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

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Image Reconstruction in Optical Interferometry

This tutorial paper describes the problem of image reconstruction from interferometric data with a particular focus on the specific problems encountered at optical (visible/IR) wavelengths. The challenging issues in image reconstruction from interferometric data are introduced in the general framework of inverse problem approach. This framework is then used to describe existing image reconstruction algorithms in radio interferometry and the new methods specifically developed for optical interferometry.Comment: accepted for publication in IEEE Signal Processing Magazin

A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured "noises". As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data

Extended object reconstruction in adaptive-optics imaging: the multiresolution approach

We propose the application of multiresolution transforms, such as wavelets (WT) and curvelets (CT), to the reconstruction of images of extended objects that have been acquired with adaptive optics (AO) systems. Such multichannel approaches normally make use of probabilistic tools in order to distinguish significant structures from noise and reconstruction residuals. Furthermore, we aim to check the historical assumption that image-reconstruction algorithms using static PSFs are not suitable for AO imaging. We convolve an image of Saturn taken with the Hubble Space Telescope (HST) with AO PSFs from the 5-m Hale telescope at the Palomar Observatory and add both shot and readout noise. Subsequently, we apply different approaches to the blurred and noisy data in order to recover the original object. The approaches include multi-frame blind deconvolution (with the algorithm IDAC), myopic deconvolution with regularization (with MISTRAL) and wavelets- or curvelets-based static PSF deconvolution (AWMLE and ACMLE algorithms). We used the mean squared error (MSE) and the structural similarity index (SSIM) to compare the results. We discuss the strengths and weaknesses of the two metrics. We found that CT produces better results than WT, as measured in terms of MSE and SSIM. Multichannel deconvolution with a static PSF produces results which are generally better than the results obtained with the myopic/blind approaches (for the images we tested) thus showing that the ability of a method to suppress the noise and to track the underlying iterative process is just as critical as the capability of the myopic/blind approaches to update the PSF.Comment: In revision in Astronomy & Astrophysics. 19 pages, 13 figure