Recovery from Linear Measurements with Complexity-Matching Universal Signal Estimation

We study the compressed sensing (CS) signal estimation problem where an input signal is measured via a linear matrix multiplication under additive noise. While this setup usually assumes sparsity or compressibility in the input signal during recovery, the signal structure that can be leveraged is often not known a priori. In this paper, we consider universal CS recovery, where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself. Inspired by Kolmogorov complexity and minimum description length, we focus on a maximum a posteriori (MAP) estimation framework that leverages universal priors to match the complexity of the source. Our framework can also be applied to general linear inverse problems where more measurements than in CS might be needed. We provide theoretical results that support the algorithmic feasibility of universal MAP estimation using a Markov chain Monte Carlo implementation, which is computationally challenging. We incorporate some techniques to accelerate the algorithm while providing comparable and in many cases better reconstruction quality than existing algorithms. Experimental results show the promise of universality in CS, particularly for low-complexity sources that do not exhibit standard sparsity or compressibility.Comment: 29 pages, 8 figure

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

A Tutorial on Speckle Reduction in Synthetic Aperture Radar Images

Speckle is a granular disturbance, usually modeled as a multiplicative noise, that affects synthetic aperture radar (SAR) images, as well as all coherent images. Over the last three decades, several methods have been proposed for the reduction of speckle, or despeckling, in SAR images. Goal of this paper is making a comprehensive review of despeckling methods since their birth, over thirty years ago, highlighting trends and changing approaches over years. The concept of fully developed speckle is explained. Drawbacks of homomorphic filtering are pointed out. Assets of multiresolution despeckling, as opposite to spatial-domain despeckling, are highlighted. Also advantages of undecimated, or stationary, wavelet transforms over decimated ones are discussed. Bayesian estimators and probability density function (pdf) models in both spatial and multiresolution domains are reviewed. Scale-space varying pdf models, as opposite to scale varying models, are promoted. Promising methods following non-Bayesian approaches, like nonlocal (NL) filtering and total variation (TV) regularization, are reviewed and compared to spatial- and wavelet-domain Bayesian filters. Both established and new trends for assessment of despeckling are presented. A few experiments on simulated data and real COSMO-SkyMed SAR images highlight, on one side the costperformance tradeoff of the different methods, on the other side the effectiveness of solutions purposely designed for SAR heterogeneity and not fully developed speckle. Eventually, upcoming methods based on new concepts of signal processing, like compressive sensing, are foreseen as a new generation of despeckling, after spatial-domain and multiresolution-domain method

Bayesian plug & play methods for inverse problems in imaging.

