The ability to detect man-made materials with known spectral signatures in hyperspectral images has many important applications. In this paper, we present and compare two, linear mixing model-based, algorithms for the detection of low probability of occurrence targets with known spectral signatures. One involves the estimation of the target abundance in each pixel, a form of spectral unmixing, and the other binary hypothesis testing using the generalized likelihood ratio test (GLRT). In an effort to improve detection, we investigate the effects of placing the Sum-to-One (STO) constraint on the abundances of the materials present in each pixel. Both theoretical and experimental results will be presented such that the benefits of the STO constraint can be directly compared. We shall demonstrate that, in theory, the enforcement of STO constraint improves detection performance. For abundance estimation based detectors, the constraint reduces the variance of the estimate. For GLRT detectors, the STO constraint increases the signal to interference plus noise ratio (SINR). Unfortunately, we do not see the same improvements with real data. In fact, enforcing the constraint leads to a performance degradation, in most cases we have investigated. It turns out that the abundance estimation based detector moves the full pixels, subpixels, and background pixels closer to each other; which makes reliable detection more difficult. With regard to the constrained GLRT detector, there is an introduction of bias to the background pixels, which naturally results to a deterioration in detection performance.