Statistical clustering and Mineral Spectral Unmixing in Aviris Hyperspectral Image of Cuprite, NV

Hyperspectral Imaging is a technique for obtaining a spectrum in each position of a large array of spatial positions so that a recognizable image is obtained at each of a set of discrete wavelengths. The images might be of a rock in the laboratory, a field study site from an aircraft or a rover camera, or a whole planet from a spacecraft or Earth-based telescope. By analyzing the spectral features (generally neighborhoods of local minima in the spectra) one can map materials. A simplistic explanation for this being is that specific chemical bonds in different materials manifest themselves as absorption features at different wavelengths and by mapping where those bonds occur in the spectra one can uniquely identify what is called the unique spectral signature of the material . The factors affecting spectra of natural materials and the causes of absorption features are several and combine in complex ways. They are not the focus of this paper but a comprehensive tutorial can be found in [3]. Spectral unmixing is the procedure by which the measured spectrum of a pixel is decomposed into a collection of constituent spectra, or endmembers, and a set of corresponding fractions, or abundances, that indicate the proportion of each endmember present in the pixel. In the case of rocks or soils the endmembers can be consistent with the minerals present in the geologic surface observed. In this work we present a novel technique of endmember selection from a database of minerals based on simple convex optimization techniques. Spectral unmixing can assume linear or nonlinear combination of the endmembers depending on the nature of the surface observed [9]. Unfortunately nonlinear schemes can be impractical for hyperspectral imaging because multiple views from different angles of the same scene are required [11]. Linear Spectral Unmixing is based on the assumption that the spectrum of each pixel of the scene is a convex combination of the spectra of its component minerals [13]. Deterministic modeling of the mixture lacks the ability to explain the statistical variability of the spectra within a class due for example to illumination differences, altimetry, grain size of the material and other causes. Several attempts have been made to correct this problem: these approaches allow the endmembers of the mixture to be random variables (mostly Gaussians) [6], [14]. The present work assumes a mixture of multidimensional pdf’s for the statistical distribution of the spectra of the single pixels composing the scene. Each pdf (a multinomial Gaussian) represents the likelihood of a certain mineral mixture in the scene. We make use of the Gauss Mixture Vector