1. Introduction
In electrical impedance tomography (EIT) the goal is to reconstruct an unknown conductivity distribution inside a physical body. This is based on noninvasive measurements of electric current and voltage at electrodes placed on the boundary of the object. According to the idealized continuum model (CM) of EIT, such measurements correspond to knowing the infinitedimensional NeumanntoDirichlet (ND) boundary map for the conductivity equation that models the behavior of the electric potential inside the object of interest. Alternatively, if one is able to perform the measurements of EIT for the same object with two different conductivities, the CM assumes the available data is the difference of the respective ND maps, i.e. the socalled relative measurements. This is the case with timedifference imaging or by numerically simulating the reference measurements for, say, the unit conductivity. Quite often the relative measurements are supposed to originate from two conductivities that coincide in some interior neighborhood of the object boundary, which is also the setting considered in this work.
In the framework of the CM, an isotropic conductivity, i.e. the strictly positive scalarvalued coefficient in the conductivity equation, is uniquely identified by the corresponding boundary measurements under dimensiondependent regularity assumptions. In two dimensions, it is enough to assume the conductivity is essentially bounded [1], whereas the most general uniqueness results in higher dimension require Lipschitz continuity [6]. There also exist several reconstruction methods that are based on the (slightly unrealistic) CM such as the method [32, 33, 36], the monotonicity method [11, 21, 22, 5], the factorization method [31, 3, 2, 19, 15], and the enclosure method [27, 28, 29]. It is worth noting that all these reconstruction techniques actually assume the availability of relative CM measurements.
In all realworld settings for EIT, it is only possible to perform measurements with a finite number of finitesized electrodes, making it impossible to exactly record continuum data in practice. In consequence, more accurate electrode models should arguably be used to treat such realistic measurement setups. The complete electrode model (CEM) includes the actual shapes and positions of the employed electrodes as well as the contact impedances at the electrodeobject interfaces in the forward model [7, 41]. The limit of the CEM when the contact impedances tend to zero is called the shunt model [7]; a bit counterintuitively, the shunt model has been shown to exhibit worse numerical behavior than the CEM [9]. For small or pointlike electrodes and relative measurements, the injected currents can be relatively accurately modeled as Dirac delta distributions according to the point electrode model (PEM) [17].
Although the aforementioned electrode models accurately predict realworld measurements under appropriate assumptions on the measurement configuration, very little is actually known about the unique identifiability of conductivities in their frameworks: among the only such results, the unique identifiability has been proven in [20] for the CEM assuming the conductivity belongs to a suitable a priori known finitedimensional set of piecewise analytic functions. In fact, it is intuitively acceptable to assume there is no uniqueness for any combination of an infinitedimensional family of conductivities and an electrode model producing finitedimensional data. For the PEM it is even possible to stably construct conductivity perturbations that really are invisible to measurements by a given finite set of electrodes [8]. However, since it is also possible to approximate (absolute) CM measurements by those of the CEM when the number of electrodes tends to infinity and the electrodes cover the object boundary in a controlled manner, one can argue that the unique identifiability results for the CM transfer to the framework of the CEM in the sense of limits. To be slightly more precise, it has been shown in [24] that when the number of electrodes is increasing (and their size appropriately decreasing) the ND map corresponding to a smooth enough conductivity can be approximated in the space of bounded linear operators on square integrable functions, so that the associated discrepancy tends to zero essentially as the maximal distance between adjacent electrodes.
As mentioned above, many of the reconstruction methods designed for the CM assume relative measurements as their input, and so the ability to approximate relative continuum data by electrode models is important for the practical implementation of sophisticated mathematical reconstruction algorithms. In fact, electrode measurement variants have previously been introduced for, e.g., the factorization method, the monotonicity method, and the method (see, e.g., [34, 13, 14, 23, 30]), but the accuracy of the associated techniques for approximating relative continuum data based on the available electrode measurements has not always been carefully analyzed. Indeed, the best result in this direction the authors are aware of is the one in [24], which does not, in particular, exploit the extra structure carried by relative measurements if the considered two conductivities coincide in an interior neighborhood of the object boundary. See also [4] for a recent Bayesian approach for moving between electrode and continuum measurements in EIT.
