Tailored interior and boundary parameter transformations for iterative inversion in electrical impedance tomography

Electrical impedance tomography is a non-invasive method for imaging the electrical conductivity of an object from electrode measurements on its surface. The underlying mathematical problem is highly nonlinear, severely ill-posed, and several model parameters are usually not known accurately. Despite the strong nonlinearity, iterative Newton-type methods are widely used to tackle the problem numerically. This work presents and analyzes tailored transformations for the conductivity and for electrode parameters which are favourable in two regards: they remove the constrainedness of the unknown parameters and simultaneously decrease the nonlinearity of the underlying problem. We study the impact of various transformations on the nonlinearity of the problem and demonstrate improved speed of convergence for Newton-type methods while avoiding local minima in the solution space. The presented transformations can conveniently be incorporated into existing iterative solvers as they improve stability and do not require hand-tuned regularization parameters or line-search strategies, thereby bridging a gap between a variety of established conductivity estimation methods and practical applications.


Background
The primary mathematical problem behind electrical impedance tomography (EIT) is the recovery of the spatial conductivity distribution inside an object from electrode measurements on its surface, called inverse conductivity problem (ICP). To solve the ICP numerically, the boundary configuration of the measurement setup, i.e. the surface shape, the electrode sizes and positions and their contact impedances, must be known. In contrast to other imaging modalities such as computed tomography or magnetic resonance imaging, the ICP is highly nonlinear, highly sensitive to measurement and modelling errors, and the boundary configuration is not known accurately. As a consequence, methods have been developed to solve an extended ICP, that is, to recover the boundary configuration along with the conductivity from the electrode measurements [1][2][3]. All methods rely on the Fréchet differentiability of the forward operator which maps the conductivity and the model parameters to the observed current-to-voltage behaviour at the electrodes.
The forward operator is nonlinear in all its input parameters: the current-to-voltage behaviour depends nonlinearly on the conductivity, the electrode contact impedances, their size and position, and the boundary shape. Moreover, all parameters are bounded by physical restrictions: the conductivity and the electrode contact impedances are strictly positive, the electrodes have finite positive sizes, and their positions are restricted such that they do not touch or overlap.

Motivation for using transformations
At first glance, it might not be intuitive to solve the extended ICP with Newton-type methods, which solve parameter problems locally by iterative linearization and may have pitfalls when parameters are constrained. In fact, a wide variety of techniques was developed to avoid high linearization errors, out-of-bounds parameters and oscillatory solutions. These include line-search strategies (e.g. in [2, section 7.2]), 'hand tuning' of regularization parameters for the Tikhonov-Phillips regularization or in the Bayesian framework (e.g. in [4, algorithm 1]), inexact Newton methods [5] or-with more computational effort-cross-validation [6] and L-curve techniques [7].
A popular method for lifting the positivity constraint of the conductivity is to consider the log-conductivity, that is, to apply Newton's method to the logarithm of the conductivity, which is unconstrained. The inverse (exponential) transformation guarantees the positivity of the conductivity after each Newton update. This way, measurement, modelling or linearization errors cannot lead to a violation of the physical model.
Moreover, it was observed that parameter transformations can increase the speed of convergence of Newton-type methods. In the most trivial case of a spatially constant conductivity and in the absence of contact impedances, the electrode measurements are linear in the resistivity, the reciprocal of the conductivity. This means that Newton's method converges after one iteration, and there are stable least-squares estimates in case of measurement errors 1 [8,9]. However, the linearity is violated for non-constant conductivities, and the reciprocal again has a positivity constraint. This sparked the discussion whether conductivity, logconductivity or resistivity should be recovered (e.g. in [10]), and lead to the investigation of conductivity transformations (e.g. in [9,11,12]). Recently, theoretical foundations for the properties of a logarithmic EIT forward operator have been derived, and its advantage in terms of forward accuracy and linearization error for one-step reconstruction has been highlighted numerically, in [13]. Moreover, a transformation of the electrode contact impedances was also considered, while the boundary geometry was assumed to be known.
Moreover, using the Fréchet-differentiability of an extended forward operator with respect to the electrode configuration provides a natural extension for correcting boundary model uncertainties and was studied extensively both regarding theoretical properties [1][2][3] and in practical applications, where it was found to be superior to other techniques for handling modelling errors [14]. Thus, it is natural to investigate whether applying transformations is also beneficial for recovering boundary model parameters.
The results of this work are focused at (pointwise) conductivity and boundary parameter transformations, but can be conveniently combined with other well-established methods for improving the stability and speed of convergence of the inverse conductivity problem, such as sensitivity-based domain discretizations (investigated theoretically and numerically for various models e.g. in [15][16][17][18][19][20]), which are not addressed explicitly here.

