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Multidimensional synthetic aperture radar (SAR) imaging is a technique based on coherent SAR data combination for space (full 3-D) and space deformation-velocity (4-D) analysis. It is an extension of the concepts of SAR interferometry and differential interferometry SAR and offers new options for the analysis and monitoring of ground scenes. In this paper, we consider the problem of detecting single scatterers for localization and monitoring issues. To this end, we resort to a constant false alarm rate (CFAR) detection scheme which can be synthesized according to three different design criteria: generalized likelihood ratio test, Rao test, and Wald test. At the analysis stage, the performance of the aforementioned detector is compared to that of a previously proposed CFAR scheme, based on the multi-interferogram complex coherence and widely used in persistent scatterer interferometry. The analysis is conducted both on simulated and on real SAR data, acquired by ERS-1/2 satellites. Finally, Cramer-Rao lower bounds for the estimation of the scatterer elevation and velocity are provided.