A Scalar Function Formulation for Optical Flow

Abstract

In this work, we present results from a new formulation for determining image velocities from a time-sequence of X-ray projection images of flowing fluid. 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 flow constraint. We are interested in the study of non-rigid motion of blood which is an incompressible fluid, and as such have developed a formulation for optical flow which is applicable to such media. The formulation is particularly efficient, as the flow field is obtained from a 90 degrees rotation applied to the gradient of a scalar function. It is shown that if specific criteria are met, in addition to normal flow which is commonly recoverable, the tangential component of the flow field is also recoverable, bypassing the aperture problem. An algorithm is presented to illustrate this. Preliminary results from the optical flow formulation applied to synthetic images, as well as contrast-injected X-ray images of flowing fluid, in a cylindrical fluid phantom are presented.