Abstract: The image motion field for an observer moving through a static environment depends on the observer's translational and rotational velocities along with the distances to surface points. Given such a motion field as input we have recently introduced subspace methods for the recovery of the observer's motion and the depth structure of the scene. This class of methods involve splitting the equations describing the motion field into separate equations for the observer's translational direction, the rotational velocity, and the relative depths. The resulting equations can then be solved successively, beginning with the equations for the translational direction. Here we concentrate on this first step. In earlier work, a linear method was shown to provide a biased estimate of the translational direction. We discuss the source of this bias and show how it can be effectively removed. The consequence is that the observer's velocity and the relative depths to points in the scene can all be recovered by successively solving three linear problems.