In this paper, we tackle mimicking relative continuum data by electrode measurements in twodimensional EIT as a problem of (hardware) algorithm design: our aim is to choose (optimal) positions for the employed electrodes based on the shape of the imaged domain and the net electrode currents as functions of the continuum current pattern one would like to drive through the object boundary. We assume the examined bounded simply connected twodimensional domain has a smooth enough boundary and the target continuum current patterns exhibit based Sobolev regularity of order
. Our algorithm leads to estimates of order
in the number of electrodes for the discrepancy between the relative measurements of the CM and suitably postprocessed PEM or CEM data (provided the width of the electrodes decay appropriately in for the CEM). This result is first proven for the PEM in the unit disk with equiangular electrodes, and subsequently it is extended to more general domains with the help of conformal mappings. Finally, the required estimates for the CEM are obtained by resorting to the material in [17], where an approximative link between the CEM and the PEM is considered. In addition to the Riemann mapping theorem, the main ingredients for obtaining our estimates are sufficiently accurate interpolation and quadrature rules on the boundary of the unit disk for the relevant Sobolev spaces; in fact, the order of the obtained estimates is directly dictated by the accuracy of these rules.This paper is organized as follows. In Section 2, we introduce the CM, PEM, and CEM as well as their respective relative measurement maps. Section 3 briefly recalls the connection between Sobolev spaces and Fourier series, while Section 4 reviews trigonometric interpolation and introduces estimates related to pointwise current injection in the PEM. Section 5 provides the desired estimates for the PEM in the unit disk as Theorem 5.1. Finally, Section 6 introduces our algorithm with general domains for both the PEM and the CEM, with Theorem 6.1 and Corollary 6.2 serving as our main results. The paper is concluded by a numerical example verifying the shown convergence rate for the CEM in a simple geometry. Finally, an appendix clarifies how the constants appearing in the estimates of [17] depend on the number of electrodes.
1.1. Some notational remarks
Let denote the space of bounded linear operators between Banach spaces and , with .
For the sake of brevity, we use the following notation for finite summations with :
An analogous notation is also used for sets that are indexed by integers. For
, we systematically index the components of vectors in
from to , i.e. , and denote the mean free subspace of by .We often use generic positive constants that may change from one estimate to the next. As an example, writing indicates that such a constant only depends on the parameters , , and .
2. On forward models of EIT
In this section we review three forward models for (twodimensional) EIT. All of them correspond to the same underlying elliptic partial differential equation
(2.1) 
where is a bounded simply connected domain with a boundary. The coefficient is a complexvalued isotropic conductivity satisfying
(2.2) 
almost everywhere in for some .
Remark 2.1.
The forward problems of CM and CEM would still be well defined if were Lipschitz and only the second condition of (2.2) were satisfied (assuming the boundary current densities for the CM are regular enough). On the other hand, the PEM requires the first condition of (2.2) and also some regularity from , but piecewise smooth boundary would actually be enough [40]. Moreover, all three models could be introduced in three dimensions and for anisotropic conductivities as well. The reason for assuming twodimensionality and extra regularity from and is that they are required by our main results in their optimal form. On the other hand, anisotropic conductivities are excluded solely for the sake of notational convenience.
2.1. Continuum model
The CM assumes that one can drive any mean free normal current density through and measure the resulting boundary potential everywhere. In mathematical terms, (2.1) is accompanied by the Neumann boundary condition
(2.3) 
where is the exterior unit normal of and , with denoting the mean free subspace of the Sobolev space . To be more precise,
where denotes the sesquilinear dual evaluation between the associated Sobolev spaces.
According to the standard theory on elliptic boundary value problems [37], the combination of (2.1) and (2.3) has a unique solution . Moreover, as satisfies the Laplace equation in an interior neighborhood of , its Dirichlet trace is welldefined in [35]. By choosing the ground level of potential appropriately, one can thus introduce the ND map
whose boundedness for any is considered, e.g., in [18, Theorem A.3].
The relative ND map
(2.4) 
exhibits considerably more regularity than itself. More precisely,
(2.5) 
for all . This well known result is presented in [18, Theorem A.3] for the unit disk, but the corresponding proof applies as such to any smooth and bounded domain since the essential ingredient is the first condition in (2.2), not the shape of the domain.
According to the CM, relative EIT measurements for a conductivity satisfying (2.2) produce the boundary operator as the data. However, it is obvious that cannot be precisely retrieved based on practical EIT measurement that employ a finite set of contact electrodes. This observation motivates the introduction of more realistic electrode models for EIT.
