The William Harvey Code: Mathematical Analysis of Optical Flow Computation for Cardiac Motion

For the non-invasive imaging of moving organs, in this chapter, we investigate the generalisation of optical flow in three-dimensional Euclidean space. In computer vision, optical flow is dealt with as a local motion of pixels in a pair of successive images in a sequence of images. In a space, optical flow is defined as the local motion of the voxel of spatial distributions, such as x-ray intensity and proton distributions in living organs. Optical flow is used in motion analysis of beating hearts measured by dynamic cone beam x-ray CT and gated MRI tomography. This generalisation of optical flow defines a class of new constraints for optical-flow computation. We first develop a numerically stable optical-flow computation algorithm. The accuracy of the solution of this algorithm is guaranteed by Lax equivalence theorem which is the basis of the numerical computation of the solution for partial differential equations. Secondly, we examine numerically the effects of the divergence-free condition, which is required from linear approximation of infinitesimal deformation, for the computation of cardiac optical flow from images measured by gated MRI. Furthermore, we investigate the relation between the vector-spline constraint and the thin plate constraint. Moreover, we theoretically examine the validity of the error measure for the evaluation of computed optical flow.