Detecting targets in nonhomogeneous backgrounds is a difficult task, particularly if the detection processor must respect a Constant False Alarm Rate (CFAR). In this article, a novel approach is developed for the detection of punctual targets embedded in nonhomogeneous gamma-distributed backgrounds. The well-known CA-CFAR (Cell-Averaging CFAR) processor is generalized to nonhomogeneous backgrounds through a new CFAR thresholding method. This new CFAR thresholding method allows one the design of new CFAR processors that use other background estimators than the arithmeticmean. A new CFAR processor (Q-CFAR) adapted to nonhomogeneous backgrounds is designed, the improvement of the detection performance, and more particularly the improvement of the false alarm regulation obtained with the proposed CFAR processor is shown with Monte-Carlo experiments.