Moving objects leave extended tracks in optical images acquired with a telescope that is tracking stars or other targets. By searching images for these tracks, one can obtain statistics on populations of space debris in Earth orbit. The algorithm described here combines matched filtering with a Fourier implementation of the discrete Radon transform and can detect long linear tracks with high sensitivity and speed. Monte-Carlo simulations show that such tracks, in a background of Poisson random noise, can be reliably detected even if they are invisible to the eye. On a 2.2 GHz computer the algorithm can process a 4096 × 4096-pixel image in less than a minute.