In this paper, we present a novel method for measuring the location and estimating the dynamics of fast-moving small objects in free space. The proposed 3D localization method is realized by a space-to-time optical transform and measurement of time-of-flight. We present the underlying physical and mathematical model of the method and provide an example based on a simple configuration. In the simplest mode, the method is implemented by two plane mirrors, a spherical light pulse illuminator, and a single fast response photodetector. The 3D spatial information is retrieved from the temporal measurements by solving an inverse problem that uses a sparse approximation of the scene. System simulation shows the ability to track fast small objects that are moving in space using only a single time-resolved detector.