Scope and outline
In this work, we pick up and extend the analysis of conductivity transformations from [9]. To that end, we present transformations yielding one-sided ('log-type') and two-sided (sigmoidal) bounds to the conductivity in section 2.6. Then, we investigate the impact of transformations on the nonlinearity for constant and certain non-constant conductivities in sections 3.1 and 3.2 with the help of highly accurate semi-analytic evaluations of the forward operator.
Moreover, we carry over the transformation approach to the boundary parameters in section 2.7 and investigate its impact on the nonlinearity in section 3.3. The result is an unconstrained parameter space for the extended ICP which is favorable for Newton-type methods. It can moreover be used to control the trade-off between conductivity and boundary changes during simultaneous reconstruction, which allows the user to incorporate accuracy assumptions for the boundary configuration. Numerical results for conductivity and boundary parameter transformations within an inexact Newton reconstruction framework are shown in section 4.
In summary, the purpose of this work is to give further insight into the nonlinearity of the ICP and to provide an easy-to-use extension to existing Newton-type methods for EIT applications. MATLAB examples are provided at www.math.kit.edu/ianm3/~winkler/de.

Properties of the complete electrode model
To maintain consistent notation, we briefly restate relevant properties of the ICP and the underlying complete electrode model (CEM). The CEM describes measurements on a finite set of electrodes attached to the object surface and models contact impedances on the electrodedomain-interface. It is generally assumed to be the most realistic among the commonly used electrode models and was verified to reproduce data from tank experiments to measurement precision [21,22].

Definition of the complete electrode model
For a compact, simply connected and piecewise smooth Lipschitz domain Ω ∈ R n , n ∈ {2, 3}, we identify a set of L ∈ N 2 simply connected and pairwise separated electrodes E 1 , . . . , E L with the subset on ∂Ω that they cover, that is, closed and simply connected on ∂Ω, For n = 2, we assume w.l.o.g. that the electrodes are ordered consecutively along the boundary. We briefly restate the definition of the CEM in our notation and refer to [23, section 2.1.3.] for a more detailed introduction of the model and its weak formulation using the same notation.

Definition 2.1 (Complete electrode model and measurement operator).
Let Ω ⊂ R n and E 1 , . . . , E L ⊂ Ω as above. Further, denote by σ ∈ L ∞ + (Ω) a conductivity on Ω and by z 1 , . . . , z L ∈ R >0 a set of contact impedances. The set of equations −∇ · (σ∇u) = 0 on Ω, (1) where f = u| ∂Ω and j ν = ν · σ∇u, is called the CEM. Herein, ν denotes the boundary normal on ∂Ω. For any current vector there is a unique u ∈ H 1 (Ω) and a unique potential vector U = (U 1 , . . . , U L ) ∈ R L satisfying (1)-(4), and vice versa. The current-to-potential map is thus well-defined and one-to-one and moreover bounded, linear, and symmetric; see [22]. We call the map the forward operator of the CEM. For M ∈ N, a measurement pattern is defined as and its corresponding measurement data is defined as The operator The ICP in this setting reads Problem statement 2.2 (Inverse conductivity problem for the complete electrode model). Given a current frame I and corresponding noisy potential measurements

Differentiability and Newton-type methods for the complete electrode model
Newton-type methods in EIT rely on the Fréchet differentiability of F I [24,25]. For a single current vector I, the Fréchet derivative of F I at a conductivity σ ∈ L ∞ + (Ω) in direction η ∈ L ∞ (Ω) is denoted by F I (σ)η := U , where U is the unique solution of a weak formulation; see [25, theorem 4.1] and [24]. The standard method for computing the Fréchet derivative is to solve this variational (adjoint state) formulation using the finite element method 2 (FEM). The Fréchet derivative of F I is a concatenation of single current vector derivatives and is denoted by Given a conductivity estimate σ (k) ∈ D(F), k ∈ N, a Newton iteration for the ICP consists of the following steps: Finding an approximate solution in step (1) and choosing a stopping criterion are the main design parameters for Newton-type methods.

