Direct regularized reconstruction for the three-dimensional Calderón problem

Knudsen Kim; Rasmussen Aksel Kaastrup

doi:10.3934/ipi.2022002

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2022, Volume 16, Issue 4: 871-894. Doi: 10.3934/ipi.2022002

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Direct regularized reconstruction for the three-dimensional Calderón problem

Knudsen Kim and
Rasmussen Aksel Kaastrup^,

Technical University of Denmark, Department of Applied Mathematics and Computer Science, DK-2800 Kgs. Lyngby, Denmark

^* Corresponding author
^* Corresponding author

Received: May 31, 2021

Revised: October 31, 2021

Early access: January 2022

Published: August 2022

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Abstract

Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calderón problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data illustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.

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Mathematics Subject Classification: Primary: 35R30, 65J20; Secondary: 65N21.

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Figure 1. The piecewise constant heart-lungs phantom in a threedimensional view (A), and in the planar cross section x³ = 0 (B)

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Figure 2. Cross sections $ (x^3 = 0) $ of reconstructions using the regularized reconstruction algorithm with different choices of truncation radius $ M $, $ K = 12 $ and $ |\zeta(\xi)| = \frac{1}{4}M^{3/2} $. There is no added noise

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Figure 3. Cross sections $ (x^3 = 0) $ of reconstructions using the regularized reconstruction algorithm on noisy Dirichlet-to-Neumann maps. The noise levels correspond to relative noise levels $ \varepsilon \approx 0.1\% $ with $ \mathrm{SNR} = 12\cdot 10^3 $ (top left), $ \varepsilon \approx 0.01\% $ with $ \mathrm{SNR} = 123\cdot 10^3 $ (top right), $ \varepsilon \approx 0.001\% $ with $ \mathrm{SNR} = 1172\cdot 10^3 $ (bottom left) and $ \varepsilon \approx 0.0001\% $ with $ \mathrm{SNR} = 11299\cdot 10^3 $ (bottom right). The parameters used are $ K = 11 $ and $ |\zeta(\xi)| = \frac{1}{3\sqrt{2}}M^{3/2} $

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Figure 4. Regularized reconstruction using noisy Dirichlet-to-Neumann maps with $ \varepsilon = 10^{-2} $, which corresponds to approximately $ 1\% $ relative noise and $ \mathrm{SNR} = 1.17\cdot 10^3 $. Plot (A) shows the cross sections $ x^3 = 0 $, $ x^2 = -0.6 $, $ x^2 = {-0.05} $ and $ x^2 = 0.6 $, whereas plot (B) shows the plane corresponding to $ x^3 = 0 $. The parameters used are $ M = 9 $, $ K = 11 $ and $ |\zeta(\xi)| = \frac{1}{3\sqrt{2}}M^{3/2} $

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Figure 5. The truncation radii as predicted by theory $ M = (-1/11\log(\varepsilon))^{-1/p} $ for $ p = 3/2 $, and the chosen truncation radii for the noisy reconstructions of Figure 3 and 4

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Table 1. Summary of piecewise constant heart-lungs phantom consisting of three inclusions

Inclusion	Center	Radii	Axes	Conductivity
Ball	$ (-0.09,-0.55,0) $	$ r = 0.273 $		2
Left spheroid	$ 0.55(-\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0) $	$ r_1 = 0.468 $, $ r_2 = 0.234 $, $ r_3 = 0.234 $	$(\cos(\frac{5\pi}{12}),\sin(\frac{5\pi}{12}),0)$, $(-\sin(\frac{5\pi}{12}),\cos\frac{5\pi}{12}),0)$, $(0,0,1)$	0.5
Right spheroid	$0.45(\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0)$	$r_1 = 0.546$, $r_2 = 0.273$, $r_3 = 0.273$	$(\cos(\frac{5\pi}{12}),-\sin(\frac{5\pi}{12}),0)$, $(\sin(\frac{5\pi}{12}), \cos(\frac{5\pi}{12}), 0)$, $(0,0,1)$	0.5

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