FFT-Accelerated Transformation-Domain Image Reconstruction for Electrical Impedance Tomography

Ziheng Zhou, Maokun Li, Xudong Chen, Zhun Wei, Ke Zhang, Zhimeng Xu, Zhizhang Chen

Abstract: Electrical impedance tomography (EIT) is a promising imaging technique that recovers the conductivity distribution inside a domain from noninvasive electrical measurements on the boundary. In this work, to accelerate the solving of EIT problems in arbitrarily shaped domains and with a large number of unknowns ( ${N}$ ), we propose a fast integral-equation-based inversion method. First, by applying the Schwarz–Christoffel (SC) conformal transformation, we map the arbitrarily shaped domain of an EIT problem to a rectangle, on which Green’s function can be derived analytically in Fourier series representation. Leveraging such a mathematical structure of Green’s function, we then propose a fast Fourier transform (FFT)-based algorithm to compute the multiplication of the associated impedance matrix with vectors, where the time complexity is substantially reduced from ${O}$ ( ${N}^{2}$ ) to ${O}$ ( $N\log (N)$ ) and the memory complexity is reduced from ${O}$ ( ${N}^{2}$ ) to ${O}$ ( ${N}$ ). Using the contrast source inversion method along with the accelerated matrix–vector multiplications, the conductivity profile can be reconstructed much more efficiently in the rectangular transformation domain. As validated by numerical and experimental tests, the proposed FFT-accelerated transformation-domain EIT image reconstruction method can offer significantly reduced computational and memory complexity without sacrificing the image quality.