Andrew J. Hesford

Fast inverse scattering solutions using the distorted Born iterative method and the multilevel fast multipole algorithm

The distorted Born iterative method (DBIM) computes iterative solutions to nonlinear inverse scattering problems through successive linear approximations. By decomposing the scattered field into a superposition of scattering by an inhomogeneous background and by a material perturbation, large or high-contrast variations in medium properties can be imaged through iterations that are each subject to the distorted Born approximation. However, the need to repeatedly compute forward solutions still imposes a very heavy computational burden. To ameliorate this problem, the multilevel fast multipole algorithm (MLFMA) has been applied as a forward solver within the DBIM. The MLFMA computes forward solutions in linear time for volumetric scatterers. The typically regular distribution and shape of scattering elements in the inverse scattering problem allow the method to take advantage of data redundancy and reduce the computational demands of the normally expensive MLFMA setup. Additional benefits are gained by employing Kaczmarz-like iterations, where partial measurements are used to accelerate convergence. Numerical results demonstrate both the efficiency of the forward solver and the successful application of the inverse method to imaging problems with dimensions in the neighborhood of ten wavelengths.

On Preconditioning and the Eigensystems of Electromagnetic Radiation Problems

A formulation of the method of moments (MoM) impedance matrix is presented that facilitates discussion of the behavior of its eigenvalues and eigenvectors. This provides insight into the difficulties of producing iterative solutions to electromagnetic radiation problems, which typically involve nonuniform meshes. Based on this analysis, a localized self-box inclusion (SBI) preconditioner is developed to overcome the aforementioned issues. Numerical results are shown using a parallel multilevel fast multipole algorithm (MLFMA) library, coupled with an implementation of the SBI preconditioner. Using these parallel libraries allows the solution of very large problems, due to both excessive size and poor conditioning. A model of an XM antenna, mounted atop an automobile above a very large ground plane, establishes the effectiveness of these methods for more than 3.5 million unknowns.

A frequency-domain formulation of the Fréchet derivative to exploit the inherent parallelism of the distorted Born iterative method

With its consideration for nonlinear scattering phenomena, the distorted Born iterative method (DBIM) is known to provide images superior to those of linear tomographic methods. However, the complexity involved with the production of superior images has prevented DBIM from overtaking simpler imaging schemes in commercial applications. The iterative process and need to solve the forward-scattering problem multiple times make DBIM a slow algorithm compared to diffraction tomography. Fortunately, as computer prices continue to decline, it is becoming easier to assemble large, distributed computer clusters from low-cost personal computer systems. These are well-suited to DBIM inversions, and offer great promise in accelerating the method. Traditional frequency-domain DBIM formulations produce an image by inverting the Fréchet derivative. If the derivative is treated as a matrix, it is costly to construct and awkward to invert on distributed computer systems. This paper presents an interpretation of the Fréchet derivative that is ideal for parallel-computing applications. As a bonus, this formulation reduces the storage requirements of DBIM implementations, making it possible to invert larger problems on a fixed system.

The fast multipole method and Fourier convolution for the solution of acoustic scattering on regular volumetric grids

The fast multipole method (FMM) is applied to the solution of large-scale, three-dimensional acoustic scattering problems involving inhomogeneous objects defined on a regular grid. The grid arrangement is especially well suited to applications in which the scattering geometry is not known a priori and is reconstructed on a regular grid using iterative inverse scattering algorithms or other imaging techniques. The regular structure of unknown scattering elements facilitates a dramatic reduction in the amount of storage and computation required for the FMM, both of which scale linearly with the number of scattering elements. In particular, the use of fast Fourier transforms to compute Greens function convolutions required for neighboring interactions lowers the often-significant cost of finest-level FMM computations and helps mitigate the dependence of FMM cost on finest-level box size. Numerical results demonstrate the efficiency of the composite method as the number of scattering elements in each finest-level box is increased.

A mesh-free approach to acoustic scattering from multiple spheres nested inside a large sphere by using diagonal translation operators

A multiple-scattering approach is presented to compute the solution of the Helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere. The solution is represented in terms of partial-wave expansions, and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces. This approach yields high-order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape. Calculations are accelerated by using diagonal translation operators to compute the interactions between spheres when the operators are numerically stable. Numerical results are presented to demonstrate the accuracy and efficiency of the method.

