Determination of Displacement From an Image Sequence Based on Time-Reversal Invariance

This paper addresses the under-constrained issue for displacement determination in an image sequence to propose a fully constrained system of equations. The differentiation between the under-constrained issue and the issue of motion tracking by featureless morphologies (aperture problem) is clarified based on the refinements of logic for this particular physical system. This system is found to be time-reversal invariant because the motion can be observed from two frame images regardless of the order of the two images. A fully constrained system is derived based only on the brightness constancy constraint without any approximation, additional constraint, or assumption. The system includes the time-reversal invariance of displacement or average velocity equations and brightness constancy constraint equations for optical or heat flow computation. A unified adaptive framework proposed in the author's previous works is employed for solving the nonlinear system of equations. The fully constrained system cannot be used to derive motion vectors in featureless regions (aperture problem), and it is also proved theoretically that there is no solution to the fully constrained system of equations in a featureless region. It confirms that the under-constrained issue is different from and independent from the aperture problem. The goal of this paper is to infer motion vectors consistent with physical observation (actually tracked motion) by optimizing both forward and backward motion-compensated predictions rather than to find physical motion in featureless regions. A series of simulation images and real-world thermal images is used to examine and demonstrate the performance of the fully constrained system.