Sparsity Regularization in Dynamic Elastography

We consider the inverse problem of continuum mechanics with the tissue deformation described by a mixed displacement–pressure finite element formulation. The mixed formulation is used to model nearly incompressible materials by simultaneously solving for both elasticity and pressure distributions. To improve numerical conditioning, a common solution to this problem is to use regularization to constrain the solutions of the inverse problem. We present a sparsity regularization technique that uses the discrete cosine transform to transform the elasticity and pressure fields to a sparse domain in which a smaller number of unknowns is required to represent the original field. We evaluate the approach by solving the dynamic elastography problem for synthetic data using such a mixed finite element technique, assuming time harmonic motion, and linear, isotropic and elastic behavior for the tissue. We compare our simulation results to those obtained using the more common Tikhonov regularization. We show that the sparsity regularization is less dependent on boundary conditions, less influenced by noise, requires no parameter tuning and is computationally faster. The algorithm has been tested on magnetic resonance elastography data captured from a CIRS elastography phantom with similar results as the simulation.