arXiv subjects

Audrey Repetti

Publications and source records attributed to Audrey Repetti.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Plug-and-Play method for Dynamic Imaging Via Kalman Smoothing

State-space models (SSM) are common in signal processing, where Kalman smoothing (KS) methods are state-of-the-art. However, traditional KS techniques lack expressivity as they do not incorporate spatial prior information. Recently, [1] proposed an ADMM algorithm that handles the state-space fidelity term with KS while regularizing the object via a sparsity-based prior with proximity operators. Plug-and-Play (PnP) methods are a popular type of iterative algorithms that replace proximal operators encoding prior knowledge with powerful denoisers such as deep neural networks. These methods are widely used in image processing, achieving state-of-the-art results. In this work, we build on the KS-ADMM method, combining it with deep learning to achieve higher expressivity. We propose a PnP algorithm based on KS-ADMM iterations, efficiently handling the SSM through KS, while enabling the use of powerful denoising networks. Simulations on a 2D+t imaging problem show that the proposed PnP-KS-ADMM algorithm improves the computational efficiency over standard PnP-ADMM for large numbers of timesteps.

eess.IV↗

arXiv2025

A Distributed Plug-and-Play MCMC Algorithm for High-Dimensional Inverse Problems

Markov Chain Monte Carlo (MCMC) algorithms are standard approaches to solve imaging inverse problems and quantify estimation uncertainties, a key requirement in absence of ground-truth data. To improve estimation quality, Plug-and-Play MCMC algorithms, such as PnP-ULA, have been recently developed to accommodate priors encoded by a denoising neural network. Designing scalable samplers for high-dimensional imaging inverse problems remains a challenge: drawing and storing high-dimensional samples can be prohibitive, especially for high-resolution images. To address this issue, this work proposes a distributed sampler based on approximate data augmentation and PnP-ULA to solve very large problems. The proposed sampler uses lightweight denoising convolutional neural network, to efficiently exploit multiple GPUs on a Single Program Multiple Data architecture. Reconstruction performance and scalability are evaluated on several imaging problems. Communication and computation overheads due to the denoiser are carefully discussed. The proposed distributed approach noticeably combines three very precious qualities: it is scalable, enables uncertainty quantification, for a reconstruction performance comparable to other PnP methods.

stat.ME↗

arXiv2025

A Unified Framework for Lifted Training and Inversion Approaches

The training of deep neural networks predominantly relies on a combination of gradient-based optimisation and back-propagation for the computation of the gradient. While incredibly successful, this approach faces challenges such as vanishing or exploding gradients, difficulties with non-smooth activations, and an inherently sequential structure that limits parallelisation. Lifted training methods offer an alternative by reformulating the nested optimisation problem into a higher-dimensional, constrained optimisation problem where the constraints are no longer enforced directly but penalised with penalty terms. This chapter introduces a unified framework that encapsulates various lifted training strategies, including the Method of Auxiliary Coordinates, Fenchel Lifted Networks, and Lifted Bregman Training, and demonstrates how diverse architectures, such as Multi-Layer Perceptrons, Residual Neural Networks, and Proximal Neural Networks fit within this structure. By leveraging tools from convex optimisation, particularly Bregman distances, the framework facilitates distributed optimisation, accommodates non-differentiable proximal activations, and can improve the conditioning of the training landscape. We discuss the implementation of these methods using block-coordinate descent strategies, including deterministic implementations enhanced by accelerated and adaptive optimisation techniques, as well as implicit stochastic gradient methods. Furthermore, we explore the application of this framework to inverse problems, detailing methodologies for both the training of specialised networks (e.g., unrolled architectures) and the stable inversion of pre-trained networks. Numerical results on standard imaging tasks validate the effectiveness and stability of the lifted Bregman approach compared to conventional training, particularly for architectures employing proximal activations.