2.2. Point electrode model
In our framework, the PEM assumes that pointlike electrodes are attached to at the distinct locations for some . The net currents , with a zeromean condition imposed, are driven through the electrodes and the relative potentials are measured at these same positions. In other words, the employed current patterns are of the form
(2.6) 
where , , is the Dirac delta distribution supported at on the onedimensional boundary . Observe that , , for any .
Recalling the relative ND map from (2.4), the measurements of PEM can be modeled by the pointwise currenttovoltage map
(2.7) 
where and the ground level of potential is chosen so that the relative potentials at the electrodes have zero mean. Due to (2.5) and the Sobolev embedding theorem, the definition (2.7) is unambiguous.
According to the PEM, relative EIT measurements for a conductivity satisfying (2.2) produce (a noisy version of) the finitedimensional linear map as the data. Although the pointlike electrodes of the PEM cannot completely accurately model the finitesized ones used in practical EIT measurements, it has been shown that the discrepancy between relative measurements modeled by the CEM and the PEM behave asymptotically as in the maximal diameter of the electrodes [17].
2.3. Complete electrode model
The CEM is arguably the most accurate model for EIT [7, 41]. In our twodimensional setting, mutually disjoint electrodes are attached to the object boundary. They are identified with the nonempty open and connected subsets of that they cover, and their midpoints with respect to the arclength of are , respectively. The contact impedances at the electrodeobject interfaces are modeled by the complex numbers with positive real parts.
In the forward problem of the CEM, the conductivity equation (2.1) is combined with the boundary conditions
(2.8)  
where models the net currents through the electrodes, is the electric potential within , and carries the constant potentials at the electrodes. The combination of (2.1) and (2.8) uniquely defines the interiorelectrode potential pair [41]. Take note that forcing the electrode potential vector to belong to corresponds to a particular choice for the ground level of potential.
The absolute measurements of the CEM are modeled by the electrode currenttovoltage map
and the corresponding relative measurement map is
(2.9) 
According to the CEM, relative EIT measurements for a conductivity satisfying (2.2) produce (a noisy version of) the finitedimensional linear map as the data.
3. On periodic distributions, periodic Sobolev spaces, and Fourier series
This section reviews some basic facts on periodic distributions and Sobolev spaces based on [39, Sections 5.2 and 5.3]. We present the theory in terms of periodic functions and distributions rather than 1periodic ones as in [39], mainly because we will initially focus on the unit disk when considering EIT.
We denote by the space of smooth functions with compact support and by its dual space, i.e. the space of distributions on . The dual pairing between these spaces is denoted as .
Definition 3.1.
A distribution is called periodic if
where for . Moreover, the spaces of periodic smooth functions and distributions are defined as
respectively.
As hinted by the notation, can be identified with the dual space of : Following [39, Section 5.2], we may introduce such that its translates form a partition of unity, i.e.
(3.1) 
The dual pairing between and is then defined by
(3.2) 
A short computation reveals that the definition in (3.2) is independent of the choice of with the property (3.1). We refer to [39, Section 5.2] for a more careful analysis of the duality between and .
The definition (3.2) enables introducing Fourier coefficients for periodic distributions. To streamline the notation, we denote the trigonometric monomials by , .
Definition 3.2.
The Fourier coefficients of are defined as
(3.3) 
By identifying as a subspace of in the usual manner, it is easy to see that (3.3) provides a generalization for the standard Fourier coefficients defined for periodic functions.
Example 3.3.
We identify the standard Dirac delta distribution , , with its periodic counterpart
(3.4) 
It follows immediately from (3.3) that
(3.5) 
Let us then define the periodic Sobolev spaces.
Definition 3.4.
The periodic Sobolev space with a smoothness index is defined as
where
(3.6) 
and . The meanfree subspace of is
It is obvious that (as well as its closed subspace ) is a Hilbert space when equipped with the inner product
(3.7) 
We write and , which is motivated by Parseval’s theorem guaranteeing that for all , that is, .
The bracket denotes the sesquilinear dual pairing, acting as an extension of the inner product on . By mimicking the construction in [12, Section 1.1], one can use this same pairing for realizing the duality between and as well.
Remark 3.5.
