arXiv subjects

Nicolas Papadakis

Publications and source records attributed to Nicolas Papadakis.

At least 19 recordsLinked to original sources

arXiv2026

Gradient Step Plug-and-Play Model for Dental Cone-Beam CT Reconstruction

The goal of this work is to reduce the effect of photon noise in dental cone-beam CT reconstruction. We consider an inverse problem formulation and develop a databased prior. To this end, we simulate fan-beam acquisitions and add photon noise to the projection data. The prior is obtained by training a gradient-step denoiser using reconstructed simulated acquisitions. The trained model is integrated into a plug-and-play gradient-step algorithm to reconstruct images from simulated projections. Experiments on synthetic data demonstrate the denoising capabilities of the trained model, while qualitative evaluations on real images showcase the algorithm's performance and generalization ability.

cs.AI↗

arXiv2026

i-DEQ: A stable inertial deep equilibrium model for image restoration

Deep Equilibrium Models (DEQs) are an established framework for image restoration that learn a problem-adapted regularization by solving a fixed-point (i.e. equilibrium) problem. While flexible and expressive, DEQs are often hindered by high computational cost and training instability. We propose an inertial DEQ (i-DEQ) that learns an explicit nonconvex regularization within the DEQ formulation. By using momentum within the fixed-point iterations, i-DEQ has convergence guarantees and accelerated rates. Moreover, we observe that i-DEQ is significantly more stable during the training and robust to rough initialization than DEQs. Numerical experiments on various linear and nonlinear inverse problems demonstrate that i-DEQ achieves reconstruction quality comparable to state-of-the-art methods, while reducing DEQ's inference time by a factor of two.

math.OC↗

arXiv2026

On the Convergence of Proximal Algorithms for Weakly-convex Min-max Optimization

We study alternating first-order algorithms with no inner loops for solving nonconvex-strongly-concave min-max problems. We show the convergence of the alternating gradient descent--ascent algorithm method by proposing a substantially simplified proof compared to previous ones. It allows us to enlarge the set of admissible step-sizes. Building on this general reformulation, we also prove the convergence of a doubly proximal algorithm in the weakly convex-strongly concave setting. Finally, we show how this new result opens the way to new applications of min-max optimization algorithms for solving regularized imaging inverse problems with neural networks in a plug-and-play manner.

math.OC↗

arXiv2026

Reinterpreting EMML as Mirror Descent for Constrained Maximum Likelihood Estimation

The Expectation--Maximization Maximum Likelihood (EMML) algorithm belongs to the Expectation--Maximization family and is widely used for image reconstruction problems under Poisson noise. In this paper, we reinterpret EMML as a mirror descent method applied to a reparametrized objective function. This perspective allows us to incorporate convex constraints into the algorithm through appropriately chosen Bregman projections, while preserving the multiplicative structure of the EMML updates to ensure computational efficiency. We then establish the convergence of the resulting algorithm toward a solution of the constrained maximum-likelihood problem. Numerical experiments on hyperspectral unmixing problems demonstrate that the constrained EMML converges in fewer iterations than the classical EMML.

math.OC↗

arXiv2025

Equivariant Denoisers for Plug and Play Image Restoration

One key ingredient of image restoration is to define a realistic prior on clean images to complete the missing information in the observation. State-of-the-art restoration methods rely on a neural network to encode this prior. Typical image distributions are invariant to some set of transformations, such as rotations or flips. However, most deep architectures are not designed to represent an invariant image distribution. Recent works have proposed to overcome this difficulty by including equivariance properties within a Plug-and-Play paradigm. In this work, we propose two unified frameworks named Equivariant Regularization by Denoising (ERED) and Equivariant Plug-and-Play (EPnP) based on equivariant denoisers and stochastic optimization. We analyze the convergence of the proposed algorithms and discuss their practical benefit.

eess.IV↗

arXiv2025

On the Moreau envelope properties of weakly convex functions

In this document, we present the main properties satisfied by the Moreau envelope of weakly convex functions. The Moreau envelope has been introduced in convex optimization to regularize convex functionals while preserving their global minimizers. However, the Moreau envelope is also defined for the more general class of weakly convex function and can be a useful tool for optimization in this context. The main properties of the Moreau envelope have been demonstrated for convex functions and are generalized to weakly convex function in various works. This document summarizes the vast literature on the properties of the Moreau envelope and provides the associated proofs.

math.OC↗

arXiv2025

From the Gradient-Step Denoiser to the Proximal Denoiser and their associated convergent Plug-and-Play algorithms

In this paper we analyze the Gradient-Step Denoiser and its usage in Plug-and-Play algorithms. The Plug-and-Play paradigm of optimization algorithms uses off the shelf denoisers to replace a proximity operator or a gradient descent operator of an image prior. Usually this image prior is implicit and cannot be expressed, but the Gradient-Step Denoiser is trained to be exactly the gradient descent operator or the proximity operator of an explicit functional while preserving state-of-the-art denoising capabilities.

