American Institute of Mathematical Sciences

February  2019, 13(1): 93-116. doi: 10.3934/ipi.2019006

The regularized monotonicity method: Detecting irregular indefinite inclusions

 1 Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark 2 Eniram Oy (A Wärtsilä company), Itälahdenkatu 22a, 00210 Helsinki, Finland

Received  January 2018 Revised  August 2018 Published  December 2018

Fund Project: This research is funded by grant 4002-00123 Improved Impedance Tomography with Hybrid Data from The Danish Council for Independent Research | Natural Sciences.

In inclusion detection in electrical impedance tomography, the support of perturbations (inclusion) from a known background conductivity is typically reconstructed from idealized continuum data modelled by a Neumann-to-Dirichlet map. Only few reconstruction methods apply when detecting indefinite inclusions, where the conductivity distribution has both more and less conductive parts relative to the background conductivity; one such method is the monotonicity method of Harrach, Seo, and Ullrich [17,15]. We formulate the method for irregular indefinite inclusions, meaning that we make no regularity assumptions on the conductivity perturbations nor on the inclusion boundaries. We show, provided that the perturbations are bounded away from zero, that the outer support of the positive and negative parts of the inclusions can be reconstructed independently. Moreover, we formulate a regularization scheme that applies to a class of approximative measurement models, including the Complete Electrode Model, hence making the method robust against modelling error and noise. In particular, we demonstrate that for a convergent family of approximative models there exists a sequence of regularization parameters such that the outer shape of the inclusions is asymptotically exactly characterized. Finally, a peeling-type reconstruction algorithm is presented and, for the first time in literature, numerical examples of monotonicity reconstructions for indefinite inclusions are presented.

Citation: Henrik Garde, Stratos Staboulis. The regularized monotonicity method: Detecting irregular indefinite inclusions. Inverse Problems & Imaging, 2019, 13 (1) : 93-116. doi: 10.3934/ipi.2019006
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References:
Illustration of case (a) in the proof of Theorem 2.3.(ⅱ).
Profile of a conductivity distribution $\gamma$ which does not satisfy the assumption $\sup({{\kappa _ - }})<\inf(\gamma_0)$
(a) Two dimensional numerical phantom with positive part ${{D_ + }}$ (square and pentagon) and negative part ${{D_ - }}$ (ball). (b) Reconstruction of ${{D_ + }}$ from noiseless datum with $\alpha_0 = 6.56\times 10^{-4}$. (c) Reconstruction of ${{D_ - }}$ from noiseless datum with $\alpha_0 = 2.36\times 10^{-3}$. (d) Reconstruction of ${{D_ + }}$ with $0.5\%$ noise and $\alpha_0 = 5.00\times 10^{-3}$. (e) Reconstruction of ${{D_ - }}$ with $0.5\%$ noise and $\alpha_0 = 2.30\times 10^{-3}$
(a) Two dimensional numerical phantom with positive part ${{D_ + }}$ (wedge) and negative part ${{D_ - }}$ (ball). (b) Reconstruction of ${{D_ + }}$ from noiseless datum with $\alpha_0 = 9.00\times 10^{-4}$. (c) Reconstruction of ${{D_ - }}$ from noiseless datum with $\alpha_0 = 6.72\times 10^{-4}$. (d) Reconstruction of ${{D_ + }}$ with $0.5\%$ noise and $\alpha_0 = 4.20\times 10^{-3}$. (e) Reconstruction of ${{D_ - }}$ with $0.5\%$ noise and $\alpha_0 = 3.26\times 10^{-3}$
(a) Three dimensional numerical phantom with positive part ${{D_ + }}$ (ball) and negative part ${{D_ - }}$ (L-shape). (b) Reconstruction of ${{D_ + }}$ from noiseless datum with $\alpha_0 = 7.50\times 10^{-5}$. (c) Reconstruction of ${{D_ - }}$ from noiseless datum with $\alpha_0 = 2.90\times 10^{-4}$. (d) Reconstruction of ${{D_ + }}$ with $0.5\%$ noise and $\alpha_0 = 2.40\times 10^{-4}$. (e) Reconstruction of ${{D_ - }}$ with $0.5\%$ noise and $\alpha_0 = 2.50\times 10^{-4}$
For $k = 8$, $16$, and $32$ electrodes of size $\pi/k$, upper bounds of reconstructions of ${{D_ + }}$ are shown, using regularization parameter $\alpha_0 = 0$. ${{D_ + }}$ is the ball outlined on the right side of the domain and ${{D_ - }}$ is on the left
). $h = 2\pi/k$ is the corresponding maximal extended electrode diameter from [20] and [9,Theorems 2 and 3]">Figure 7.  For $k = 4, 5, \dots, 64$ electrodes of size $\pi/k$, the distance $d_k$ from $\partial\Omega$ to upper bounds of reconstructions of ${{D_ + }}$ is plotted (cf. Figure 6). $h = 2\pi/k$ is the corresponding maximal extended electrode diameter from [20] and [9,Theorems 2 and 3]
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