We may identify with the standard Sobolev space on the boundary of the unit disk . Indeed, via the identification the topologies of the two spaces coincide, and hence we may use their properties interchangeably. For more information, see [35, Remark 7.6] that characterizes for any smooth bounded domain with the help of an eigensystem for the Laplace–Beltrami operator on ; observe also that , , can be identified on
with an eigenfunction for the Laplace–Beltrami operator, which is essentially the second angular derivative, with the corresponding eigenvalue being
. In particular, the elements of can be identified with continuous periodic functions for due to the Sobolev embedding theorem, and the (periodic) Dirac delta distribution belongs to for any . The latter could also be easily deduced by combining (3.5) and (3.6).Let us complete this section by briefly considering the convergence of Fourier series in different spaces. The partial sums of a Fourier series are defined by
According to [39, Theorem 5.2.1 and Section 5.3], in as for any of the choices , . Moreover,
(3.8) 
meaning that the trigonometric monomials form an orthogonal basis of for any . It now follows immediately from (3.7) that
(3.9) 
defines the orthogonal projection of onto for any .
4. Approximations based on equidistant interpolation points
This section considers two techniques for approximating a periodic function based on its pointwise values on a uniform grid over a single period: trigonometric interpolation and forming a weighted linear combination of Dirac delta distributions placed at the quadrature points. The latter is needed when continuum current patterns are transformed into pointwise currents for the point electrode model, whereas the former gives a natural technique for extending (relative) point electrode measurements to smooth functions over the whole object boundary.
4.1. Interpolation by trigonometric polynomials
We follow [39, Section 8.1–8.3]
, always choosing an odd number of interpolation points to avoid certain asymmetry when it comes to the degrees of the employed trigonometric polynomials and Fourier coefficients.
Let be the complex vector space of trigonometric polynomials of degree at most and let be its mean free subspace, that is,
Due to the definition of the inner product of in (3.7), the formula
(4.1) 
defines the orthogonal projection of onto for any . Moreover, is obviously the orthogonal projection of onto , and a comparison with (3.9) easily leads to the conclusion that is the orthogonal projection of onto . According to [39, Theorem 8.2.1 and Exercise 8.2.1],
(4.2) 
for any such that .
As a side note, which is not directly connected to electrode models, we immediately obtain estimates for the discrepancy introduced when is replaced by its natural finitedimensional approximations with respect to a trigonometric basis of . Take note that, up to the numerical errors introduced by the employed forward solver, is a finitedimensional operator one would naturally use in optimizationbased reconstruction algorithms with finite amount of data in the framework of the CM. On the other hand, is a matrix approximation for with respect to a trigonometric basis of , more suitable for direct reconstruction methods.
Proposition 4.1.
For and with ,
Proof.
The assertion is a direct consequence of (4.2):
where we also used the fact as the projection is orthogonal. ∎
Let us then consider trigonometric interpolation of periodic functions based on their values at equidistant points over a single period. We denote the interpolation points by
The trigonometric interpolation basis is then defined via
We are particularly interested in the following special properties of these functions:
(4.3) 
which are straightforward consequences of the following identity that holds for any ,
(4.4) 
The second line of (4.4) is trivial, while the first one follows from the formula for a truncated geometric series. In particular, (4.3) guarantees that are linearly independent, and thus form a basis for .
Remark 4.2.
If one interprets as angular coordinates, then
(4.5) 
are the corresponding interpolation points (i.e. point electrodes), on the boundary of the unit disk . In particular, consists of arcs of equal length.
We are now ready to introduce the trigonometric interpolation operator , , defined in the natural manner
(4.6) 
As its name suggests , , due to (4.3). Take note that the definition (4.6) is unambiguous since any , , is continuous by the Sobolev embedding theorem; see Remark 3.5. As is a basis for , it must hold that
(4.7) 
because this is the only way to give as a linear combination of at the interpolation points by virtue of (4.3). In other words, is a nonorthogonal projection of onto .
Based on (4.3) and (4.7), the periodic trapezoidal quadrature rule, with the quadrature points chosen to be the interpolation points , can be applied to exactly evaluate inner products of functions in :
(4.8) 
Moreover, by choosing and in (4.8) and employing (4.3) for one more time, we obtain
Hence,
(4.9) 
In particular, the latter condition obviously holds for all in the mean free subspace , and hence the former also holds for such , as could have been directly verified using (4.4) as well.
