Accelerating image reconstruction in ultrasound transmission tomography using L-BFGS algorithm

In ultrasound transmission tomography, image reconstruction is an inverse problem which is solved iteratively based on a forward model that simulates the wave propagation of ultrasound. A commonly used forward model is paraxial approximation of the Helmholtz equation, which is time-consuming. Hence developing optimizers that minimize the number of forward solutions is crucial to achieve clinically acceptable reconstruction time, while the state-of-the-art methods in this field such as Gauss-Newton conjugate gradient (CG) and nonlinear CG are not capable of reaching this goal. To that end, we focus on Jacobian-free optimizers or accelerators in this paper, since the computation of the Jacobian is expensive. We investigate the limited memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm as a preconditioning technique due to its ability to efficiently approximate inverse Hessian without performing forward model or its adjoint. We show L-BFGS can reach a speedup of more than one order of magnitude for the noise-free case, while the method still halves the reconstruction time in presence of noise in the data. The performance drop is explained by perturbed gradients due to noise in the data. We also show when used alone as a quasi-Newton method, L-BFGS is competitive with the accelerated CG based methods regarding the number of iterations, and outperforms them regarding reconstruction time.