This paper considers the problem of detecting a subpixel target in a hyperspectral image with a uniform background. A uniform background region is defined as an area of a hyperspectral image composed of a single material except for possibly a target material. It is shown that in the uniform background case the likelihood ratio test (LRT), which is the optimal signal detection algorithm, can be evaluated. It is shown that the LRT derived in this paper is a special case of the constrained signal detector (CSD). Though no other detector outperforms the LRT, it is possible for other algorithms to equal its performance. It is shown in what special cases the orthogonal subspace projection (OSP) detector and the matched filter detector (MFD) perform the same as the CSD.