Direct Passive Navigation: Analytical Solution for Quadratic Patches

Abstract

In this paper, we solve the problem of recovering the motion of an observer relative to a surface which can be locally approximated by a quadratic patch directly from image brightness values. We do not compute the optical flow as an intermediate step. We use the coefficients of the Taylor series expansion of the intensity function in two frames to determine 15 intermediate parameters, termed the essential parameters, from a set of linear equations. We then solve analytically for the motion and structure parameters from a set of nonlinear equations in terms of these intermediate parameters. We show that the solution is always unique, unlike some earlier results that reported two-fold ambiguities in some special cases.