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We consider the problem of estimating the $2D$ vector displacement field in aheterogeneous elastic solid deforming under plane stress conditions. Theproblem is motivated by applications in quasistatic elastography. From preciseand accurate measurements of one component of the $2D$ vector displacementfield and very limited information of the second component, the methodreconstructs the second component quite accurately. No a priori knowledge ofthe heterogeneous distribution of material properties is required. This methodrelies on using a special form of the momentum equations to filter ultrasounddisplacement measurements to produce more precise estimates. We verify themethod with applications to simulated displacement data. We validate the methodwith applications to displacement data measured from a tissue mimickingphantom, and in-vivo data; significant improvements are noticed in the filtereddisplacements recovered from all the tests. In verification studies, error inlateral displacement estimates decreased from about $50\%$ to about $2\%$, andstrain error decreased from more than $250\%$ to below $2\%$.

Recovering vector displacement estimates in quasistatic elastography using sparse relaxation of the momentum equation