A method of computing trajectories of objects by using velocity data, particularly as acquired with phase-contrast magnetic resonance (MR) imaging, is presented. Starting from a specified location at one time point, the method recursively estimates the trajectory. The effects of measurement noise and eddy current-induced velocity offsets are analyzed. When the motion is periodic, trajectories can be computed by integrating in both the forward and backward temporal directions, and a linear combination of these trajectories minimizes the effect of velocity offsets and maximizes the precision of the combined trajectory. For representative acquisition parameters and signal-to-noise ratios, the limitations due to measurement noise are acceptable. In a phantom with reciprocal rotation, the measured and true trajectories agreed to within 3.3%. Sample trajectory estimates of human myocardial regions are encouraging.