Recovering boundary parameters
In practical applications, the boundary parameters of the CEM, i.e. the contact impedances, the electrode positions and the boundary shape, are rarely known accurately. Thus, it can be necessary to recover them simultaneously with the conductivity from measured data. For spacial dimension n = 2, these four quantities are coupled by conformal equivalence: when allowing non-constant contact impedances, there are distinct settings which have an identical measurement operator.
Conformal equivalence was already used in [26] for Calderón's model on the unit disk and extended to the CEM in [20,22,27]. This was used in [20] and [28] to compensate modelling errors by electrode movements, an effect which was further investigated in [29]. For that reason, we fix the boundary geometry in the scope of this work to avoid under-determination due to conformal equivalence. However, we want to optimize for the electrode geometry, thus we consider an extension of the classical EIT problem.
In awareness of a fixed, but possibly inaccurate boundary shape model, we consider the reconstruction result as an approximation to a conformal equivalent of the true, unknown setting. For that reason, we add the individual electrode sizes as an additional degree of freedom since even if they might be known accurately, the sizes of their conformal images are not.

Definition 2.3 (Extended forward operator).
We define an extended forward operator F depending on the conductivity, the electrode locations, sizes and contact impedances by denote the electrode contact impedances. The corresponding extended measurement operator is denoted by In [2, theorem 4.1 and corollary 4.2], the Fréchet derivative of the forward operator with respect to an electrode perturbation is given about the origin, that is, at their current shape. In the 2D case, perturbing the electrode boundary simply means to move the end-points of an electrode. Moreover, the integral over the electrode boundary appearing in [2, corollary 4.2] is simply a point-evaluation of the interior potential u at the electrode edges in this case. Since The explicit formulas in our notation are given in the following for the 2D case. To that end, let for an electrode mid-point θ l ∈ [0, |∂Ω|) and an electrode half-width ω l > 0, l = 1, . . . , L, where γ : [0, |∂Ω|] → R 2 is a continuous, piecewise smooth parametrization of ∂Ω with γ(0) = γ(|∂Ω|) and γ (θ) 2 = 1 almost everywhere. To avoid cumbersome notation, we consider the argument θ of γ modulo the length of the boundary |∂Ω|, where required.

Fréchet derivative of the extended operator
By equally perturbing both electrode edges in the same direction, a Fréchet derivative with respect to the electrode movement is given about its current position. For the lth electrode, it is given component-wise (omitting the arguments) by where u (k) , U (k) are the solutions of (1)-(4) for the kth current I (k) of the -frame and for conductivity σ. When perturbing both electrode edges by the same amount in opposite direction, a Fréchet derivative with respect to the electrode size is given about its current size. For the lth electrode size, it reads In particular, F θ l and F |E l | only differ by one sign in the sum. The Fréchet derivative of the forward operator with respect to the contact impedance z l of the lth electrode, studied e.g. in [3], is given by is the solution of (1)-(4) for the current vector I (k) . The integral can be approximated e.g. with a quadrature rule on the point values of FEM approximations of u (k) and u (m) , k, m = 1, . . . , L.
The Fréchet derivative for an extended measurement operator with arbitrary measurement pattern I can be obtained by a multiplication with the particular currents, i.e.