cs.LG↗

arXiv2025

From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling

We consider the problem of sampling distributions stemming from non-convex potentials with Unadjusted Langevin Algorithm (ULA). We prove the stability of the discrete-time ULA to drift approximations under the assumption that the potential is strongly convex at infinity. In many context, e.g. imaging inverse problems, potentials are non-convex and non-smooth. Proximal Stochastic Gradient Langevin Algorithm (PSGLA) is a popular algorithm to handle such potentials. It combines the forward-backward optimization algorithm with a ULA step. Our main stability result combined with properties of the Moreau envelope allows us to derive the first proof of convergence of the PSGLA for non-convex potentials. We empirically validate our methodology on synthetic data and in the context of imaging inverse problems. In particular, we observe that PSGLA exhibits faster convergence rates than Stochastic Gradient Langevin Algorithm for posterior sampling while preserving its restoration properties.

stat.ML↗

arXiv2025

LATINO-PRO: LAtent consisTency INverse sOlver with PRompt Optimization

Text-to-image latent diffusion models (LDMs) have recently emerged as powerful generative models with great potential for solving inverse problems in imaging. However, leveraging such models in a Plug & Play (PnP), zero-shot manner remains challenging because it requires identifying a suitable text prompt for the unknown image of interest. Also, existing text-to-image PnP approaches are highly computationally expensive. We herein address these challenges by proposing a novel PnP inference paradigm specifically designed for embedding generative models within stochastic inverse solvers, with special attention to Latent Consistency Models (LCMs), which distill LDMs into fast generators. We leverage our framework to propose LAtent consisTency INverse sOlver (LATINO), the first zero-shot PnP framework to solve inverse problems with priors encoded by LCMs. Our conditioning mechanism avoids automatic differentiation and reaches SOTA quality in as little as 8 neural function evaluations. As a result, LATINO delivers remarkably accurate solutions and is significantly more memory and computationally efficient than previous approaches. We then embed LATINO within an empirical Bayesian framework that automatically calibrates the text prompt from the observed measurements by marginal maximum likelihood estimation. Extensive experiments show that prompt self-calibration greatly improves estimation, allowing LATINO with PRompt Optimization to define new SOTAs in image reconstruction quality and computational efficiency. The code is available at https://latino-pro.github.io

cs.CV↗

arXiv2024

Convergence Analysis of a Proximal Stochastic Denoising Regularization Algorithm

Plug-and-Play methods for image restoration are iterative algorithms that solve a variational problem to recover a clean image from a degraded observation. These algorithms are known to be flexible to changes of degradation and to perform state-of-the-art restoration. Recently, significant efforts have been made to explore new stochastic algorithms based on the Plug-and-Play or REgularization by Denoising (RED) frameworks, such as SNORE, which is a convergent stochastic gradient descent algorithm. A variant of this algorithm, named SNORE Prox, reaches state-of-the-art performances, especially for inpainting tasks. However, the convergence of SNORE Prox, that can be seen as a stochastic proximal gradient descent, has not been analyzed so far. In this paper, we prove the convergence of SNORE Prox under non convex assumptions.

eess.IV↗

arXiv2024

Equivariant Denoisers for Image Restoration

One key ingredient of image restoration is to define a realistic prior on clean images to complete the missing information in the observation. State-of-the-art restoration methods rely on a neural network to encode this prior. Moreover, typical image distributions are invariant to some set of transformations, such as rotations or flips. However, most deep architectures are not designed to represent an invariant image distribution. Recent works have proposed to overcome this difficulty by including equivariance properties within a Plug-and-Play paradigm. In this work, we propose a unified framework named Equivariant Regularization by Denoising (ERED) based on equivariant denoisers and stochastic optimization. We analyze the convergence of this algorithm and discuss its practical benefit.

eess.IV↗

arXiv2024

Optimization with First Order Algorithms

These notes focus on the minimization of convex functionals using first-order optimization methods, which are fundamental in many areas of applied mathematics and engineering. The primary goal of this document is to introduce and analyze the most classical first-order optimization algorithms. We aim to provide readers with both a practical and theoretical understanding in how and why these algorithms converge to minimizers of convex functions. The main algorithms covered in these notes include gradient descent, Forward-Backward splitting, Douglas-Rachford splitting, the Alternating Direction Method of Multipliers (ADMM), and Primal-Dual algorithms. All these algorithms fall into the class of first order methods, as they only involve gradients and subdifferentials, that are first order derivatives of the functions to optimize. For each method, we provide convergence theorems, with precise assumptions and conditions under which the convergence holds, accompanied by complete proofs. Beyond convex optimization, the final part of this manuscript extends the analysis to nonconvex problems, where we discuss the convergence behavior of these same first-order methods under broader assumptions. To contextualize the theory, we also include a selection of practical examples illustrating how these algorithms are applied in different image processing problems.