cs.LG↗

arXiv2024

Africanus III. pfb-imaging -- a flexible radio interferometric imaging suite

The popularity of the CLEAN algorithm in radio interferometric imaging stems from its maturity, speed, and robustness. While many alternatives have been proposed in the literature, none have achieved mainstream adoption by astronomers working with data from interferometric arrays operating in the big data regime. This lack of adoption is largely due to increased computational complexity, absence of mature implementations, and the need for astronomers to tune obscure algorithmic parameters. This work introduces pfb-imaging: a flexible library that implements the scaffolding required to develop and accelerate general radio interferometric imaging algorithms. We demonstrate how the framework can be used to implement a sparsity-based image reconstruction technique known as (unconstrained) SARA in a way that scales with image size rather than data volume and features interpretable algorithmic parameters. The implementation is validated on terabyte-sized data from the MeerKAT telescope, using both a single compute node and Amazon Web Services computing instances.

astro-ph.IM↗

arXiv2024

Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms

In this work we study the behavior of the forward-backward (FB) algorithm when the proximity operator is replaced by a sub-iterative procedure to approximate a Gaussian denoiser, in a Plug-and-Play (PnP) fashion. In particular, we consider both analysis and synthesis Gaussian denoisers within a dictionary framework, obtained by unrolling dual-FB iterations or FB iterations, respectively. We analyze the associated minimization problems as well as the asymptotic behavior of the resulting FB-PnP iterations. In particular, we show that the synthesis Gaussian denoising problem can be viewed as a proximity operator. For each case, analysis and synthesis, we show that the FB-PnP algorithms solve the same problem whether we use only one or an infinite number of sub-iteration to solve the denoising problem at each iteration. To this aim, we show that each "one sub-iteration" strategy within the FB-PnP can be interpreted as a primal-dual algorithm when a warm-restart strategy is used. We further present similar results when using a Moreau-Yosida smoothing of the global problem, for an arbitrary number of sub-iterations. Finally, we provide numerical simulations to illustrate our theoretical results. In particular we first consider a toy compressive sensing example, as well as an image restoration problem in a deep dictionary framework.

math.OC↗

arXiv2024

A lifted Bregman strategy for training unfolded proximal neural network Gaussian denoisers

Unfolded proximal neural networks (PNNs) form a family of methods that combines deep learning and proximal optimization approaches. They consist in designing a neural network for a specific task by unrolling a proximal algorithm for a fixed number of iterations, where linearities can be learned from prior training procedure. PNNs have shown to be more robust than traditional deep learning approaches while reaching at least as good performances, in particular in computational imaging. However, training PNNs still depends on the efficiency of available training algorithms. In this work, we propose a lifted training formulation based on Bregman distances for unfolded PNNs. Leveraging the deterministic mini-batch block-coordinate forward-backward method, we design a bespoke computational strategy beyond traditional back-propagation methods for solving the resulting learning problem efficiently. We assess the behaviour of the proposed training approach for PNNs through numerical simulations on image denoising, considering a denoising PNN whose structure is based on dual proximal-gradient iterations.

math.OC↗

arXiv2024

Learning truly monotone operators with applications to nonlinear inverse problems

This article introduces a novel approach to learning monotone neural networks through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone inclusion problems, commonly encountered in image processing tasks. The Forward-Backward-Forward (FBF) algorithm is employed to address these problems, offering a solution even when the Lipschitz constant of the neural network is unknown. Notably, the FBF algorithm provides convergence guarantees under the condition that the learned operator is monotone. Building on plug-and-play methodologies, our objective is to apply these newly learned operators to solving non-linear inverse problems. To achieve this, we initially formulate the problem as a variational inclusion problem. Subsequently, we train a monotone neural network to approximate an operator that may not inherently be monotone. Leveraging the FBF algorithm, we then show simulation examples where the non-linear inverse problem is successfully solved.

math.OC↗

arXiv2024

A multilevel framework for accelerating uSARA in radio-interferometric imaging

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.

math.OC↗

arXiv2023

Unfolded proximal neural networks for robust image Gaussian denoising

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.

cs.CV↗

arXiv2023

A new non-convex framework to improve asymptotical knowledge on generic stochastic gradient descent

