Mission parameters derived from optical flow

Optical flow refers to the apparent motion of objects in the image plane, due to either camera or object motion. Applications of optical flow include robotics, image enhancement by means of frame integration, moving-target indication and passive navigation. The purpose of this paper is the simplest and clearest formulation of the dependence of optical flow on the mission and trajectory parameters. Once the canonical equations are established, their invertability is addressed: to what extent can mission parameters be obtained from optical flow? Analysis shows that there are eight independent mission parameters (excluding focal length), so there need to be exactly four relationships among the 12 flow coefficients. These are explicitly exhibited. It is then shown that the solution for the eight mission parameters in terms of the remaining eight independent coefficients hinges on a cubic resolvent. The roots of this resolvent are closely connected to the velocity-to-altitude ratio, and the solution can be constructed in terms of these. This solution generally turns out to be double valued with both sets of mission parameters producing identical optical flows. Fortunately, one of these values can usually be eliminated as inappropriate within the context. The recognition of the dual solution and its explicit equations are believed to be a new contribution. To make the paper more self-contained, algorithms, window selection, correlation measures, and data editing are covered. Much of this material has been previously published by others. The paper concludes with a discussion of the focus of expansion. It is shown that only camera-rotation-free parameters or their conjugates give rise to a focus of expansion. Explicit equations for these parameters are given.