Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography

Abstract

The inverse problem of electrical impedance tomography is severely ill-posed, meaning that, only limited information about the conductivity can in practice be recovered from boundary measurements of electric current and voltage. Recently it was shown that a simple monotonicity property of the related Neumann-to-Dirichlet map can be used to characterize shapes of inhomogeneities in a known background conductivity. In this paper we formulate a monotonicity-based shape reconstruction scheme that applies to approximative measurement models, and regularizes against noise and modelling error. We demonstrate that for admissible choices of regularization parameters the inhomogeneities are detected, and under reasonable assumptions, asymptotically exactly characterized. Moreover, we rigorously associate this result with the complete electrode model, and describe how a computationally cheap monotonicity-based reconstruction algorithm can be implemented. Numerical reconstructions from both simulated and real-life measurement data are presented.

Appendices

Appendix A: A lemma on the convergence of infima/suprema

Lemma 2

Let J be an arbitrary index set and $\{a_j\}_{j\in J}, \{a_j(h)\}_{j\in J} \subset \mathbb {R}$, $h > 0$, be sequences such that $\inf _{j\in J}a_j>-\infty$ and

$$\begin{aligned} \lim _{h\rightarrow 0}\sup _{j\in J}|a_j - a_j(h)| = 0. \end{aligned}$$

Denoting $a = \inf _{j\in J} a_j$ and $a(h) = \inf _{j\in J} a_j(h)$ we have

$$\begin{aligned} \lim _{h\rightarrow 0} a(h) = a. \end{aligned}$$

(51)

Proof

Let us first show that the limit in (51) exists. Given an arbitrary $\varepsilon > 0$, there exists an $h_{\varepsilon } > 0$ such that $\sup _{j\in J}|a_j - a_j(h)| \le \varepsilon /2$ for all $h \in (0,h_{\varepsilon })$. Let $h, h' \in (0,h_{\varepsilon })$ then $\sup _{j\in J}|a_j(h)-a_j(h')|\le \varepsilon$, and fix a sequence $\{j(k)\}_{k=1}^\infty \subseteq J$ such that $a_{j(k)}(h)$ converges to a(h). Hence

$$\begin{aligned} a(h') \le \liminf _{k\rightarrow \infty } a_{j(k)}(h') \le \liminf _{k\rightarrow \infty } a_{j(k)}(h) + \varepsilon = a(h) + \varepsilon . \end{aligned}$$

By symmetry with respect to h and $h'$, it follows that $\{a(h)\}_{h>0}$ is a Cauchy sequence.

It still remains to show that the limit coincides with a. For any $\varepsilon > 0$, there exists $j_{\varepsilon }\in J$ and $h_{\varepsilon } > 0$ such that $|a_{j_{\varepsilon }} - a|\le \varepsilon /2$ and $\sup _{j\in J}|a_j - a_j(h)| \le \varepsilon /2$ for $h\in (0,h_{\varepsilon })$, respectively. Thus for $h\in (0,h_{\varepsilon })$

$$\begin{aligned} a(h) \le a_{j_{\varepsilon }}(h) \le a_{j_{\varepsilon }} + \varepsilon /2 \le a + \varepsilon . \end{aligned}$$

For $h\in (0,h_{\varepsilon })$ pick $j_{\varepsilon }'$ such that $|a_{j_{\varepsilon }'}(h)-a(h)|\le \varepsilon /2$ then

$$\begin{aligned} a\le a_{j_{\varepsilon }'} \le a_{j_{\varepsilon }'}(h) + \varepsilon /2 \le a(h)+\varepsilon . \end{aligned}$$

Altogether we have shown for any $\varepsilon > 0$ that $|a(h) - a| \le \varepsilon$ for $h\in (0,h_{\varepsilon })$. $\square$

