We develop a powerful probabilistic framework for the local characterization of surfaces and edges in range images. We use the geometrical nature of the data to derive an analytic expression for the joint probability density function (pdf) for the random variables used to model the ranges of a set of pixels in a local neighborhood of an image. We decompose this joint pdf by considering independently the cases where two real world points corresponding to two neighboring pixels are locally on the same real world surface or not. In particular, we show that this joint pdf is linked to the Voigt pdf and not to the Gaussian pdf as it is assumed in some applications. We apply our framework to edge detection and develop a locally adaptive algorithm that is based on a probabilistic decision rule. We show in an objective evaluation that this new edge detector performs better than prior art edge detectors. This proves the benefits of the probabilistic characterization of the local neighborhood as a tool to improve applications that involve range images.