Newton's method for the extended inverse conductivity problem
A Newton iteration for the extended ICP can be defined analogous to the classical ICP. Denote by σ (k) the conductivity estimate after the kth Newton iteration and further by the contact impedance estimates and electrode configuration estimate after the kth iteration. The task of Newton's method for the extended ICP in the k + 1th iteration is to find a Newton update and such that E

Transformed forward operator
Instead of linearizing the measurement operator directly, we want to apply the Newton step to a transformation of the conductivity coefficient. To that end, we consider injective Assuming, for simplicity, that the conductivity can be evaluated pointwise, we consider a transformed forward operator F * defined as where t * operates pointwise on σ, i.e. σ * (x) = t * (σ(x)) for x ∈ Ω. This extends straightforward to general σ ∈ L ∞ + (Ω). The transformed measurement operator (F * ) I is defined analogously. Table 1 displays the identity transformation t Id , the negative log-conductivity t log , the resistivity transformation (reciprocal conductivity) t ρ , the σ-ρ-transformation t σρ as well as the tan-log-transformation t tl . The latter transformation is special since its domain enforces the conductivity to be strictly bounded from two sides, 0 < c min σ c max < ∞. The σ-ρ-transformation is a convex combination of the negative conductivity and the resistivity, namely As the sum of two strictly decreasing functions, it is oneto-one. The choice of α satisfies a certain optimality condition, although the exact value is uncritical. Moreover, this transformation has the asymptotic of the identity for σ → ∞ and the asymptotic of the resistivity for σ → 0. For details, see [23, section 4.2].

Transformed extended forward operator
Similarly to the conductivity constraints, we have restrictions for the electrode geometry: all electrodes have positive size, do not change order, and do not overlap. To implicitly enforce these conditions when applying Newton type updates, we again consider transformations from bounded to unbounded parameter spaces. We want to perform electrode changes independently of the neighboring electrodes, thus we introduce per-iteration barriers at the electrode gap centers. The gap centers are parametrized by This situation is depicted in figure 1, where the electrode center and their parameters θ l are depicted in red, while the gap centers and their parameters θ l are depicted in blue. Now we allow each electrode E l , l = 1, . . . , L, to move and change size within the boundary section i.e. we want the updated parameters θ (k+1) l and ω (k+1) l to satisfy after the kth Newton step. This is a coupled restriction for each electrode size and position. To decouple the conditions, we further split this available space into one part for the electrode movement and another part for the electrode size change, as shown in figure 2. Therein, the yellow sections of the boundary depict the parts of the boundary 'reserved' for the electrode resizing, and the green sections depict the parts for the electrode movement.
For the electrode sizes and the parameters of the locations, we thus obtain restrictions from below and from above. One possibility to handle this restriction is to use a sigmoidal transformation for the parameters, mapping the interval boundaries to ±∞. First, we introduce a lower bound ζ for the updated half-width w (k+1) l of the lth electrode after the kth iteration to ensure numerical stability for the FEM forward solutions 3 Figure 1. Parametrization of the electrodes on the boundary. The gap centers γ( θ 1 ), . . . , γ( θ L ) will act as barriers for the electrode movement and deformation in each Newton iteration.
for each l = 1, . . . , L, all possibly modulo |∂Ω|. Using these quantities, we restrict the electrode movement in the kth Newton step to and the electrode resizing to In figure 2, introduced quantities simply parametrize the boundaries of the yellow and green sections for the electrode resizing and movement, respectively. One possibility for defining a sigmoidal transformation is to choose again the tangent, i.e.
for the position update and for size update of the lth electrode in the kth iteration, respectively. These transformations map the admissible interval for the position and size updates to the real line.

Nonlinearity of the forward operator
The nonlinearity of the ICP for non-constant conducitivities is obvious: for example, when σ χB → ∞ on a subset B ⊂ Ω of the domain with positive distance to the boundary, we still obtain a finite Neumann-Dirichlet problem in the limit case, and thus F → ∞. We next demonstrate the impact of several conductivity transformations on the nonlinearity of F. To that end, we generate highly accurate references using semi-analytic solutions of the CEM on a circular domain with (a) a homogeneous conductivity, (b) (possibly non-centered) circular inhomogeneities, as introduced in [20]. We choose a generic electrode setup with L = 16 equally distributed electrodes with |E| l = 2π 32 , l = 1, . . . , L, and z = 10 −4 , on the unit disk Ω = B 1 (O). On this domain we define, for any ξ > 0, for the homogeneous conductivity case (a) and for some circular inclusion B ⊂ Ω for case (b). Then, we study the behaviour of F(σ ξ ) − F(σ 1 ) Fro as well as the conductivity nonlinearity indicator for various (pointwise) conductivity transformations t * : (0, ∞) → R. If F is linear in the transformed conductivity t * , then κ cond is constant in ξ (the converse is not necessarily true). In that sense, the variation of κ cond in ξ is an indicator for the nonlinearity of F in the transformed conductivity.

