In this paper, we show how to recover the motion of an observer relative to a planar surface directly from image brightness derivatives. We do not compute the optical flow as an intermediate step. We derive a set of nine non-linear equations using a least-squares formulation. A simple iterative scheme allows us to find either of two possible solutions of these equations. An initial pass over the relevant image region is used to accumulate a number of moments of the image brightness derivatives. All of the quantities used in the iteration can be efficiently computed from these totals, without the need to refer back to the image. A new, compact notation allows is to show easily that there are at most two planar solutions. Key words: Passive Navigation, Optical flow, Structure and Motion, Least Squares, Planar surface, Non-linear Equations, Dial Solution, Planar Motion Field Equation.
