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The purpose of this paper is to investigate Nesterov’s accelerated gradient (NAG) method for the reconstruction of speed of sound and attenuation images in ultrasound computed tomography. The inverse problem of reconstruction is tackled via minimizing the deviation between exact measurements and the predicted measurements based on a paraxial approximation of the Helmholtz equation which simulates the ultrasound wave forward propagation. To solve this optimization problem, NAG is performed and compared with other algorithms. Also, a line search method is used to compute the step size for each iteration since finding proper step sizes is crucial for the convergence of such optimization algorithms. The strong Wolfe conditions are adopted as the termination condition for line search. We have compared five algorithms, namely Gauss-Newton conjugate gradient, gradient descent, NAG, gradient descent with line search, and NAG with line search. On one hand, NAG with line search has the fastest convergence rate in respect to the number of used iterations compared to the other methods. However, due to the increased computational complexity of line search for each iteration, it requires extra computational time. On the other hand, NAG with a fixed step size for all iterations is the fastest method among all the tested methods regarding computational time.

Ultrasound transmission tomography can offer quantitative characterization of breast tissue by reconstructing its sound speed and attenuation images. The image reconstruction process is a nonlinear inverse problem which we iteratively solve using the paraxial approximation of wave equation. The problem is tackled via the Gauss-Newton method yielding a set of linear systems and iteratively solving these linear systems. In this paper, we study multigrid methods for solving these linear systems. We test three multigrid schemes including V-cycle, W-cycle, and full multigrid (FMG), with up to four resolution levels. At each grid level, we use the conjugate gradient (CG) method as a standard solver. Our interest is at first by how far these schemes allow us to reduce the computations alone. For performance evaluation, we compare the multigrid methods with fixed-grid CG method where we directly apply CG on the finest grid. Results show that all tested multigrid methods have an accelerating effect in terms of needing fewer CG iterations on the finest grid. The best-case reduction is 32% for V-cycle, 33% for W-cycle, and 27% for FMG. This means that our multigrid scheme has the potential for significantly reducing reconstruction time for large-scale 2D or 3D images, where the computation cost on the finest grid is very high.