIAM Imaging Science

Non-negative matrix factorization is a problem of dimensionality reduction and source separation of data that has been widely used in many fields since it was studied in depth in 1999 by Lee and Seung, including in compression of data, document clustering, processing of audio spectrograms and astronomy. In this work we have adapted a minimization scheme for convex functions with non-differentiable constraints called PALM to solve the NMF problem with solutions that can be smooth and/or sparse, two properties frequently desired.

The present work investigates the segmentation of textures by formulating it as a strongly convex optimization problem, aiming to favor piecewise constancy of fractal features (local variance and local regularity) widely used to model real-world textures in numerous applications very different in nature. Two objective functions combining these two features are compared, referred to as joint and coupled, promoting either independent or co-localized changes in local variance and regularity. To solve the resulting convex nonsmooth optimization problems, because the processing of large size images and databases are targeted, two categories of proximal algorithms (dual forward-backward and primal-dual), are devised and compared. An in-depth study of the objective functions, notably of their strong convexity, memory and computational costs, permits to propose significantly accelerated algorithms. A class of synthetic models of piecewise fractal texture is constructed and studied. They enable, by means of large-scale Monte-Carlo simulations, to quantify the benefits in texture segmentation of combining local regularity and local variance (as opposed to regularity only) while using strong-convexity accelerated primal-dual algorithms. Achieved results also permit to discuss the gains/costs in imposing co-localizations of changes in local regularity and local variance in the problem formulation. Finally, the potential of the proposed approaches is illustrated on real-world textures taken from a publicly available and documented database., Comment: To appear in 2021 in Applied and Computational Harmonic Analysis

Origin-Destination matrix estimation is a keystone for traffic representation and analysis. Traditionally estimated thanks to traffic counts, surveys and socio-economic models, recent technological advances permit to rethink the estimation problem. Road user identification technologies, such as connected GPS, Bluetooth or Wifi detectors bring additional information, that is, for a fraction of the users, the origin, the destination and to some extend the itinerary taken. In the present work, this additional information is used for the estimation of a more comprehensive traffic representation tool: the link-origin-destination matrix. Such three-dimensional matrices extend the concept of traditional origin-destination matrices by also giving information on the traffic assignment. Their estimation is solved as an inverse problem whose objective function represents a trade-off between important properties the traffic has to satisfy. This article presents the theory and how to implement such method on real dataset. With the case study of Brisbane City where over 600 hundreds Bluetooth detectors have been installed it also illustrates the opportunities such link-origin-destination matrices create for traffic analysis.

This work focuses on learning optimization problems with quadratical interactions between variables, which go beyond the additive models of traditional linear learning. We investigate more specifically two different methods encountered in the literature to deal with this problem: "hierNet" and structured-sparsity regularization, and study their connections. We propose a primal-dual proximal algorithm based on an epigraphical projection to optimize a general formulation of these learning problems. The experimental setting first highlights the improvement of the proposed procedure compared to state-of-the-art methods based on fast iterative shrinkage-thresholding algorithm (i.e. FISTA) or alternating direction method of multipliers (i.e. ADMM), and then, using the proposed flexible optimization framework, we provide fair comparisons between the different hierarchical penalizations and their improvement over the standard $\ell_1$-norm penalization. The experiments are conducted both on synthetic and real data, and they clearly show that the proposed primal-dual proximal algorithm based on epigraphical projection is efficient and effective to solve and investigate the problem of hierarchical interaction learning.

Self-similarity is widely considered the reference framework for modeling the scaling properties of real-world data. However, most theoretical studies and their practical use have remained univariate. Operator Fractional Brownian Motion (OfBm) was recently proposed as a multivariate model for self-similarity. Yet it has remained seldom used in applications because of serious issues that appear in the joint estimation of its numerous parameters. While the univariate fractional Brownian motion requires the estimation of two parameters only, its mere bivariate extension already involves 7 parameters which are very different in nature. The present contribution proposes a method for the full identification of bivariate OfBm (i.e., the joint estimation of all parameters) through an original formulation as a non-linear wavelet regression coupled with a custom-made Branch & Bound numerical scheme. The estimation performance (consistency and asymptotic normality) is mathematically established and numerically assessed by means of Monte Carlo experiments. The impact of the parameters defining OfBm on the estimation performance as well as the associated computational costs are also thoroughly investigated.

Piecewise constant denoising can be solved either by deterministic optimization approaches, based on the Potts model, or by stochastic Bayesian procedures. The former lead to low computational time but require the selection of a regularization parameter, whose value significantly impacts the achieved solution, and whose automated selection remains an involved and challenging problem. Conversely, fully Bayesian formalisms encapsulate the regularization parameter selection into hierarchical models, at the price of high computational costs. This contribution proposes an operational strategy that combines hierarchical Bayesian and Potts model formulations, with the double aim of automatically tuning the regularization parameter and of maintaining computational effciency. The proposed procedure relies on formally connecting a Bayesian framework to a l2-Potts functional. Behaviors and performance for the proposed piecewise constant denoising and regularization parameter tuning techniques are studied qualitatively and assessed quantitatively, and shown to compare favorably against those of a fully Bayesian hierarchical procedure, both in accuracy and in computational load.

