Temporal evolution of the Covid19 pandemic reproduction number: Estimations from proximal optimization to Monte Carlo sampling.

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 interval-based 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.

Spatial and temporal regularization to estimate COVID-19 reproduction number R(t): Promoting piecewise smoothness via convex optimization.

Among the different indicators that quantify the spread of an epidemic such as the on-going COVID-19, stands first the reproduction number which measures how many people can be contaminated by an infected person. In order to permit the monitoring of the evolution of this number, a new estimation procedure is proposed here, assuming a well-accepted model for current incidence data, based on past observations. The novelty of the proposed approach is twofold: 1) the estimation of the reproduction number is achieved by convex optimization within a proximal-based inverse problem formulation, with constraints aimed at promoting piecewise smoothness; 2) the approach is developed in a multivariate setting, allowing for the simultaneous handling of multiple time series attached to different geographical regions, together with a spatial (graph-based) regularization of their evolutions in time. The effectiveness of the approach is first supported by simulations, and two main applications to real COVID-19 data are then discussed. The first one refers to the comparative evolution of the reproduction number for a number of countries, while the second one focuses on French departments and their joint analysis, leading to dynamic maps revealing the temporal co-evolution of their reproduction numbers.

Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model.

The Mumford-Shah model is a standard model in image segmentation, and due to its difficulty, many approximations have been proposed. The major interest of this functional is to enable joint image restoration and contour detection. In this work, we propose a general formulation of the discrete counterpart of the Mumford-Shah functional, adapted to nonsmooth penalizations, fitting the assumptions required by the Proximal Alternating Linearized Minimization (PALM), with convergence guarantees. A second contribution aims to relax some assumptions on the involved functionals and derive a novel Semi-Linearized Proximal Alternated Minimization (SL-PAM) algorithm, with proved convergence. We compare the performances of the algorithm with several nonsmooth penalizations, for Gaussian and Poisson denoising, image restoration and RGB-color denoising. We compare the results with state-of-the-art convex relaxations of the Mumford-Shah functional, and a discrete version of the Ambrosio-Tortorelli functional. We show that the SL-PAM algorithm is faster than the original PALM algorithm, and leads to competitive denoising, restoration and segmentation results.

Nonlinear denoising for characterization of solid friction under low confinement pressure.

The present work investigates paper-paper friction dynamics by pulling a slider over a substrate. It focuses on the transition between stick-slip and inertial regimes. Although the device is classical, probing solid friction with the fewest contact damage requires that the applied load should be small. This induces noise, mostly impulsive in nature, on the recorded slider motion and force signals. To address the challenging issue of describing the physics of such systems, we promote here the use of nonlinear filtering techniques relying on recent nonsmooth optimization schemes. In contrast to linear filtering, nonlinear filtering captures the slider velocity asymmetry and, thus, the creep motion before sliding. Precise estimates of the stick and slip phase durations can thus be obtained. The transition between the stick-slip and inertial regimes is continuous. Here we propose a criterion based on the probability of the system to be in the stick-slip regime to quantify this transition. A phase diagram is obtained that characterizes the dynamics of this frictional system under low confinement pressure.

Nonsmooth Convex Optimization for Structured Illumination Microscopy Image Reconstruction.

In this paper, we propose a new approach for structured illumination microscopy image reconstruction. We first introduce the principles of this imaging modality and describe the forward model. We then propose the minimization of nonsmooth convex objective functions for the recovery of the unknown image. In this context, we investigate two data-fitting terms for Poisson-Gaussian noise and introduce a new patch-based regularization method. This approach is tested against other regularization approaches on a realistic benchmark. Finally, we perform some test experiments on images acquired on two different microscopes.

Sparse Support Vector Machine for Intrapartum Fetal Heart Rate Classification.

Fetal heart rate (FHR) monitoring is routinely used in clinical practice to help obstetricians assess fetal health status during delivery. However, early detection of fetal acidosis that allows relevant decisions for operative delivery remains a challenging task, receiving considerable attention. This contribution promotes sparse support vector machine classification that permits to select a small number of relevant features and to achieve efficient fetal acidosis detection. A comprehensive set of features is used for FHR description, including enhanced and computerized clinical features, frequency domain, and scaling and multifractal features, all computed on a large (1288 subjects) and well-documented database. The individual performance obtained for each feature independently is discussed first. Then, it is shown that the automatic selection of a sparse subset of features achieves satisfactory classification performance (sensitivity 0.73 and specificity 0.75, outperforming clinical practice). The subset of selected features (average depth of decelerations MADdtrd, baseline level ß0 , and variability H) receives simple interpretation in clinical practice. Intrapartum fetal acidosis detection is improved in several respects: A comprehensive set of features combining clinical, spectral, and scale-free dynamics is used; an original multivariate classification targeting both sparse feature selection and high performance is devised; state-of-the-art performance is obtained on a much larger database than that generally studied with description of common pitfalls in supervised classification performance assessments.

A nonlocal structure tensor-based approach for multicomponent image recovery problems.

Nonlocal 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 nonlocal variations, jointly for the different components, through various l(1, p)-matrix-norms with p ≥ 1. To facilitate the choice of the hyperparameters, 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 because of the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for color, multispectral, and hyperspectral images. The results demonstrate the interest of introducing a nonlocal ST regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods, such as the alternating direction method of multipliers.

2D Prony-Huang transform: a new tool for 2D spectral analysis.

This paper provides an extension of the 1D Hilbert Huang transform for the analysis of images using recent optimization techniques. The proposed method consists of: 1) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure and 2) 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 2D mode decompositions based on nonsmooth convex optimization: 1) a genuine 2D approach, which constrains the local extrema of the IMFs and 2) a pseudo-2D approach, which separately constrains the extrema of lines, columns, and diagonals. The spectral analysis step is an optimization strategy based on Prony annihilation property and applied on small square patches of the IMFs. The resulting 2D Prony­Huang transform is validated on simulated and real data.

Parallel proximal algorithm for image restoration using hybrid regularization.

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet-domain regularization brings other artefacts, e.g., ringing. However, a tradeoff can be made by introducing a hybrid regularization including several terms not necessarily acting in the same domain (e.g., spatial and wavelet transform domains). While this approach was shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, 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.