A Discrete Dipole Approximation Solver Based on the COCG-FFT Algorithm and Its Application to Microwave Breast Imaging

Abstract

We introduce the discrete dipole approximation (DDA) for efficiently calculating the two-dimensional electric field distribution for our microwave tomographic breast imaging system. For iterative inverse problems such as microwave tomography, the forward field computation is the time limiting step. In this paper, the two-dimensional algorithm is derived and formulated such that the iterative conjugate orthogonal conjugate gradient (COCG) method can be used for efficiently solving the forward problem. We have also optimized the matrix-vector multiplication step by formulating the problem such that the nondiagonal portion of the matrix used to compute the dipole moments is block-Toeplitz. The computation costs for multiplying the block matrices times a vector can be dramatically accelerated by expanding each Toeplitz matrix to a circulant matrix for which the convolution theorem is applied for fast computation utilizing the fast Fourier transform (FFT). The results demonstrate that this formulation is accurate and efficient. In this work, the computation times for the direct solvers, the iterative solver (COCG), and the iterative solver using the fast Fourier transform (COCG-FFT) are compared with the best performance achieved using the iterative solver (COCG-FFT) in C++. Utilizing this formulation provides a computationally efficient building block for developing a low cost and fast breast imaging system to serve under-resourced populations.

An Overview of Microwave Imaging for Breast Tumor Detection

“…Contrast agent used was gold nanoparticle that could enhance the dielectric contrast between malignant and fibrous tissues. In 2019, Discrete Dipole Approximation (DDA) was developed for efficiently calculating the two-dimensional electric field distribution which helps to accelerate the forward problem computations [164]. In 2019, efforts were also made to replace the VNA for the first time in tomography by developing a 4-channel VNA based on software defined radio technology [165].…”

Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix

“…The 2D DDA forward solver and the corresponding Jacobian matrix are embedded in the log-magnitude and phase format reconstruction algorithm. The fundamental aspects of the reconstruction algorithm and the 2D DDA were previously investigated by the authors [ 27 , 42 , 51 , 52 ] and are employed for this study.…”

“…More recently, the discrete dipole approximation (DDA) has been introduced as a means of dramatically reducing the forward solution computation time and the associated memory requirements [ 42 ]. This has the potential to reduce the forward solution time of a 2D imaging problem by an order of magnitude or more.…”

Recent Microwave Technologies

“…The detailed derivation of the vector form of the Bessel beam is illustrated below. State-of-the-art generation techniques of OAM wave using different antenna types from GHz to THz [54][55][56][57][58][59].…”

“…While our 2D FEM-based approach can usually produce images in roughly 5 minutes, we have explored ways to further reduce the time. In particular, we have been experimenting with the discrete dipole approximation (DDA) to improve efficiency [59]. This technique is generally not suitable when there are large, high contrast scatterers in the field of viewespecially other bulky antennas; however, because our monopole antennas only slightly perturb the fields, and almost not at all when submerged in a lossy bath, this algorithm is ideal for our approach.…”