We propose an automatic, privacy-preserving, fall detection method for indoor environments, based on the usage of the Microsoft Kinect® depth sensor, in an "on-ceiling" configuration, and on the analysis of depth frames. All the elements captured in the depth scene are recognized by means of an Ad-Hoc segmentation algorithm, which analyzes the raw depth data directly provided by the sensor. The system extracts the elements, and implements a solution to classify all the blobs in the scene. Anthropometric relationships and features are exploited to recognize one or more human subjects among the blobs. Once a person is detected, he is followed by a tracking algorithm between different frames. The use of a reference depth frame, containing the set-up of the scene, allows one to extract a human subject, even when he/she is interacting with other objects, such as chairs or desks. In addition, the problem of blob fusion is taken into account and efficiently solved through an inter-frame processing algorithm. A fall is detected if the depth blob associated to a person is near to the floor. Experimental tests show the effectiveness of the proposed solution, even in complex scenarios.