Stochastic gradient optimization methods are broadly used to minimize non-convex smooth objective functions, for instance when training deep neural networks. However, theoretical guarantees on the asymptotic behaviour of these methods remain scarce. Especially, ensuring almost-sure convergence of the iterates to a stationary point is quite challenging. In this work, we introduce a new Kurdyka-Lojasiewicz theoretical framework to analyze asymptotic behavior of stochastic gradient descent (SGD) schemes when minimizing non-convex smooth objectives. In particular, our framework provides new almost-sure convergence results, on iterates generated by any SGD method satisfying mild conditional descent conditions. We illustrate the proposed framework by means of several toy simulation examples. We illustrate the role of the considered theoretical assumptions, and investigate how SGD iterates are impacted whether these assumptions are either fully or partially satisfied.

math.OC↗

arXiv2023

A primal-dual data-driven method for computational optical imaging with a photonic lantern

Optical fibres aim to image in-vivo biological processes. In this context, high spatial resolution and stability to fibre movements are key to enable decision-making processes (e.g., for microendoscopy). Recently, a single-pixel imaging technique based on a multicore fibre photonic lantern has been designed, named computational optical imaging using a lantern (COIL). A proximal algorithm based on a sparsity prior, dubbed SARA-COIL, has been further proposed to solve the associated inverse problem, to enable image reconstructions for high resolution COIL microendoscopy. In this work, we develop a data-driven approach for COIL. We replace the sparsity prior in the proximal algorithm by a learned denoiser, leading to a plug-and-play (PnP) algorithm. The resulting PnP method, based on a proximal primal-dual algorithm, enables to solve the Morozov formulation of the inverse problem. We use recent results in learning theory to train a network with desirable Lipschitz properties, and we show that the resulting primal-dual PnP algorithm converges to a solution to a monotone inclusion problem. Our simulations highlight that the proposed data-driven approach improves the reconstruction quality over variational SARA-COIL method on both simulated and real data.

eess.IV↗

arXiv2023

A Plug-and-Play Method with Inpainting Network for Bayesian Uncertainty Quantification in Imaging

We contribute to an uncertainty quantification problem in imaging that evaluates a hypothesis test questioning the existence of local "artefacts" appearing in the maximum a posteriori (MAP) estimate (obtained from standard numerical tools). Such a method, called Bayesian uncertainty quantification by optimization (BUQO), was introduced a few years ago as an efficient and scalable alternative to sampling methods when per-pixel error-bars are not needed. BUQO formulates a hypothesis test for probing the existence of local structures in the MAP estimate as a minimization problem, that can be solved efficiently with standard optimization algorithms. In this context, BUQO requires a "mathematical" definition of the "local artefact". This definition can be interpreted as an inpainting of the structure. However, only simple hand-crafted techniques have been proposed so far due to the complexity of the problem. In this work, we propose a data-driven alternative to BUQO where the inpainting procedure in the algorithm is performed using a convolutional inpainting neural network (NN). This results in a plug-and-play algorithm, based on the primal-dual Condat-Vu iterations,where the inpainting procedure is performed with a NN. The proposed approach is assessed on two image reconstruction problems inspired by medicine. We specifically perform simulations on two Fourier undersampling problems (discrete and non-uniform) encountered in magnetic resonance imaging, as well as a computed tomography problem using the Radon measurement operator.

stat.ME↗

arXiv2023

A stochastic use of the Kurdyka-Lojasiewicz property: Investigation of optimization algorithms behaviours in a non-convex differentiable framework

Stochastic differentiable approximation schemes are widely used for solving high dimensional problems. Most of existing methods satisfy some desirable properties, including conditional descent inequalities, and almost sure (a.s.) convergence guarantees on the objective function, or on the involved gradient. However, for non-convex objective functions, a.s. convergence of the iterates, i.e., the stochastic process, to a critical point is usually not guaranteed, and remains an important challenge. In this article, we develop a framework to bridge the gap between descent-type inequalities and a.s. convergence of the associated stochastic process. Leveraging a novel Kurdyka-Lojasiewicz property, we show convergence guarantees of stochastic processes under mild assumptions on the objective function. We also provide examples of stochastic algorithms benefiting from the proposed framework and derive a.s. convergence guarantees on the iterates.

