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Spectral mixtures observed in hyperspectral imagery often display nonlinear mixing effects. Since most traditional unmixing techniques are based upon the linear mixing model, they perform poorly in finding the correct endmembers and their abundances in the case of nonlinear spectral mixing. In this paper, we present an unmixing algorithm that is capable of extracting endmembers and determining their abundances in hyperspectral imagery under nonlinear mixing assumptions. The algorithm is based upon simplex volume maximization, and uses shortest-path distances in a nearest-neighbor graph in spectral space, hereby respecting the nontrivial geometry of the data manifold in the case of nonlinearly mixed pixels. We demonstrate the algorithm on an artificial data set, the AVIRIS Cuprite data set, and a hyperspectral image of a heathland area in Belgium.