  We consider the problem of estimating the $2D$ vector displacement field in a
heterogeneous elastic solid deforming under plane stress conditions. The
problem is motivated by applications in quasistatic elastography. From precise
and accurate measurements of one component of the $2D$ vector displacement
field and very limited information of the second component, the method
reconstructs the second component quite accurately. No a priori knowledge of
the heterogeneous distribution of material properties is required. This method
relies on using a special form of the momentum equations to filter ultrasound
displacement measurements to produce more precise estimates. We verify the
method with applications to simulated displacement data. We validate the method
with applications to displacement data measured from a tissue mimicking
phantom, and in-vivo data; significant improvements are noticed in the filtered
displacements recovered from all the tests. In verification studies, error in
lateral displacement estimates decreased from about $50\%$ to about $2\%$, and
strain error decreased from more than $250\%$ to below $2\%$.
