Three-dimensional-position-sensitive semiconductor detectors provide both energy and locations of gamma-ray interactions in the detector. From these data, Compton-imaging methods can reconstruct the spatial source distribution around the detector as a function of energy. Here, the generalized likelihood ratio test (GLRT) and the source-intensity test (SIT) are applied to automate detection and identification of point sources of radiation in distributed background by using the combined energy and direction information from these detectors. Throughout, the model makes no assumptions of the background intensity or energy distribution. However, directional information enables detection and identification without prior knowledge of the background spectrum and the detection of isotopes also appearing in the background. Spatially, background is modeled as uniform, but only slight degradation in performance was observed when the background was twice as intense in one hemisphere. Experimental detection and identification performance as a function of false-alarm probability was measured for three weak isotopes using experimental data collected on a 20 mm x 20 mm x 15 mm CdZnTe detector. Both the GLRT and SIT performed better than the commercial software Genie 2000 (Canberra), which uses only spectral information, when the possible source direction was unknown. Prior knowledge of the possible source direction further improves both detection and identification.