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In the LMM for hyperspectral images, all the image spectra lie on a high-dimensional simplex with corners called endmembers. Given a set of endmembers, the standard calculation of fractional abundances with constrained least squares typically identifies the spectra as combinations of most, if not all, endmembers. We assume instead that pixels are combinations of only a few endmembers, yielding abundance vectors that are sparse. We introduce sparse demixing (SD), which is a method that is similar to orthogonal matching pursuit, for calculating these sparse abundances. We demonstrate that SD outperforms an existing L1 demixing algorithm, which we prove to depend adversely on the angles between endmembers. We combine SD with dictionary learning methods to calculate automatically endmembers for a provided set of spectra. Applying it to an airborne visible/infrared imaging spectrometer image of Cuprite, NV, yields endmembers that compare favorably with signatures from the USGS spectral library.