Thèse de Doctorat de Mathématiques Appliquées (Université de Paris)Tesis de Doctorado en Ingeniería Eléctrica (Universidad de la República)This thesis deals with Bayesian methods for solving ill-posed inverse problems in imaging with learnt image priors. The first part of this thesis (Chapter 3) concentrates on two particular problems, namely joint denoising and decompression and multi-image super-resolution. After an extensive study of the noise statistics for these problem in the transformed (wavelet or Fourier) domain, we derive two novel algorithms to solve this particular inverse problem. One of them is based on a multi-scale self-similarity prior and can be seen as a transform-domain generalization of the celebrated non-local bayes algorithm to the case of non-Gaussian noise. The second one uses a neural-network denoiser to implicitly encode the image prior, and a splitting scheme to incorporate this prior into an optimization algorithm to find a MAP-like estimator. The second part of this thesis concentrates on the Variational AutoEncoder (VAE) model and some of its variants that show its capabilities to explicitly capture the probability distribution of high-dimensional datasets such as images. Based on these VAE models, we propose two ways to incorporate them as priors for general inverse problems in imaging : • The first one (Chapter 4) computes a joint (space-latent) MAP estimator named Joint Posterior Maximization using an Autoencoding Prior (JPMAP). We show theoretical and experimental evidence that the proposed objective function satisfies a weak bi-convexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other non-convex MAP approaches which more often get stuck in spurious local optima. • The second one (Chapter 5) develops a Gibbs-like posterior sampling algorithm for the exploration of posterior distributions of inverse problems using multiple chains and a VAE as image prior. We showhowto use those samples to obtain MMSE estimates and their corresponding uncertainty.Cette thèse traite des méthodes bayésiennes pour résoudre des problèmes inverses mal posés en imagerie avec des distributions a priori d’images apprises. La première partie de cette thèse (Chapitre 3) se concentre sur deux problèmes partic-uliers, à savoir le débruitage et la décompression conjoints et la super-résolutionmulti-images. Après une étude approfondie des statistiques de bruit pour ces problèmes dans le domaine transformé (ondelettes ou Fourier), nous dérivons deuxnouveaux algorithmes pour résoudre ce problème inverse particulie. L’un d’euxest basé sur une distributions a priori d’auto-similarité multi-échelle et peut êtrevu comme une généralisation du célèbre algorithme de Non-Local Bayes au cas dubruit non gaussien. Le second utilise un débruiteur de réseau de neurones pourcoder implicitement la distribution a priori, et un schéma de division pour incor-porer cet distribution dans un algorithme d’optimisation pour trouver un estima-teur de type MAP. La deuxième partie de cette thèse se concentre sur le modèle Variational Auto Encoder (VAE) et certaines de ses variantes qui montrent ses capacités à capturer explicitement la distribution de probabilité d’ensembles de données de grande dimension tels que les images. Sur la base de ces modèles VAE, nous proposons deuxmanières de les incorporer comme distribution a priori pour les problèmes inverses généraux en imagerie: •Le premier (Chapitre 4) calcule un estimateur MAP conjoint (espace-latent) nommé Joint Posterior Maximization using an Autoencoding Prior (JPMAP). Nous montrons des preuves théoriques et expérimentales que la fonction objectif proposée satisfait une propriété de bi-convexité faible qui est suffisante pour garantir que notre schéma d’optimisation converge vers un pointstationnaire. Les résultats expérimentaux montrent également la meilleurequalité des solutions obtenues par notre approche JPMAP par rapport à d’autresapproches MAP non convexes qui restent le plus souvent bloquées dans desminima locaux. •Le second (Chapitre 5) développe un algorithme d’échantillonnage a poste-riori de type Gibbs pour l’exploration des distributions a posteriori de problèmes inverses utilisant des chaînes multiples et un VAE comme distribution a priori. Nous montrons comment utiliser ces échantillons pour obtenir desestimations MMSE et leur incertitude correspondante.En esta tesis se estudian métodos bayesianos para resolver problemas inversos mal condicionados en imágenes usando distribuciones a priori entrenadas. La primera parte de esta tesis (Capítulo 3) se concentra en dos problemas particulares, a saber, el de eliminación de ruido y descompresión conjuntos, y el de superresolución a partir de múltiples imágenes. Después de un extenso estudio de las estadísticas del ruido para estos problemas en el dominio transformado (wavelet o Fourier),derivamos dos algoritmos nuevos para resolver este problema inverso en particular. Uno de ellos se basa en una distribución a priori de autosimilitud multiescala y puede verse como una generalización al dominio wavelet del célebre algoritmo Non-Local Bayes para el caso de ruido no Gaussiano. El segundo utiliza un algoritmo de eliminación de ruido basado en una red neuronal para codificar implícitamente la distribución a priori de las imágenes y un esquema de relajación para incorporar esta distribución en un algoritmo de optimización y así encontrar un estimador similar al MAP. La segunda parte de esta tesis se concentra en el modelo Variational AutoEncoder (VAE) y algunas de sus variantes que han mostrado capacidad para capturar explícitamente la distribución de probabilidad de conjuntos de datos en alta dimensión como las imágenes. Basándonos en estos modelos VAE, proponemos dos formas de incorporarlos como distribución a priori para problemas inversos genéricos en imágenes : •El primero (Capítulo 4) calcula un estimador MAP conjunto (espacio imagen y latente) llamado Joint Posterior Maximization using an Autoencoding Prior (JPMAP). Mostramos evidencia teórica y experimental de que la función objetivo propuesta satisface una propiedad de biconvexidad débil que es suficiente para garantizar que nuestro esquema de optimización converge a un punto estacionario. Los resultados experimentales también muestran la mayor calidad de las soluciones obtenidas por nuestro enfoque JPMAP con respecto a otros enfoques MAP no convexos que a menudo se atascan en mínimos locales espurios. •El segundo (Capítulo 5) desarrolla un algoritmo de muestreo tipo Gibbs parala exploración de la distribución a posteriori de problemas inversos utilizando múltiples cadenas y un VAE como distribución a priori. Mostramos cómo usar esas muestras para obtener estimaciones de MMSE y su correspondiente incertidumbr

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Review of the mathematical foundations of data fusion techniques in surface metrology

The recent proliferation of engineered surfaces, including freeform and structured surfaces, is challenging current metrology techniques. Measurement using multiple sensors has been proposed to achieve enhanced benefits, mainly in terms of spatial frequency bandwidth, which a single sensor cannot provide. When using data from different sensors, a process of data fusion is required and there is much active research in this area. In this paper, current data fusion methods and applications are reviewed, with a focus on the mathematical foundations of the subject. Common research questions in the fusion of surface metrology data are raised and potential fusion algorithms are discussed