CONTEXT-BASED ENDMEMBER DETECTION FOR HYPERSPECTRAL IMAGERY Alina Zare and Paul Gader University of Florida Department of Computer and Information Science and Engineering Gainesville, FL 32611 USA ABSTRACT An endmember detection algorithm that simultaneously par- titions an input data set into distinct contexts, estimates end- members, number of endmembers, and abundances for each partition is presented. In contrast to previous endmember detection algorithms based on the Convex Geometry Model, this method is capable of describing non-convex sets of hy- perspectral pixels. Endmembers are found for non-convex regions by partitioning the set of pixels into convex regions using the Dirichlet Process and determining unique endmem- bers for each region. This novel endmember detection method naturally produces a classifier with a reject class. The al- gorithm can effectively identify to which context a test data point belongs and identify test pixels for which the associated context is unknown. Results are shown on AVIRIS Indian Pines Hyperspectral data. The results show the classification capability of this context-based endmember algorithm. Index Terms— Hyperspectral, Endmember, Spectral Un- mixing, Convex Geometry Model, Context, Dirichlet. 1. INTRODUCTION The convex geometry model assumes that each pixel in a hy- perspectral image can be described with a convex combina- tion of the endmembers [1]. x i = M  k=1 a ik e k +  i i =1,...,N (1) where N is the number of pixels, M is the number of end- members,  i is an error term, a ik is the abundance of end- member k in pixel i, and e k is the k th endmember. The abun- dances of this model satisfy the constraints in Equation 2, a ik ≥ 0 ∀k =1,...,M ; M  k=1 a ik =1. (2) Several endmember detection algorithms are described in the literature. Previous endmember detection methods include methods that rely on the pixel purity assumption and search for endmembers within the data set [2], are based on Non- Negative Matrix Factorization [3, 4], use Independent Com- ponents Analysis [5, 6], and others. These methods generally search for a single set of endmembers to describe a hyperspec- tral scene. The presented algorithm partitions the input hyper- spectral set into distinct contexts using the Dirichlet Process. Each context has its own set of endmembers. Section 2 describes the Robust Sparsity Promoting Iter- ated Constrained Endmembers (R-SPICE) algorithm used to compute the endmembers within each convex region. Sec- tion 3 describes the Piece-Wise Convex Endmember (PCE) algorithm used to partition the data and find a unique set of endmembers for each partition. 2. ROBUST SPICE Robust SPICE (R-SPICE) is a novel algorithm used to deter- mine endmembers and the number of endmembers. R-SPICE is an extension of the Sparsity Promoting Iterated Constrained Endmembers (SPICE) algorithm [7]. SPICE simultaneously determines endmember spectra, estimates abundance values, and determines the number of endmembers. SPICE deter- mines the number of endmembers by using a sparsity promot- ing Laplacian prior on the abundance values. The R-SPICE algorithm differs from SPICE by weighting each pixel based on its similarity to the other pixels in the data set. There- fore, outliers have little effect on endmember estimation. The weights on each pixel cause the results of the R-SPICE al- gorithm to be more stable over a wide range of parameter values as compared to SPICE. The objective function in R- SPICE is ln (p (X|A, E) p (E) p (A)) where p (X|A, E) is the likelihood of the data given the endmembers and abun- dances, p (E) is the prior on the endmembers, and p (A) is the sparsity promoting prior on the abundances. The data term is given by a weighted squared error term between the input data points and the estimated endmembers and abundances, p (X|A, E)= exp    -1 2 N  i=1 w i ν  x i - M  k=1 a ik e k  T  x i - M  k=1 a ik e k     (3)