On computing all solutions to the motion estimation problem with exact or noisy data

摘要

A novel method of etimating 3D motion parameters (all solutions) from the image locations and velocities of several points is described. Given perspective projection, we successively eliminate variables and reduce the problem to a polynomial equation in a single variable. The advantages of the latter are that (i) we know exactly how many roots it has, (ii) we can compute them all — often with great precision. (iii) they need no initial estimates (thus eliminating blind search), and (iv) the problem of identifying the global minimum becomes trivial. The motion equations reduce to a pair of quartics in two variables given five exact data points. They in turn reduce to a polynomial degree of 16 in a single variable. If the linear motion is constrained to be on a known plain, only four data points suffice and the problem simplifies to a quartic in a single variable. In the least squares formalism, the problem still reduces to the solutions of a polynomial, albeit of higher degree. Normally troublesome cases like degenerate surfaces and vanishingly small liner velocity V need no separate treatment.