A Scalar Function Formulation for Optical Flow

In this work, we present results from a new formulation for determining image velocities from a time-sequence of X-ray projection images of owing uid. Starting with the conservation of mass principle, and physics of X-ray projection, we derive a motion constraint equation for projection imaging, a practical special case of which is shown to be the Horn and Schunck's optical ow constraint. We are interested in the study of non-rigid motion of blood which is an incompressible uid, and as such have developed a formulation for optical ow which is applicable to such media. The formulation is particularly eecient, as the ow eld is obtained from a 90 degrees rotation applied to the gradient of a scalar function. It is shown that if speciic criteria are met, in addition to normal ow which is commonly recoverable, the tangential component of the ow eld is also recoverable, bypassing the aperture problem. An algorithm is presented to illustrate this. Preliminary results from the optical ow formulation applied to synthetic images, as well as contrast-injected X-ray images of owing uid, in a cylindrical uid phantom are presented.