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.

In the context of change-point detection, addressed by Total Variation minimization strategies, an efficient on-the-fly algorithm has been designed leading to exact solutions for univariate data. In this contribution, an extension of such an on-the-fly strategy to multivariate data is investigated. The proposed algorithm relies on the local validation of the Karush-Kuhn-Tucker conditions on the dual problem. Showing that the non-local nature of the multivariate setting precludes to obtain an exact on-the-fly solution, we devise an on-the-fly algorithm delivering an approximate solution, whose quality is controlled by a practitioner-tunable parameter, acting as a trade-off between quality and computational cost. Performance assessment shows that high quality solutions are obtained on-the-fly while benefiting of computational costs several orders of magnitude lower than standard iterative procedures. The proposed algorithm thus provides practitioners with an efficient multivariate change-point detection on-the-fly procedure.

Texture segmentation constitutes a standard image processing task, crucial to many applications. The present contribution focuses on the particular subset of scale-free textures and its originality resides in the combination of three key ingredients: First, texture characterization relies on the concept of local regularity ; Second, estimation of local regularity is based on new multiscale quantities referred to as wavelet leaders ; Third, segmentation from local regularity faces a fundamental bias variance trade-off: In nature, local regularity estimation shows high variability that impairs the detection of changes, while a posteriori smoothing of regularity estimates precludes from locating correctly changes. Instead, the present contribution proposes several variational problem formulations based on total variation and proximal resolutions that effectively circumvent this trade-off. Estimation and segmentation performance for the proposed procedures are quantified and compared on synthetic as well as on real-world textures.

Sparsity-inducing penalties are useful tools to design multiclass support vector machines (SVMs). In this paper, we propose a convex optimization approach for efficiently and exactly solving the multiclass SVM learning problem involving a sparse regularization and the multiclass hinge loss formulated by Crammer and Singer. We provide two algorithms: the first one dealing with the hinge loss as a penalty term, and the other one addressing the case when the hinge loss is enforced through a constraint. The related convex optimization problems can be efficiently solved thanks to the flexibility offered by recent primal-dual proximal algorithms and epigraphical splitting techniques. Experiments carried out on several datasets demonstrate the interest of considering the exact expression of the hinge loss rather than a smooth approximation. The efficiency of the proposed algorithms w.r.t. several state-of-the-art methods is also assessed through comparisons of execution times.

This work proposes an extension of the 1-D Hilbert Huang transform for the analysis of images. The proposed method consists in (i) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure, and (ii) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2-D mode decompositions based on non-smooth convex optimization: a "Genuine 2-D" approach, that constrains the local extrema of the IMFs, and a "Pseudo 2-D" approach, which constrains separately the extrema of lines, columns, and diagonals. The spectral analysis step is based on Prony annihilation property that is applied on small square patches of the IMFs. The resulting 2-D Prony-Huang transform is validated on simulated and real data., Comment: 24 pages, 7 figures

Non-Local Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the non-local variations, jointly for the different components, through various $\ell_{1,p}$ matrix norms with $p \ge 1$. To facilitate the choice of the hyper-parameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented thanks to the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for multispectral and hyperspectral images. The results demonstrate the interest of introducing a non-local structure tensor regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods.

We propose a proximal approach to deal with a class of convex variational problems involving nonlinear constraints. A large family of constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For such constraints, the associated projection operator generally does not have a simple form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed half-space constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set. In this paper, we focus on a family of constraints involving linear transforms of distance functions to a convex set or $\ell_{1,p}$ norms with $p\in \{1,2,\infty\}$. In these cases, the projection onto the epigraph of the involved function has a closed form expression. The proposed approach is validated in the context of image restoration with missing samples, by making use of constraints based on Non-Local Total Variation. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems. A second application to a pulse shape design problem is provided in order to illustrate the flexibility of the proposed approach.

A fruitful approach for solving signal deconvolution problems consists of resorting to a frame-based convex variational formulation. In this context, parallel proximal algorithms and related alternating direction methods of multipliers have become popular optimization techniques to approximate iteratively the desired solution. Until now, in most of these methods, either Lipschitz differentiability properties or tight frame representations were assumed. In this paper, it is shown that it is possible to relax these assumptions by considering a class of non necessarily tight frame representations, thus offering the possibility of addressing a broader class of signal restoration problems. In particular, it is possible to use non necessarily maximally decimated filter banks with perfect reconstruction, which are common tools in digital signal processing. The proposed approach allows us to solve both frame analysis and frame synthesis problems for various noise distributions. In our simulations, it is applied to the deconvolution of data corrupted with Poisson noise or Laplacian noise by using (non-tight) discrete dual-tree wavelet representations and filter bank structures.

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, which is how to find a good regularizer. While total variation introduces staircase effects, wavelet domain regularization brings other artefacts, e.g. ringing. However, a compromise can be found by introducing a hybrid regularization including several terms non necessarily acting in the same domain (e.g. spatial and wavelet transform domains). We adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel ProXimal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behavior of the algorithm as well as promising results concerning the use of hybrid regularization techniques.

The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be non-smooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.