IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 51, NO. 5, MAY 2013 2853 Piecewise Convex Multiple-Model Endmember Detection and Spectral Unmixing Alina Zare, Member, IEEE, Paul Gader, Fellow, IEEE, Ouiem Bchir, and Hichem Frigui, Member, IEEE Abstract—A hyperspectral endmember detection and spectral unmixing algorithm that finds multiple sets of endmembers is presented. Hyperspectral data are often nonconvex. The Piece- wise Convex Multiple-Model Endmember Detection algorithm accounts for this using a piecewise convex model. Multiple sets of endmembers and abundances are found using an iterative fuzzy clustering and spectral unmixing method. The results indicate that the piecewise convex representation estimates endmembers that better represent hyperspectral imagery composed of multiple regions where each region is represented with a distinct set of endmembers. Index Terms—Clustering functional forms, endmember, fuzzy, hyperspectral, image analysis, non-linear unmixing, piece-wise convex, scene analysis, scene segmentation, unmixing. I. I NTRODUCTION T HE standard model used to perform hyperspectral unmix- ing and endmember extraction is the linear mixing model as shown in x i = M  k=1 p ik e k + ǫ i , i =1,...,N (1) where N is the number of pixels in the image, M is the number of endmembers, ǫ i is an error term, p ik is the proportion of endmember k in pixel i, and e k is the kth endmember [1]. The proportions of this model satisfy the constraints in p ik ≥ 0 ∀k =1,...,M M  k=1 p ik =1. (2) Generally, when applying the linear mixing model with a single set of endmembers, input pixels are assumed to be linear mixtures of all endmembers in the scene. Many endmember Manuscript received March 15, 2012; revised June 21, 2012 and August 2, 2012; accepted August 26, 2012. Date of publication November 15, 2012; date of current version April 18, 2013. A. Zare is with the Department of Electrical and Computer Engineering, Uni- versity of Missouri, Columbia, MO 65211 USA (e-mail: zarea@missouri.edu). P. Gader is with the Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail: pgader@cise.ufl.edu). O. Bchir is with the Computer Science Department, College of Computer and Information Sciences, King Saud University, Riyadh 11543, Saudi Arabia (e-mail: o0bchi01@louisville.com). H. Frigui is with the Department of Computer Engineering and Com- puter Science, University of Louisville, Louisville, KY 40292 USA (e-mail: h.frigui@louisville.edu). 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.2012.2219058 detection and spectral unmixing algorithms which rely on the linear mixing model have been proposed in the literature [1]–[3]. A number of these methods make the pixel purity assumption and assume that the endmembers can be found within the data set [4]–[10]. Methods have also been developed based on nonnegative matrix factorization [11]–[14], indepen- dent component analysis [15], [16], and others [17]–[21]. In the results shown here, comparisons are made to the vertex com- ponent analysis (VCA) and iterative constrained endmembers (ICE) algorithms [6], [19]. The VCA algorithm is an iterative algorithm that relies on the pixel purity assumption and selects endmembers from the input scene. The ICE algorithm estimates “virtual” endmembers and is, therefore, capable of estimating endmembers for highly mixed scenes. However, all of these methods (including VCA and ICE) search for a single set of endmembers and, therefore, a single convex region to describe a hyperspectral scene. Since these algorithms assume a single convex region, they often cannot find appropriate endmembers for nonconvex data sets. Piecewise Convex Multiple-Model Endmember Detection (PCOMMEND) extends the linear mixing model to multiple sets of endmembers. Consider a scene that contains multiple distinct regions that do not share common materials. Each region in this scene is composed of pixels that are linear mixtures of a distinct set of endmembers, and each of these endmember sets defines a simplex. Then, the set of all image spectra will consist of a union of all of the simplices. The union of simplices is unlikely to be convex. For scenes of this type, a piecewise convex model with multiple sets of endmembers is more appropriate than the linear mixing model with a single convex region. Spectral unmixing methods that account for spectral vari- ability or use per-pixel specific endmember sets to unmix hyperspectral data have been previously developed in the lit- erature [2]. These methods include a Monte Carlo unmixing approach that determines the mean and variance of abun- dance values computed using randomly selected endmembers [22]. Also, “endmember bundles” have been investigated to address endmember variability. An endmember bundle is a set of spectra from one material. These bundles are often formed from field measurements, pulled from spectral libraries, or pulled from an imaged scene [23], [24]. The multiple- endmember spectral mixture analysis algorithm has been used extensively to account for endmember variability and allow for per-pixel specific endmembers [25]. Methods based on the normal compositional model, which account for spectral vari- ability of the endmembers by representing endmembers using Gaussian distributions, have also been developed in [26]–[29]. 0196-2892/$31.00 © 2012 IEEE