Assumptions of linear motion with constant velocity or acceleration are valid in many situations. Can we arrive at such constraints for general non-rigid motion of points in space? That would be most beneficial. We derive algebraic constraints for such a situation in this section.
Our approach is based on the following observation. A moving 3D point traces out a contour or curve in space over time. This contour gets mapped to a contour in the image. If we have multiple views of the same object motion, their consistency reduces to the matching of corresponding shapes. Any contour-matching approach can be used for this step. If the 3D motion of the point is restricted to a plane, the problem reduces to planar contour recognition.
The matching constraints using vector Fourier representation presented in [10] can yield algebraic constraints on points moving arbitrarily, as long as the motion is planar. The analysis of the motion of the object can be carried out by studying the contours traced out in the various views. Many surveillance applications involve studying the motion of an object (like vehicles on the ground) from cameras that are far away from them (at top of tall buildings or on satellites). The trajectory of the objects is restricted to a plane and the cameras are affine in practice in this case. We demonstrate how the necessary conditions for matching a planar contour across multiple views [10] can yield a view and time independent constraint for arbitrary planar motion. We first consider the case of linear motion before considering arbitrary non-rigid motion.