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

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 $2\%$.