Recovery of shear modulus in elastography using an adjoint method with B-spline representation

Shear modulus recovery in elastography with harmonic excitation is posed as an inverse problem. Large aperture is assumed. By minimizing the residual error norm of the displacement field from the signature field using a gradient-based algorithm, we recovered the B-spline represented shear modulus. The adjoint method is used to calculate the gradient efficiently. The finite element method is used for calculating both the primal and adjoint solutions. It is demonstrated that representation of the shear modulus by B-spline eliminates the need for additional regularization and hence renders the method stable and robust. The effect of noisy data is considered. The optimal B-spline corresponds to a minimal solution norm and balances the relative residual norm with the signal-to-noise ratio.