On the visual mathematics of tracking

摘要

A mathematical theory for visual tracking of a three-dimensional target of known local shape moving rigidly in 3D is presented here, and it is shown how a monocular observer can track an initially foveated object and keep it stationary in the centre of the visual field assuming that the target is always visible during the tracking phase. Our attempt is to develop correspondence-free tracking schemes and get rid of the limitations inherent in the optical flow formalism. A general tracking criterion, the Tracking Constraint, is derived, which reduces tracking to an appropriate optimization problem. A correspondence-free scheme is devised, that depends on global information about the scene in order to bypass the ill-posed problem of computing the spatial derivatives of the image intensity function, and amounts to the solution of a linear system of equations in order to estimate the 3D motion of the target. Finally, it must be emphasized that — as hinted in the title — this work is devoted entirely to the vision aspects of tracking, and does not get involved in control aspects. In addition, the principal difference between this technique and existing ones is that here tracking is performed in 3D as opposed to 2D. Experimental results with synthetic and real images demonstrate the feasibility of the approach and its robustness against noise.