Biomechanical imaging (BMI) is the process of non-invasively measuring the spatial distribution of mechanical properties of biological tissues. The most common approach uses ultrasound to non-invasively measure soft tissue deformations. The measured deformations are then used in an inverse problem to infer local tissue mechanical properties. Thus quantifying local tissue mechanical properties can enable better medical diagnosis, treatment, and understanding of various diseases. A major difficulty with ultrasound biomechanical imaging is getting accurate measurements of all components of the tissue displacement vector field. One component of the displacement field, that parallel to the direction of sound propagation, is typically measured accurately and precisely; the others are available at such low precision that they may be disregarded in the first instance. If all components were available at high precision, the inverse problem for mechanical properties could be solved directly, and very efficiently. When only one component is available, the inverse problem solution is necessarily iterative, and relatively speaking, computationally inefficient. The goal of this thesis, therefore, is to develop a processing method that can be used to recover the missing displacement data with sufficient precision to allow the direct reconstruction of the linear elastic modulus distribution in tissue. This goal was achieved by using a novel spatial regularization to adaptively enforce and locally relax a special form of momentum conservation on the measured deformation field. The new processing method was implemented with the Finite Element Method (FEM). The processing method was tested with simulated data, measured data from a tissue mimicking phantom, and in-vivo clinical data of breast masses, and in all cases it was able to recover precise estimates the full 2D displacement and strain fields. The recovered strains were then used to calculate the material property distribution directly.