USCT Image Reconstruction: Acceleration using Gauss-Newton Preconditioned Conjugate Gradient

Ultrasound transmission tomography offers quantitative characterization of the tissue or materials by their speed of sound and attenuation. Reconstruction of such images is an inverse problem which is solved iteratively based on a forward model of the Helmholtz equation by paraxial approximation and thus is time-consuming. Hence, developing optimizers that decrease this time, in particular reducing the number of forward propagations is of high relevance in order to bring this technology into clinical practice. In this paper, we solve the inverse problem of reconstruction in a two-level strategy, by an outer and an inner loop. At each iteration of the outer loop, the system is linearized and this linear subproblem is solved in the inner loop with a preconditioned conjugate gradient (CG). A standard Cholesky preconditioning method based on the system matrix is compared with a matrix-free Quasi-Newton update approach, where a preconditioned matrix-vector product is computed at the beginning of every CG iteration. We also use a multigrid scheme with multi-frequency reconstruction to get a convergent rough reconstruction at a lower frequency and then refine it on a higher-resolution grid. The Cholesky preconditioning reduces the number of CG iterations by approx. 70%~85%; but the computation time for determining the system matrix for the Cholesky preconditioner is dominating, offsetting the gains of the reduction of iterations. The matrix-free preconditioning method saves approx. 30% of the computation time on average for single-frequency and multi-frequency reconstruction. For the robust multifrequency reconstruction, we test three breast-like numerical phantoms resulting in a deviation of 0.13 m/s on average in speed of sound reconstruction and a deviation of 5.4% on average in attenuation reconstruction, from the ground truth simulation.

[1]  F. Natterer,et al.  A propagation-backpropagation method for ultrasound tomography , 1995 .

[2]  M. Feit,et al.  Light propagation in graded-index optical fibers. , 1978, Applied optics.

[3]  J. Shewchuk.  An Introduction to the Conjugate Gradient Method Without the Agonizing Pain , 1994 .

[4]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[5]  A. Fichtner.  Full Seismic Waveform Modelling and Inversion , 2011 .

[6]  Lea Althaus.  On acoustic tomography using paraxial approximations , 2016 .

[7]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[8]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[9]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.

[10]  Jorge Nocedal,et al.  Algorithm 809: PREQN: Fortran 77 subroutines for preconditioning the conjugate gradient method , 2001, TOMS.

[11]  Jorge Nocedal,et al.  Automatic Preconditioning by Limited Memory Quasi-Newton Updating , 1999, SIAM J. Optim..

[12]  James F. Greenleaf,et al.  CLINICAL IMAGING WITH TRANSMISSIVE ULTRASONIC COMPUTERIZED TOMOGRAPHY , 1981 .

[13]  H. Gemmeke,et al.  3D ultrasound computer tomography for medical imaging , 2007 .

[14]  H. Trotter.  On the product of semi-groups of operators , 1959 .

[15]  Wei Xu,et al.  Efficient (Partial) Determination of Derivative Matrices via Automatic Differentiation , 2013, SIAM J. Sci. Comput..

[16]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[17]  Wei Xu,et al.  Jacobian-Free Implicit Inner-Iteration Preconditioner for Nonlinear Least Squares Problems , 2016, J. Sci. Comput..

[18]  Keiichi Morikuni,et al.  Inner-Iteration Krylov Subspace Methods for Least Squares Problems , 2013, SIAM J. Matrix Anal. Appl..