To complete this subsection, we recall the following estimate from [39, Theorem 8.3.1]:
(4.10) 
where
(4.11) 
Comparing this to (4.2), one sees that provides asymptotically in , and up to a multiplicative constant, as good an approximation for the (embedding) identity operator as the orthogonal projection , if the conditions in (4.10) are satisfied.
Remark 4.3.
According to (4.9), the interpolant of a continuous has vanishing mean if and only if . In particular, for by virtue of (4.7). Since the PEM for the unit disk with electrodes placed at the equiangular points employs currents in , we may thus identify electrode current patterns with trigonometric polynomials in . Indeed, the mapping
(4.12) 
defines a bijection between and . This gives a natural method for going back and forth between admissible electrode current patterns in and admissible continuum current patterns in . Take note that if and only if for a continuous function . Moreover,
by virtue of (4.3), and thus
(4.13) 
if is equipped with the Euclidean norm.
4.2. Pointwise approximation of smooth enough functions
In this subsection, we review how periodic continuous functions can be approximated in weak Sobolev topologies by linear combinations of Dirac delta distributions supported at . To this end, we define a point evaluation operator for and (cf. Remark 3.5):
where, as always, is identified with its periodic extension. It is worth noting that the multiplier is the mesh parameter corresponding to the employed grid .
Lemma 4.4.
Proof.
Let , , and recall from Section 3 that denotes the sesquilinear dual bracket. By using the definition of and applying (4.8) to the interpolated functions and , we deduce
Employing (4.10) and the continuity of the embedding now yields
Finally, taking the supremum over with gives the sought for bound. ∎
When considering the point electrode model of EIT in the next section, it is mandatory that any linear combination of Dirac deltas defining a current pattern has vanishing mean, that is, the coefficients defining the linear combination must sum to zero. It is trivial to check that has zero mean if and only if . In particular, we thus have , , based on the remark after (4.9). To ensure the zero mean condition holds more generally, we need to combine with , , defined by
(4.14) 
It is straightforward to check that for any and . Moreover, is the identity operator when restricted to , and thus on ; see again the comment succeeding (4.9).
The following lemma considers the approximation of identity by on , , i.e., on a space of admissible continuous current patterns for the continuum model of EIT.
Lemma 4.5.
Proof.
We first demonstrate that proving the assertion boils down to writing a uniform estimate with respect to for the norm of , . Indeed,
(4.15) 
where the second step follows from the mean free function not seeing the constant , cf. (4.14). Since the first term on the righthand side of (4.2) can be handled by Lemma 4.4, we only need to worry about the second one.
5. Equiangular point electrodes for the unit disk
Let us return to the setting of Section 2 with being the unit disk; the case of a more general twodimensional smooth and simply connected domain will be analyzed in Section 6 below, including considerations related to the CEM. In particular, we assume satisfies the conditions in (2.2) with replaced by . In what follows, we make the identification , which means we can use and , , interchangeably; see Remark 3.5.
Assume one would like to apply the continuum current pattern , , and measure the corresponding relative potential on the whole boundary , but due to practical restrictions the measurements need to be carried out with infinitesimal electrodes at the (interpolation) points defined in (4.5). Our method for approximating the smooth output of the CM is as follows:

Introduce
(5.1) as the electrode current pattern for the PEM.

Perform the measurements of the PEM to retrieve .

Build an approximation for the continuum boundary potential via trigonometric interpolation, that is, , where the bijection is defined by (4.12).
Although it is not necessarily completely evident, another way of writing the above approximation procedure is
(5.2) 
Let us clarify this claim. First of all, it is easy to check that
for from (5.1) and the corresponding defined as in (2.6) with replaced by and by . Thus equals the vector obtained by evaluating at the point electrodes up to the addition of a multiple of (cf. (2.7)). It is easy to check that this component in the direction of only affects the component of the trigonometric polynomial in the direction of the constant function, meaning in particular that
for some constant . Taking the orthogonal projection onto the mean free trigonometric polynomials thus proves (5.2).
To summarize, considering the accuracy of the above introduced procedure for mimicking CM measurements for the unit disk, with a finite number of electrodes modeled by the PEM, is equivalent to proving estimates for the discrepancy between the CM measurement map and its modified version .
Theorem 5.1.
For and ,
(5.3) 
where the positive constant is independent of .
Proof.
Let us consider an arbitrary and . Since
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