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.
Highlights
The mortality rate due to the breast cancer in women worldwide has led numerous research groups to investigate early diagnosis programs
We focus on microwave tomography (MWT) which has been tested in several breast imaging clinical trials and has provided relevant diagnostic information regarding diagnosis of cancer and monitoring of tumor progression during neoadjuvant chemotherapy [13, 14]
While the interpretive code struggles computation time-wise with constructs such as loops, it contains highly optimized matrix operations which can often overcome such disadvantages. These examinations show that the computation time for the 2D-discrete dipole approximation (DDA) is significantly decreased in the conjugate orthogonal conjugate gradient method (COCG)-FFT approach and that the best performance is achieved in C++ using an open source C++ package, FFTW, for fast Fourier transform calculations [37]
Summary
The mortality rate due to the breast cancer in women worldwide has led numerous research groups to investigate early diagnosis programs. For most numerical techniques, solving the 3D imaging problem can be computationally expensive and requires use of multiprocessor computers working over many hours to even days to generate single images [16, 22] While these 3D efforts are useful and necessary to advance the science of microwave imaging, the practical barriers to implementation, including measurement data costs and computation time, have greatly hindered its translation into the clinic and limited them primarily to simulation studies. Largely due to continuing computer efficiency advances, 2D techniques are poised to be viable alternatives for conventional modalities in underresourced settings where cost and portability are significant concerns In this context, reducing memory requirements and computation time for the 2D algorithm is an important concern. These examinations show that the computation time for the 2D-DDA is significantly decreased in the COCG-FFT approach and that the best performance is achieved in C++ using an open source C++ package, FFTW, for fast Fourier transform calculations [37]
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