math.OC↗

arXiv2023

Uncertainty Quantification in CT pulmonary angiography

Computed tomography (CT) imaging of the thorax is widely used for the detection and monitoring of pulmonary embolism (PE). However, CT images can contain artifacts due to the acquisition or the processes involved in image reconstruction. Radiologists often have to distinguish between such artifacts and actual PEs. Our main contribution comes in the form of a scalable hypothesis testing method for CT, to enable quantifying uncertainty of possible PEs. In particular, we introduce a Bayesian Framework to quantify the uncertainty of an observed compact structure that can be identified as a PE. We assess the ability of the method to operate under high noise environments and with insufficient data.

stat.AP↗

arXiv2022

A Variational Inequality Model for Learning Neural Networks

Neural networks have become ubiquitous tools for solving signal and image processing problems, and they often outperform standard approaches. Nevertheless, training neural networks is a challenging task in many applications. The prevalent training procedure consists of minimizing highly non-convex objectives based on data sets of huge dimension. In this context, current methodologies are not guaranteed to produce global solutions. We present an alternative approach which foregoes the optimization framework and adopts a variational inequality formalism. The associated algorithm guarantees convergence of the iterates to a true solution of the variational inequality and it possesses an efficient block-iterative structure. A numerical application is presented.

math.OC↗

arXiv2022

A Distributed Block-Split Gibbs Sampler with Hypergraph Structure for High-Dimensional Inverse Problems

Sampling-based algorithms are classical approaches to perform Bayesian inference in inverse problems. They provide estimators with the associated credibility intervals to quantify the uncertainty on the estimators. Although these methods hardly scale to high dimensional problems, they have recently been paired with optimization techniques, such as proximal and splitting approaches, to address this issue. Such approaches pave the way to distributed samplers, splitting computations to make inference more scalable and faster. We introduce a distributed Split Gibbs sampler (SGS) to efficiently solve such problems involving distributions with multiple smooth and non-smooth functions composed with linear operators. The proposed approach leverages a recent approximate augmentation technique reminiscent of primal-dual optimization methods. It is further combined with a block-coordinate approach to split the primal and dual variables into blocks, leading to a distributed block-coordinate SGS. The resulting algorithm exploits the hypergraph structure of the involved linear operators to efficiently distribute the variables over multiple workers under controlled communication costs. It accommodates several distributed architectures, such as the Single Program Multiple Data and client-server architectures. Experiments on a large image deblurring problem show the performance of the proposed approach to produce high quality estimates with credibility intervals in a small amount of time. Supplementary material to reproduce the experiments is available online.

stat.ME↗

arXiv2021

Cygnus A jointly calibrated and imaged via non-convex optimisation from VLA data

Radio interferometric (RI) data are noisy under-sampled spatial Fourier components of the unknown radio sky affected by direction-dependent antenna gains. Failure to model these antenna gains accurately results in a radio sky estimate with limited fidelity and resolution. The RI inverse problem has been recently addressed via a joint calibration and imaging approach which consists in solving a non-convex minimisation task, involving suitable priors for the DDEs, namely temporal and spatial smoothness, and sparsity for the unknown radio map via an $\ell_1$-norm prior, in the context of realistic RI simulations. Building on these developments, we propose to promote sparsity of the radio map via a log-sum prior, enforcing sparsity more strongly than the $\ell_1$-norm. The resulting minimisation task is addressed via a sequence of non-convex minimisation tasks composed of re-weighted $\ell_1$ image priors, which are solved approximately. We demonstrate the efficiency of the approach on RI observations of the celebrated radio galaxy Cygnus~A obtained with the Karl G. Jansky Very Large Array at X, C, and S bands. More precisely, we showcase that the approach enhances data fidelity significantly while achieving high resolution high dynamic range radio maps, confirming the suitability of the priors considered for the unknown DDEs and radio image. As a clear qualitative indication of the high fidelity achieved by the data and the proposed approach, we report the detection of three background sources in the vicinity of Cyg~A, at S band.

astro-ph.IM↗

arXiv2021

A practical preconditioner for wide-field continuum imaging of radio interferometric data

The celebrated CLEAN algorithm has been the cornerstone of deconvolution algorithms in radio interferometry almost since its conception in the 1970s. For all its faults, CLEAN is remarkably fast, robust to calibration artefacts and in its ability to model point sources. We demonstrate how the same assumptions that afford CLEAN its speed can be used to accelerate more sophisticated deconvolution algorithms.

astro-ph.IM↗