Appendix B: Linearization of the CEM and the CM

Proposition 5

The operators $\Lambda (\gamma ) \in \mathcal {L}(L_\diamond ^2(\partial \varOmega ))$ and $R(\gamma )\in \mathcal {L}(\mathbb {R}_\diamond ^k)$ are analytic in $\gamma \in L_+^\infty (\varOmega )$. In particular, they are infinitely many times Fréchet differentiable. Furthermore, if $\eta$ is compactly supported in $\varOmega$, then the boundary value problems

$$\begin{aligned}&\left\{ \begin{array}{ll} \displaystyle {\nabla \cdot (\gamma \nabla u') = -\nabla \cdot (\eta \nabla u ) \quad } &{}\mathrm{in}\;\; \varOmega , \\ {\displaystyle {\nu \cdot \gamma \nabla u'} = 0 }\quad &{}\mathrm{on}\;\;\partial \varOmega , \end{array} \right. \end{aligned}$$

(52)

$$\begin{aligned}&\left\{ \begin{array}{ll} \displaystyle {\nabla \cdot (\gamma \nabla v') = -\nabla \cdot (\eta \nabla v) \quad } &{}\mathrm{in}\;\; \varOmega , \\ {\displaystyle {\nu \cdot \gamma \nabla v'} = 0 }\quad &{}\mathrm{on}\;\;{\partial \varOmega }{\setminus } \bigcup _{j=1}^k \overline{E_j},\\ {\displaystyle v' + z{\nu \cdot \gamma \nabla v'} = V'_j } \quad &{}\mathrm{on}\;\; E_j, \\ {\displaystyle \int _{E_j}\nu \cdot \gamma \nabla v'\, dS = 0}, \quad &{} j=1,2,\ldots k, \\ \end{array} \right. \end{aligned}$$

(53)

uniquely determine the Fréchet derivatives via

$$\begin{aligned} (\Lambda '(\gamma )\eta )f = u'|_{\partial \Omega }, \qquad (R'(\gamma )\eta )I = V', \end{aligned}$$

respectively. Above u and (v, V) are the unique weak solutions of (2) and (29), respectively.

Proof

For clarity, we only consider the CEM case as the CM can be handled analogously [6]. Given $(v,V)\in H^1(\varOmega )\oplus \mathbb {R}_\diamond ^k$ and $\eta \in L_+^\infty (\varOmega )$, the variational problem

$$\begin{aligned} \int _\varOmega \gamma \nabla v' \cdot \nabla w \, dx + \sum _{j=1}^k \int _{E_j}\frac{1}{z}(v'-V_j')(w-W_j)\,dS = -\int _\varOmega \eta \nabla v \cdot \nabla w \,dx \end{aligned}$$

(54)

for all $(w,W)\in H^1(\varOmega ) \oplus \mathbb {R}_\diamond ^k$, is uniquely solvable. Moreover, if (v, V) weakly solves (29), then $V' = (R'(\gamma )\eta )I$ [32]. Clearly, if $\eta$ is compactly supported, the right-hand side of (54) does not induce any boundary terms and hence $(v',V')$ satisfies (53).

Define the mapping

$$\begin{aligned} D = D(\eta ):\mathbb {R}_\diamond ^k \rightarrow \mathbb {R}_\diamond ^k, \quad V \mapsto V' \end{aligned}$$

as the solution operator to (54). Consider the expansion

$$\begin{aligned} V(\gamma + \eta ) = V(\gamma ) + \tilde{V}(\eta ), \end{aligned}$$

where we denote $V(\gamma ) = R(\gamma )I$ and $V(\gamma +\eta ) = R(\gamma + \eta )I$. A direct calculation using the variational formulation with the associated internal potentials reveals

$$\begin{aligned} \tilde{V}(\eta ) = D(\eta )V(\gamma +\eta ) = D(\eta )\tilde{V}(\eta ) + D(\eta )V(\gamma ). \end{aligned}$$

As $\Vert D(\eta )\Vert \le C\Vert \eta \Vert _{L^\infty (\varOmega )}$, the associated Neumann-series converges for small enough $\eta$. Consequently,

$$\begin{aligned} V(\gamma +\eta ) = V(\gamma ) + \tilde{V}(\eta ) = V(\gamma ) + (\mathop {Id }- D(\eta ))^{-1} D(\eta )V(\gamma ) = \sum _{m=0}^\infty D(\eta )^m V(\gamma ). \end{aligned}$$

(55)

Different order Fréchet derivatives can be inductively derived using (55) and the fact that $D(\eta )$ is linear. $\square$