5. The Ill-Posed Problem in DIC

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