Abstract-Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or wavelet-based regularization). It happens that the convergence rate of these IST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is ill-conditioned or ill-posed. In this paper, we introduce two-step IST (TwIST) algorithms, exhibiting much faster convergence rate than IST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers ( norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over IST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples.

Hyperspectral instruments acquire electromagnetic energy scattered within their ground instantaneous field view in hundreds of spectral channels with high spectral resolution. Very often, however, owing to low spatial resolution of the scanner or to the presence of intimate mixtures (mixing of the materials at a very small scale) in the scene, the spectral vectors (collection of signals acquired at different spectral bands from a given pixel) acquired by the hyperspectral scanners are actually mixtures of the spectral signatures of the materials present in the scene.Given a set of mixed spectral vectors, spectral mixture analysis (or spectral unmixing) aims at estimating the number of reference materials, also called endmembers, their spectral signatures, and their fractional abundances. Spectral unmixing is, thus, a source separation problem.This paper presents an overview of the principal research directions in hyperspectral unmixing. The paper is organized into six main topics: i) mixing models, ii) signal subspace identification, iii) geometrical-based spectral unmixing, iv) statistical-based spectral unmixing, v) sparse regressionbased unmixing, and vi) spatial-contextual information. For each topic, we summarize what is the mathematical problem involved and give relevant pointers to state-of-the-art algorithms to address these problems. MIXING MODELSSpectral unmixing is an important problem in hyperspectral data exploitation. Depending on the mixing scales at each pixel and on the geometry of the scene, the observed mixture is either linear or nonlinear [1], [2]. Linear mixing holds when the mixing scale is macroscopic and the incident light interacts with just one material, as it happens in checkerboard-type scenes [4]. Nonlinear mixing holds when the light suffers multiple scattering involving different materials [5].• In a linear mixing scenario, the acquired spectral vectors are a linear combination of the endmember signatures present in the scene, weighted by the respective fractional abundances. The exploitation of this model, in spite of its simplicity, has fostered a huge amount of research leading to a plethora of unmixing algorithms developed under the geometrical or the statistical frameworks.• In a nonlinear mixing scenario, the model for the scattered light is much more complex than its linear counterpart. Radiative transfer theory (RTT) is a well established model for the transfer of energy as photons interacts with the materials in the scene. The core of the RTT is a differential equation describing radiance collected by the sensor. It can be derived via the conservation of energy and the knowledge of the phase function, which represents the probability of light with a given propagation direction be scattered into a specified angle solid around a given scattering direction.In this work, we provide an overview of current trends and techniques for analyzing hyperspectral data using spectral unmixing. Although previous efforts exist in the literature [1], none of them have been specific...

Abstract-We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an ℓ2 data-fidelity term and a nonsmooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization or total-variation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence has been proved. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is faster than the current state of the art methods.

Abstract-Hyperspectral remote sensing technology has advanced significantly in the past two decades. Current sensors onboard airborne and spaceborne platforms cover large areas of the Earth surface with unprecedented spectral, spatial, and temporal resolutions. These characteristics enable a myriad of applications requiring fine identification of materials or estimation of physical parameters. Very often, these applications rely on sophisticated and complex data analysis methods. The sources of difficulties are, namely, the high dimensionality and size of the hyperspectral data, the spectral mixing (linear and nonlinear), and the degradation mechanisms associated to the measurement process such as noise and atmospheric effects. This paper presents a tutorial/overview cross section of some relevant hyperspectral data analysis methods and algorithms, organized in six main topics: data fusion, unmixing, classification, target detection, physical parameter retrieval, and fast computing. In all topics, we describe the state-of-the-art, provide illustrative examples, and point to future challenges and research directions.

Abstract-We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the state-of-the-art.