Reduced-rank approximations to the far-field transform in the gridded fast multipole method

The fast multipole method (FMM) has been shown to have a reduced computational dependence on the size of finest-level groups of elements when the elements are positioned on a regular grid and FFT convolution is used to represent neighboring interactions. However, transformations between plane-wave expansions used for FMM interactions and pressure distributions used for neighboring interactions remain significant contributors to the cost of FMM computations when finest-level groups are large. The transformation operators, which are forward and inverse Fourier transforms with the wave space confined to the unit sphere, are smooth and well approximated using reduced-rank decompositions that further reduce the computational dependence of the FMM on finest-level group size. The adaptive cross approximation (ACA) is selected to represent the forward and adjoint far-field transformation operators required by the FMM. However, the actual error of the ACA is found to be greater than that predicted using traditional estimates, and the ACA generally performs worse than the approximation resulting from a truncated singular-value decomposition (SVD). To overcome these issues while avoiding the cost of a full-scale SVD, the ACA is employed with more stringent accuracy demands and recompressed using a reduced, truncated SVD. The results show a greatly reduced approximation error that performs comparably to the full-scale truncated SVD without degrading the asymptotic computational efficiency associated with ACA matrix assembly.

Acoustic scattering by arbitrary distributions of disjoint, homogeneous cylinders or spheres

A T-matrix formulation is presented to compute acoustic scattering from arbitrary, disjoint distributions of cylinders or spheres, each with arbitrary, uniform acoustic properties. The generalized approach exploits the similarities in these scattering problems to present a single system of equations that is easily specialized to cylindrical or spherical scatterers. By employing field expansions based on orthogonal harmonic functions, continuity of pressure and normal particle velocity are directly enforced at each scatterer using diagonal, analytic expressions to eliminate the need for integral equations. The effect of a cylinder or sphere that encloses all other scatterers is simulated with an outer iterative procedure that decouples the inner-object solution from the effect of the enclosing object to improve computational efficiency when interactions among the interior objects are significant. Numerical results establish the validity and efficiency of the outer iteration procedure for nested objects. Two- and three-dimensional methods that employ this outer iteration are used to measure and characterize the accuracy of two-dimensional approximations to three-dimensional scattering of elevation-focused beams.

Methods for Forward and Inverse Scattering in Ultrasound Tomography

Ultrasonic computed tomography (UCT) is a potentially useful technique that has been explored for decades in the context of medical imaging. UCT can provide quantitative images of acoustical parameters such as speed of sound, attenuation, and density from measurements of pressure fields. Throughout the years, several algorithms that rely on different wave propagation models have been developed. In this chapter, the fundamentals of forward and inverse solvers for ultrasonic tomography will be described.

Comparison of temporal and spectral scattering methods using acoustically large breast models derived from magnetic resonance images.

Accurate and efficient modeling of ultrasound propagation through realistic tissue models is important to many aspects of clinical ultrasound imaging. Simplified problems with known solutions are often used to study and validate numerical methods. Greater confidence in a time-domain k-space method and a frequency-domain fast multipole method is established in this paper by analyzing results for realistic models of the human breast. Models of breast tissue were produced by segmenting magnetic resonance images of ex vivo specimens into seven distinct tissue types. After confirming with histologic analysis by pathologists that the model structures mimicked in vivo breast, the tissue types were mapped to variations in sound speed and acoustic absorption. Calculations of acoustic scattering by the resulting model were performed on massively parallel supercomputer clusters using parallel implementations of the k-space method and the fast multipole method. The efficient use of these resources was confirmed by parallel efficiency and scalability studies using large-scale, realistic tissue models. Comparisons between the temporal and spectral results were performed in representative planes by Fourier transforming the temporal results. An RMS field error less than 3% throughout the model volume confirms the accuracy of the methods for modeling ultrasound propagation through human breast.

Validation of time-domain and frequency-domain calculations of acoustic propagation from breast models derived from magnetic resonance images

Magnetic resonance images with an isotropic resolution of 200 microns were collected for two human breast specimens. The images were interpolated to achieve a resolution of 50 microns and segmented to produce images of skin, fat, muscle, ductal structures, and connective tissues in consultation with a breast pathologist. The images were then mapped to acoustic parameters of sound speed, absorption and density. Calculations of acoustic propagation of fields radiated by point sources outside of the specimens were performed using the k-space finite-difference time-domain method and the frequency-domain fast multipole method. Time-domain k-space results were Fourier transformed and the 3-MHz component was compared to 3-MHz frequency-domain calculations. For the first model, measuring 1180 × 1190 × 290 voxels, the two methods were found to agree in a representative coronal slice to within 5.0% (RMS). The second specimen, comprising 1350 × 1170 × 790 voxels, yielded temporal and frequency-domain results that ag...