Nonlinearity for homogeneous conductivities
For vanishing contact impedance and homogeneous conductivity, the forward operator is linear in the resistivity. Thus for small contact impedances z 1 σ ξ ∞ , we expect the forward operator to be almost linear. Figure 3(a) shows the behaviour of the nonlinearity indicator κ cond (ξ, t * ) versus the parameter ξ for the conductivity transformations t Id , t ρ , t log and t σρ . A non-constant line indicates nonlinearity of the forward operator with respect to the transformed conductivity parameter. We observe how the σ-ρ-transform t σρ behaves similar to the resistivity transform t ρ for small conductivities and similar to the identity t Id for large conductivities. The forward operator is almost linear in the resistivity, whereas the log-transform and the σ-ρ-transform give a good trade-off between nonlinearity and unconstrainedness. This suggests that the log-transform might be better suited for recovering small, highcontrast inclusions inside a homogeneous medium, while the σ-ρ-transform might be better suited for recovering large inhomogeneities with lower contrast.

Nonlinearity of the extended forward operator
In section 2.7, we introduced parameter transformations also for the extended problem, i.e. for recovering the electrode shapes, sizes, and contact impedances. Analogously to the conductivity nonlinearity indicator, we introduce indicators for the nonlinearity with respect to the electrode positions as and with respect to the contact impedances as respetively.
Using the same 16 electrode setup as before, we evaluate κ pos using again semi-analytic solutions and letting p = ( p 1 , 0, . . . 0) for p 1 ∈ − 2π 32 · 1.05 , + 2π 32 · 1.05 (13) (to avoid electrode gaps going to zero). The scaling parameters in the tangent-transform are set as δ 1 = δ 1 = 2π 32 . The result is shown in figure 4(a). We observe an increase of the nonlinearity indicator when the electrode approaches its left or right neighbor in the untransformed case, indicating that linearization might over-estimate electrode location changes and lead to invalid (overlapping) electrode configurations. This effect is compensated when using the tangent transform, which by design prevents illegal electrode configurations.
Similarly, we use s = (s 1 , 0, . . . , 0) for s 1 ∈ − 2π 64 · 1.05 , + 2π 32 · 1.05 (14) to perturb the size of the first electrode for the same 16 electrode setup. The resulting nonlinearity indicators κ size (s, t ω,Id ) and κ size (s, t ω,tan ) are shown in figure 4(b). The nonlinearity increases as the electrode size tends to zero, indicating that linearization may lead to an underestimation and even negative electrode sizes in the untransformed case. Again, the damping effect of the tangent transform prevents this. Finally, consider varing contact impedances with a base value of z = 1 and contact impedance deltas (on a logarithmic scale) of c = (c 1 , c 1 , . . . , c 1 ) for c 1 ∈ (10 −3 − 1, 10 +3 − 1).
The resulting nonlinearity indicators κ contact (c, t z,Id ), κ contact (c, t z,σρ ) and κ contact (c, t z,log ) are shown in figure 4(c). The identity transform does not resolve the constrainedness of the contact impedance, but κ contact (c, t z,Id ) indicates that it is almost linear in this setting. In contrast, the log-transform resolves constrainedness, but higher nonlinearity is indicated by κ contact (c, t z,log ). The σρ-transform resolves constrainedness and κ contact (c, t z,σρ ) indicates that it also keeps the nonlinearity limited in this setting. In particular, the indicator is almost constant for very high contact impedances, which may occur e.g. when electrodes are badly attached.

