Direct passive navigation.

In this correspondence, we show how to recover the motion of an observer relative to a planar surface from image brightness derivatives. We do not compute the optical flow as an intermediate step, only the spatial and temporal brightness gradients (at a minimum of eight points). We first present two iterative schemes for solving nine nonlinear equations in terms of the motion and surface parameters that are derived from a least-squares fomulation. 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 are efficiently computed from these totals without the need to refer back to the image. We then show that either of two possible solutions can be obtained in closed form. We first solve a linear matrix equation for the elements of a 3 × 3 matrix. The eigenvalue decomposition of the symmetric part of the matrix is then used to compute the motion parameters and the plane orientation. A new compact notation allows us to show easily that there are at most two planar solutions.