math.OC↗

arXiv2024

Plug-and-Play image restoration with Stochastic deNOising REgularization

Plug-and-Play (PnP) algorithms are a class of iterative algorithms that address image inverse problems by combining a physical model and a deep neural network for regularization. Even if they produce impressive image restoration results, these algorithms rely on a non-standard use of a denoiser on images that are less and less noisy along the iterations, which contrasts with recent algorithms based on Diffusion Models (DM), where the denoiser is applied only on re-noised images. We propose a new PnP framework, called Stochastic deNOising REgularization (SNORE), which applies the denoiser only on images with noise of the adequate level. It is based on an explicit stochastic regularization, which leads to a stochastic gradient descent algorithm to solve ill-posed inverse problems. A convergence analysis of this algorithm and its annealing extension is provided. Experimentally, we prove that SNORE is competitive with respect to state-of-the-art methods on deblurring and inpainting tasks, both quantitatively and qualitatively.

eess.IV↗

arXiv2023

Convergent plug-and-play with proximal denoiser and unconstrained regularization parameter

In this work, we present new proofs of convergence for Plug-and-Play (PnP) algorithms. PnP methods are efficient iterative algorithms for solving image inverse problems where regularization is performed by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD) or Douglas-Rachford Splitting (DRS). Recent research has explored convergence by incorporating a denoiser that writes exactly as a proximal operator. However, the corresponding PnP algorithm has then to be run with stepsize equal to $1$. The stepsize condition for nonconvex convergence of the proximal algorithm in use then translates to restrictive conditions on the regularization parameter of the inverse problem. This can severely degrade the restoration capacity of the algorithm. In this paper, we present two remedies for this limitation. First, we provide a novel convergence proof for PnP-DRS that does not impose any restrictions on the regularization parameter. Second, we examine a relaxed version of the PGD algorithm that converges across a broader range of regularization parameters. Our experimental study, conducted on deblurring and super-resolution experiments, demonstrate that both of these solutions enhance the accuracy of image restoration.

math.OC↗

arXiv2023

Self2Seg: Single-Image Self-Supervised Joint Segmentation and Denoising

We develop Self2Seg, a self-supervised method for the joint segmentation and denoising of a single image. To this end, we combine the advantages of variational segmentation with self-supervised deep learning. One major benefit of our method lies in the fact, that in contrast to data-driven methods, where huge amounts of labeled samples are necessary, Self2Seg segments an image into meaningful regions without any training database. Moreover, we demonstrate that self-supervised denoising itself is significantly improved through the region-specific learning of Self2Seg. Therefore, we introduce a novel self-supervised energy functional in which denoising and segmentation are coupled in a way that both tasks benefit from each other. We propose a unified optimisation strategy and numerically show that for noisy microscopy images our proposed joint approach outperforms its sequential counterpart as well as alternative methods focused purely on denoising or segmentation.

cs.CV↗

arXiv2023

Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems

Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using theBregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance.

eess.IV↗

arXiv2023

A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser

This paper presents a new convergent Plug-and-Play (PnP) algorithm. PnP methods are efficient iterative algorithms for solving image inverse problems formulated as the minimization of the sum of a data-fidelity term and a regularization term. PnP methods perform regularization by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD). To ensure convergence of PnP schemes, many works study specific parametrizations of deep denoisers. However, existing results require either unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive conditions on the parameters of the inverse problem. Observing that these limitations can be due to the proximal algorithm in use, we study a relaxed version of the PGD algorithm for minimizing the sum of a convex function and a weakly convex one. When plugged with a relaxed proximal denoiser, we show that the proposed PnP-$α$PGD algorithm converges for a wider range of regularization parameters, thus allowing more accurate image restoration.

stat.ML↗

arXiv2023

SCOTCH and SODA: A Transformer Video Shadow Detection Framework

Shadows in videos are difficult to detect because of the large shadow deformation between frames. In this work, we argue that accounting for shadow deformation is essential when designing a video shadow detection method. To this end, we introduce the shadow deformation attention trajectory (SODA), a new type of video self-attention module, specially designed to handle the large shadow deformations in videos. Moreover, we present a new shadow contrastive learning mechanism (SCOTCH) which aims at guiding the network to learn a unified shadow representation from massive positive shadow pairs across different videos. We demonstrate empirically the effectiveness of our two contributions in an ablation study. Furthermore, we show that SCOTCH and SODA significantly outperforms existing techniques for video shadow detection. Code is available at the project page: https://lihaoliu-cambridge.github.io/scotch_and_soda/

cs.CV↗