Numerical evaluations
The conductivity and boundary parameter transformations introduced in sections 2 and 3 can be incorporated into any solver making use of the linearization of the forward operator. The numerical evaluations here use the well-established CG-REGINN [30] applied to the ICP for the CEM [25], using the initializations, error model, sensitivity-based weighting scheme and stopping criterion as presented in [9], called model-aware Newton-type Inversion scheme (MANTIS). In particular, MANTIS estimates an initial constant conductivity and corre sponding electrode contact impedances based on an optimality condition, and moreover provides an estimate for the measurement noise level. Details for the assembly of the extended system equations in this setting are beyond the scope of this work and can be found in [23]. After introducing the considered setting in section 4.1, we will present EIT reconstructions of the MANTIS algorithm using the extended forward operator and the introduced transformations for a configuration simulating badly attached electrodes (section 4.3), highly fluctuating contact impedances (section 4.4), wrong boundary geometry (section 4.5) and the combination of fluctuating contact impedances and wrong boundary geometry (section 4.6).  (13). For the untransformed conductivity, the nonlinearity increases as the electrode approaches the neighboring electrodes. (b) Nonlinearity indicator κ size versus electrode size change s as defined in (14). As the electrode size decreases, the nonlinearity indicator in the untransformed case increases as well, suggesting that electrode size changes are over-estimated during linearization. (c) Nonlinearity indicator κ contact versus contact impedance z + c as defined in (15). The log and σρ transformation have similar damping behaviour near vanishing contact impedances, but the σρ transformation behaves similar to the untransformed case for high contact impedances.
For reference, we will also show the the reconstruction results when no corrections for boundary parameters are estimated at all, i.e. when the extended forward operator is not used.
The number of Newton steps k * ∈ N is a practical indicator for the nonlinearity of an inverse problem: (unregularized) linear problems converge after a single iteration. The value is given, along with the relative reconstruction error e rel 0, for each reconstruction, see [23, chapter 4.4.5. and equation (6.1)] for definitions and further background.

Setting
The setting considered in this section is a 2D domain resembling the cross section of a human thorax with a highly conductive inclusion ('heart', σ = 0.75), two resistive inclusions ('lungs', σ = 0.24), and one highly resistive inclusion ('spine', σ = 0.05) in an otherwise homogeneous background ('soft tissue', σ = 0.42). The conductivity values are chosen to agree with those of the experimental setup [31]. The domain Ω is scaled to fit into the square [−1,1] 2 , resulting in a total circumference of |∂Ω| ≈ 7.377. We assume to have a boundary model with L = 32 electrodes equi-spaced electrodes covering 50% of the surface. This is depicted in figure 5. The contact impedances are z 1 , . . . , z L = 0.05 unless stated otherwise.
Measurement data for this setting is simulated using the same FEM forward solver, but on a much finer mesh, which is not a refinement of the mesh used for reconstruction, to minimize inverse crime. Noise is added according to the noise model of [23, chapter 2.2.2.] with a relative noise level of δ rel = 0.1%.

Reconstructions for perfectly known boundary parameters
For reference, we first show the reconstruction for perfectly modelled boundary geometry and the conductivity transformations t Id and t log in figure 6. The result for t σρ is visually indistinguishable from t log and converges with k * = 16 and e rel = 14.94%.

Badly attached electrodes
Now we consider a setting with with two electrodes in the non-convex part of the boundary (marked red in figure 5) having a high contact impedance: In practice, this can happen e.g. when the electrodes are badly attached. Results without and with simultaneous reconstruction of conductivity and contact impedances are shown in figure 7. Not accounting for varying contact impedances (MANTIS initialized them with z 1 , . . . , z L = 0.082) yields highly oscillating interior conductivities shown in figures 7(a) and (d) for conductivity transformations t Id and t log , respectively. They correspond to local minima of the ICP. Recovering the contact impedances simultaneously (using t z,log or t z,tan ) removes these oscillations, tremendously improves the conductivity reconstruction and improves speed of convergence (figures 7(b), (c), (e) and (f)). For reference, using t σρ as conductivity transformation yields k * = 48, e rel = 42.97% with fixed contact impedances and k * = 28, e rel = 14.52% when using the t z,tan for the contact impedances. We observe that best results are achieved when using the t log as a conductivity transformation and t z,tan with limits [10 −4 , 10 +4 ] as a contact impedance transformation. The contact impedances are recovered accurately; see figure 8.

