On div-curl regularization for motion estimation in 3-D volumetric imaging

We consider the classical optical flow algorithm due to Horn and Schunck (1981) for estimating the motion of brightness patterns between image pairs. We use a modified smoothness condition based on the divergence and curl of the velocity field. In previous work, we have developed well-posed stochastic state-space models for these optical flow methods in two dimensions. This paper extends our results to 3-D. We first show that by using the first order div-curl spline, it is not possible to obtain a first order linear differential well-posed model in 3-D. Next, we employ the second order div-curl spline smoothness condition and develop well-posed state-space models.