Multigrid Method for Solving Linearized Systems in Ultrasound Transmission Tomography

Abstract

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.