A New Minimum-Volume Enclosing Algorithm for Endmember Identification and Abundance Estimation in Hyperspectral Data

Spectral unmixing is an important technique for hyperspectral data exploitation, in which a mixed spectral signature is decomposed into a collection of spectrally pure constituent spectra, called endmembers, and a set of correspondent fractions, or abundances, that indicate the proportion of each endmember present in the mixture. Over the last years, several algorithms have been developed for automatic or semiautomatic endmember extraction. Some available approaches assume that the input data set contains at least one pure spectral signature for each distinct material and further conduct a search for the most spectrally pure signatures in the high-dimensional space spanned by the hyperspectral data. Among these approaches, those aimed at maximizing the volume of the simplex that can be formed using available spectral signatures have found wide acceptance. However, the presence of spectrally pure constituents is unlikely in remotely sensed hyperspectral scenes due to spatial resolution, mixing phenomena, and other considerations. In order to address this issue, other available algorithms have been developed to generate virtual endmembers (not necessarily present among the input data samples) by finding the simplex with minimum volume that encloses all available observations. In this paper, we discuss maximum-volume versus minimum-volume enclosing solutions and further develop a novel algorithm in the latter category which incorporates the fractional abundance estimation as an internal step of the endmember searching process (i.e., it does not require an external method to produce endmember fractional abundances). The method is based on iteratively enclosing the observations in a lower dimensional space and removing observations that are most likely not to be enclosed by the simplex of the endmembers to be estimated. The performance of the algorithm is investigated and compared to that of other algorithms (with and without the pure pixel assumption) using synthetic and real hyperspectral data sets collected by a variety of hyperspectral imaging instruments.