A Mixed Finite Element Method for Constraining

The contribution of our paper is to present a mixed finite element method for 4 estimation of the velocity in the optical flow constraint, i.e., an advection equation. The resulting 5 inverse problem is well-known to be undetermined because the velocity vector cannot be recovered 6 from the scalar field advected unless further restrictions on the flow, or motion are imposed. If 7 we suppose, for example, that the velocity is solenoidal, a well-defined least squares problem with a 8 minimizing velocity results. Equivalently, we have imposed the constraint that the underlying motion 9 is isochoric. Unfortunately, the resulting least squares system is ill-posed and so regularization 10 via a mixed formulation of the Poisson equation is proposed. Standard results for the Poisson 11 equation demonstrate that the regularized least squares problem is well-posed and has a stable finite 12 element approximation. A numerical example demonstrating the procedure supports the analyses. 13 The example also introduces a closed form solution for the unregularized, constrained least squares 14 problem so that the approximation can be quantified. 15