SIAM student chapter at UNM
A Mixed Finite Element Method for Regularizing the Optical Flow Constraint
by Rich Lehoucq, Sandia National Laboratories
My presentation introduces a mixed finite element method for the estimation of the velocity in the optical flow constraint, i.e., an advection equation. The estimation 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 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. We then demonstrate that the resulting least squares system is ill-posed and regularize via a mixed formulation of the Poisson equation. Standard results for the Poisson equation demonstrate that the regularized system is well-posed and has a stable finite element approximation. A numerical example demonstrating the procedure is given. This is joint work with Dan Turner, also of Sandia National Labs.
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