Abstract-Linear spectral unmixing is a popular tool in remotely sensed hyperspectral data interpretation. It aims at estimating the fractional abundances of pure spectral signatures (also called as endmembers) in each mixed pixel collected by an imaging spectrometer. In many situations, the identification of the endmember signatures in the original data set may be challenging due to insufficient spatial resolution, mixtures happening at different scales, and unavailability of completely pure spectral signatures in the scene. However, the unmixing problem can also be approached in semisupervised fashion, i.e., by assuming that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In practice, this is a combinatorial problem which calls for efficient linear sparse regression (SR) techniques based on sparsity-inducing regularizers, since the number of endmembers participating in a mixed pixel is usually very small compared with the (ever-growing) dimensionality (and availability) of spectral libraries. Linear SR is an area of very active research, with strong links to compressed sensing, basis pursuit (BP), BP denoising, and matching pursuit. In this paper, we study the linear spectral unmixing problem under the light of recent theoretical results published in those referred to areas. Furthermore, we provide a comparison of several available and new linear SR algorithms, with the ultimate goal of analyzing their potential in solving the spectral unmixing problem by resorting to available spectral libraries. Our experimental results, conducted using both simulated and real hyperspectral data sets collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infrared Imaging Spectrometer and spectral libraries publicly available from the U.S. Geological Survey, indicate the potential of SR techniques in the task of accurately characterizing the mixed pixels using the library spectra. This opens new perspectives for spectral unmixing, since the abundance estimation process no longer depends on the availability of pure spectral signatures in the input data nor on the capacity of a certain endmember extraction algorithm to identify such pure signatures.

Abstract-Spectral unmixing aims at estimating the fractional abundances of pure spectral signatures (also called endmembers) in each mixed pixel collected by a remote sensing hyperspectral imaging instrument. In recent work, the linear spectral unmixing problem has been approached in semisupervised fashion as a sparse regression one, under the assumption that the observed image signatures can be expressed as linear combinations of pure spectra, known a priori and available in a library. It happens, however, that sparse unmixing focuses on analyzing the hyperspectral data without incorporating spatial information. In this paper, we include the total variation (TV) regularization to the classical sparse regression formulation, thus exploiting the spatialcontextual information present in the hyperspectral images and developing a new algorithm called sparse unmixing via variable splitting augmented Lagrangian and TV. Our experimental results, conducted with both simulated and real hyperspectral data sets, indicate the potential of including spatial information (through the TV term) on sparse unmixing formulations for improved characterization of mixed pixels in hyperspectral imagery.Index Terms-Hyperspectral imaging, sparse regression, sparse unmixing, spectral unmixing, total variation (TV) regularization.

Abstract-Sparse unmixing has been recently introduced in hyperspectral imaging as a framework to characterize mixed pixels. It assumes that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In this paper, we present a refinement of the sparse unmixing methodology recently introduced which exploits the usual very low number of endmembers present in real images, out of a very large library. Specifically, we adopt the collaborative (also called "multitask" or "simultaneous") sparse regression framework that improves the unmixing results by solving a joint sparse regression problem, where the sparsity is simultaneously imposed to all pixels in the data set. Our experimental results with both synthetic and real hyperspectral data sets show clearly the advantages obtained using the new joint sparse regression strategy, compared with the pixelwise independent approach. Index Terms-Collaborative sparse regression, hyperspectral imaging, sparse unmixing, spectral libraries.

Spectral unmixing aims at finding the spectrally pure constituent materials (also called endmembers) and their respective fractional abundances in each pixel of a hyperspectral image scene. In recent years, sparse unmixing has been widely used as a reliable spectral unmixing methodology. In this approach, the observed spectral vectors are expressed as linear combinations of spectral signatures assumed to be known a priori and present in a large collection, termed spectral library or dictionary, usually acquired in laboratory. Sparse unmixing has attracted much attention as it sidesteps two common limitations of classic spectral unmixing approaches: the lack of pure pixels in hyperspectral scenes and the need to estimate the number of endmembers in a given scene, which are very difficult tasks. However, the high mutual coherence of spectral libraries, jointly with their ever-growing dimensionality, strongly limits the operational applicability of sparse unmixing. In this paper, we introduce a two-step algorithm aimed at mitigating the aforementioned limitations. The algorithm exploits the usual low dimensionality of the hyperspectral data sets. The first step, similar to the multiple signal classification (MUSIC) array signal processing algorithm, identifies a subset of the library elements which contains the endmember signatures. Because this subset has cardinality much smaller than the initial number of library elements, the sparse regression we are led to is much more well-conditioned than the initial one using the complete library. The second step applies collaborative sparse regression (CSR), which is a form of structured sparse regression, exploiting the fact that only a few spectral signatures in the library are active.The effectiveness of the proposed approach, termed MUSIC-CSR, is extensively validated using both simulated and real hyperspectral data sets. Index TermsHyperspectral imaging, hyperspectral unmixing, spectral libraries, sparse unmixing, sparse regression, collaborative sparse regression, dictionary pruning, MUSIC, array signal processing.

Abstract. We present retrieval results of tropospheric nitrogen dioxide (NO 2 ) vertical column densities (VCDs), mapped at high spatial resolution over three Belgian cities, based on the DOAS analysis of Airborne Prism EXperiment (APEX) observations. APEX, developed by a Swiss-Belgian consortium on behalf of ESA (European Space Agency), is a pushbroom hyperspectral imager characterised by a high spatial resolution and high spectral performance. APEX data have been acquired under clear-sky conditions over the two largest and most heavily polluted Belgian cities, i.e. Antwerp and Brussels on 15 April and 30 June 2015. Additionally, a number of background sites have been covered for the reference spectra. The APEX instrument was mounted in a Dornier DO-228 aeroplane, operated by Deutsches Zentrum für Luft-und Raumfahrt (DLR). NO 2 VCDs were retrieved from spatially aggregated radiance spectra allowing urban plumes to be resolved at the resolution of 60 × 80 m 2 . The main sources in the Antwerp area appear to be related to the (petro)chemical industry while traffic-related emissions dominate in Brussels. The NO 2 levels observed in Antwerp range between 3 and 35 × 10 15 molec cm −2 , with a mean VCD of 17.4 ± 3.7 × 10 15 molec cm −2 . In the Brussels area, smaller levels are found, ranging between 1 and 20 × 10 15 molec cm −2 and a mean VCD of 7.7 ± 2.1 × 10 15 molec cm −2 . The overall errors on the retrieved NO 2 VCDs are on average 21 and 28 % for the Antwerp and Brussels data sets. Low VCD retrievals are mainly limited by noise (1σ slant error), while high retrievals are mainly limited by systematic errors. Compared to coincident car mobile-DOAS measurements taken in Antwerp and Brussels, both data sets are in good agreement with correlation coefficients around 0.85 and slopes close to unity. APEX retrievals tend to be, on average, 12 and 6 % higher for Antwerp and Brussels, respectively. Results demonstrate that the NO 2 distribution in an urban environment, and its finescale variability, can be mapped accurately with high spatial resolution and in a relatively short time frame, and the contributing emission sources can be resolved. High-resolution quantitative information about the atmospheric NO 2 horizontal variability is currently rare, but can be very valuable for (air quality) studies at the urban scale.