2014 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011 Sparse Unmixing of Hyperspectral Data Marian-Daniel Iordache, José M. Bioucas-Dias, Member, IEEE, and Antonio Plaza, Senior Member, IEEE Abstract—Linear spectral unmixing is a popular tool in re- motely sensed hyperspectral data interpretation. It aims at esti- mating 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 end- member 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 im- age 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 combina- torial 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. Index Terms—Abundance estimation, convex optimization, hyperspectral imaging, sparse regression (SR), spectral unmixing. Manuscript received March 14, 2010; revised August 18, 2010 and November 2, 2010; accepted November 28, 2010. Date of publication January 19, 2011; date of current version May 20, 2011. This work was supported by the European Community’s Marie Curie Research Training Networks Program under contract MRTN-CT-2006-035927 (Hyperspectral Imaging Network) and by the Spanish Ministry of Science and Innovation (HYPERCOMP/EODIX project, reference AYA2008-05965-C04-02). M.-D. Iordache and A. Plaza are with the Department of Technology of Com- puters and Communications, Escuela Politécnica, University of Extremadura, 10071 Cáceres, Spain (e-mail: diordache@unex.es; aplaza@unex.es). J. M. Bioucas-Dias is with the Instituto de Telecomunicações, Instituto, Superior Técnico, 1049-1 Lisbon, Portugal (e-mail: bioucas@lx.it.pt). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2010.2098413 I. I NTRODUCTION H YPERSPECTRAL imaging has been transformed from being a sparse research tool into a commodity product that is available to a broad user community [1]. The wealth of spectral information available from advanced hyperspectral imaging instruments currently in operation has opened new perspectives in many application domains, such as monitoring of environmental and urban processes or risk prevention and response, including, among others, tracking wildfires, detecting biological threats, and monitoring oil spills and other types of chemical contamination. Advanced hyperspectral instruments such as NASA’s Airborne Visible Infrared Imaging Spectrom- eter (AVIRIS) [2] are now able to cover the wavelength region from 0.4 to 2.5 μm using more than 200 spectral channels at a nominal spectral resolution of 10 nm. The resulting hyper- spectral data cube is a stack of images (see Fig. 1) in which each pixel (vector) is represented by a spectral signature or fingerprint that characterizes the underlying objects. Several analytical tools have been developed for remotely sensed hyperspectral data processing in recent years, cover- ing topics like dimensionality reduction, classification, data compression, or spectral unmixing [3], [4]. The underlying assumption governing clustering and classification techniques is that each pixel vector comprises the response of a single underlying material. However, if the spatial resolution of the sensor is not high enough to separate different materials, these can jointly occupy a single pixel. For instance, it is likely that the pixel collected over a vegetation area in Fig. 1 actually comprises a mixture of vegetation and soil. In this case, the measured spectrum may be decomposed into a linear combina- tion of pure spectral signatures of soil and vegetation, weighted by abundance fractions that indicate the proportion of each macroscopically pure signature in the mixed pixel [5]. To deal with this problem, linear spectral mixture analysis techniques first identify a collection of spectrally pure con- stituent spectra, called as endmembers in the literature, and then express the measured spectrum of each mixed pixel as a linear combination of endmembers weighted by fractions or abun- dances that indicate the proportion of each endmember present in the pixel [6]. It should be noted that the linear mixture model assumes minimal secondary reflections and/or multiple scatter- ing effects in the data collection procedure, and hence, the mea- sured spectra can be expressed as a linear combination of the spectral signatures of the materials present in the mixed pixel [see Fig. 2(a)]. Being quite opposite, the nonlinear mixture model assumes that the endmembers form an intimate mixture inside the respective pixel so that the incident radiation interacts with more than one component and is affected by multiple scattering effects [see Fig. 2(b)]. Nonlinear unmixing generally 0196-2892/$26.00 © 2011 IEEE