Fluctuating contact impedances
This setting is similar to section 4.3, but now all contact impedances are randomly chosen: The reconstructions are shown, analogously to section 4.3, in figure 9. The true and estimated contact impedances are shown in figure 10.
Again, best performance is achieved using t log as conductivity transformation and t z,tan as contact impedance transformation. Using conductivity transformation t σρ yields almost identical performance and is therefore omitted here.

Wrong boundary geometry
Now, we assume that the exact boundary geometry of the setting is unknown for reconstruction and perform the estimation on an elliptically shaped domain. When estimating the electrode sizes and positions using the extended forward operator, as described in section 2.3, we allow the estimator to converge towards a solution which is conformally equivalent to the true geometry. The computed relative error of the solution is measured versus a conformal equivalent of the thorax shaped setting on the ellipse. It was obtained using the Schwarz-Christoffel toolbox (version 2.3) provided by Tobin Driscoll on www.math.udel.edu/~driscoll/SC.
The results are shown in figure 11. Ignoring the wrong boundary geometry yields highly oscillating, completely useless solutions, regardless of the conductivity transformation used (a)-(c). In contrast, considering the boundary geometry by simultaneously estimating a conformally equivalent electrode configuration results in smooth conductivity estimations, which resemble the interior geometry of the true conductivity distribution (d)-(f). The best result is achieved when using the conductivity transformation t σρ .

Wrong contact impedances and wrong boundary geometry
Finally, we consider the thorax setting with both unknown boundary shape and unknown contact impedances. The ICP is now solved using the extended forward operator including the estimation of contact impedances, electrode positions and sizes simultaneously using the transformations t σρ , t z,tan , t θ,tan and t ω,tan , respectively.
The result is shown in figure 12 and compared to the solution of the untransformed conductivity estimation with fixed boundary parameters. In figure 12(b), we observe that the resulting conductivity estimate succeeds and is visually similar to the reconstruction of figure 6 with accurately known boundary configuration. Moreover, the relative estimation error is also on the same level. For reference, using t log in the otherwise identical extended forward operator yields slightly worse results (k * = 72, e rel = 17.97%).   Further reconstructions, including some based on data from measured tank experiments, are presented in [23].

Reconstructions using untransformed boundary parametrizations
We have studied in section 3 how electrode parameter transformations do not only lift the constrainedness, but also can reduce nonlinearity and 'over-shooting' of parameter-updates (a) Using F , t Id : k * = 37, e rel = 58.03%.     due to linearization errors. For reference, the following reconstruction were performed using the extended forward operator with untransformed electrode parametrization. To avoid illegal electrode configurations, the updated electrode positions and sizes were projected back to an admissible configuration using the boundaries depicted in figure 2, while the contact impedances were, as implicitly done before, clamped to the interval [10 −4 ,10 +4 ]. To distinguish between the impact of the interior and boundary transformation, both transformed and untransformed conductivity reconstruction is shown in figure 13. Although the reconstruction of the untransformed contact impedances for known boundary configuration succeeds (with more iterations), the reconstruction of the untransformed electrode geometry fails. The author observed that during the iterations, some electrode positions and sizes tended to 'oscillate'. Moreover, some contact impedances had negative Newton updates (before clamping) even at the lower contact impedance limit, which would it make difficult to apply simple, commonly used line search strategies.

Summary and conclusion
We presented an extended, transformed forward operator for the ICP in section 2 which makes use of the Fréchet-differentiability with respect to the boundary configuration and implicitly ensures the admissibility of the configuration after Newton updates. For that reason, it can be conveniently integrated it into established Newton-type solvers for the ICP, like CG-REGINN, without setting-specific parameter tuning or line-search heuristics.      Moreover, we have motivated the theoretical advantage in terms of reduced nonlinearity when using favourable transformations in section 3 by investigating simple settings for which semi-analytic solutions are available.
Based on the numerical results in section 4, we conclude that the estimation of the conductivity succeeds in case of inaccurately boundary shape and unknown contact impedances when using the proposed extended and transformed forward operator F in the inexact Newton scheme CG-REGINN within the MANTIS reconstruction framework. In particular, the proposed solution can handle measurements with badly attached electrodes, fluctuating contact impedances, inaccurately known boundary geometry. These solutions are on par with the solution for perfectly known boundary geometry both visually and in terms of the relative reconstruction error.