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The problem of detecting a material-of-interest in a hyperspectral image is considered. Knowledge of the background materials in the image is assumed. It is also assumed that the stochastic noise in the system has a Gaussian distribution with a known covariance matrix. Using these assumptions, along with the requirement that the material abundances in the pixel must sum to one, a filter called the constrained signal detector (CSD) is derived. The CSD is a variation of the generalized likelihood ratio test (GLRT). Where the GLRT uses maximum-likelihood estimates (MLEs) of the noise in the received signal, the CSD uses constrained least squares (CLS) noise estimates. It will be shown that the CSD is actually a scaling of the CLS target abundance estimate which has been derived elsewhere. However, the CSD computes that estimate much more efficiently then existing methods do. It is proved that the CSD outperforms the orthogonal subspace projection (OSP) detector and that the CSD is the optimal detector when there is only one background material present.