A DIRECT RECONSTRUCTION ALGORITHM FOR THE ANISOTROPIC INVERSE CONDUCTIVITY PROBLEM BASED ON CALDER ´ON’S METHOD IN THE PLANE

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A DIRECT RECONSTRUCTION ALGORITHM FOR THE ANISOTROPIC INVERSE CONDUCTIVITY PROBLEM BASED

ON CALDER ´ON’S METHOD IN THE PLANE

RASHMI MURTHY, YI-HSUAN LIN, KWANCHEOL SHIN, AND JENNIFER L. MUELLER

Abstract. A direct reconstruction algorithm based on Calder´on’s linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known a priori, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calder´on-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.

Keywords. Calder´on’s problem, anisotropic, electrical impedance tomography, quasi-conformal map, exponential solutions, inverse conductivity problem, Dirichlet-to-Neumann map

Contents

1. Introduction 1

2. Mathematical formulation 3

3. Injectivity of the inverse problem 4

4. The linearized reconstruction method 8

5. Numerical implementation 9

5.1. Numerical solution of the forward problem for data simulation 9

5.2. Computing the quasi-conformal map 12

5.3. Reconstruction 13

6. Examples 13

7. Conclusions 14

Acknowledgment 15

References 15

1. Introduction

The inverse conductivity problem was first proposed in Calderón’s pioneer work [13], in which the existence of a unique solution and a direct method of reconstruction of isotropic conductivities from the associated boundary measurements was given for the linearized problem in a bounded domain. This work inspired a large body of research on the global uniqueness question for the inverse conductivity problem and methods of reconstruction. Calderón made use of special exponen- tially growing functions known as complex geometrical optics solutions (CGOs), and these have proved useful for global uniqueness results for the inverse conductivity problem, leading also to a family of direct reconstruction algorithms known as D-bar methods. The reader is referred to [42,43] and the references therein for further reading on D-bar methods. Their relationship to Calderón’s method was investigated in [34].

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The inverse conductivity problem is the mathematical model behind a medical imaging technology known as electrical impedance tomography (EIT). EIT has the attributes of being inexpensive, non-invasive, non-ionizing and portable. Medical applications of EIT include pulmonary imaging [16,18,49,55], breast cancer detection [15,29,30,33,57], human head imaging [3,17,19,56], and others. See also the review articles [12,14,26,40] on EIT. The literature of reconstruction algorithms for the isotropic conductivity is extensive. Implementations of Calder´on’s method for isotropic conductivities include [10,44,48]. However, in reality, many tissues in the human body are anisotropic, meaning that electrical current will be conducted in spatially preferred directions. Anisotropic conductivity distributions are prevalent in the body, with significant differences in the transverse and lateral directions in the bone, skeletal and cardiac muscle, and in the brain [1,9]. However, medical EIT imaging typically neglects anisotropic properties of the conductivity and re- constructs an isotropic approximation. This can lead to artifacts in the resulting images and incorrect estimates of conductivity values.

In this work, we want to study the anisotropic conductivity equation L_Au := ∇ · (A(x)∇u) =

2

X

j,k=1

∂_x_k A^jk(x)∂_x_ju = 0 in Ω, (1.1)

where Ω ⊂ R² is a bounded Lipschitz domain and A(x) = (A^jk(x))1≤j,k≤2 is a positive definite symmetric matrix. The explicit regularity assumptions of A(x) will be characterized later (see Hypothesis2.1). In order to study inverse problems in anisotropic media, the key step is to transform the anisotropic conductivity equation (1.1) into an isotropic one, by using the isothermal coordinates, which are related to quasi-conformal maps [5]. This method is widely used in the study of the Calder´on type inverse problems in the plane.

Contrary to the isotropic case, knowledge of the Dirichlet-to-Neumann (DN) map is not sufficient to recover an anisotropic conductivity [24,36]. The non-uniqueness of the anisotropic problem stems from the fact that any diffeomorphism of Ω which keeps the boundary points fixed has the property of leaving the DN map unchanged (i.e., one can find a diffeomorphism with the boundary map being identity), despite the change in conductivity [8]. However, a uniqueness result was proved in [41]

under the assumption that the conductivity is known up to a multiplicative scalar field. The uniqueness and stability results for an anisotropic conductivity of the form A = A(x, a(x)), where a(x) is an unknown scalar function was proved in [6]. In the same paper, the uniqueness result for the interior conductivities of A were also proved by piecewise analytic perturbations of scalar term a. A priori knowledge of the preferred directions of conductivity, or the entries of the tensor of anisotropy, may be obtained, for example, from a diffusion tensor MRI, as discussed in [2].

In this work, we provide a direct reconstruction algorithm under the assumption that the anisotropic conductivity is a “small perturbation” (under the matrix no- tion) of a given 2 × 2 positive definite matrix with constant entries in the plane.

Other approaches to the reconstruction problem for anisotropic conductivities include [1,11,22,23,25,39]. The quasi-conformal map in the plane and invariance of the equations under a change of coordinates is the key to the algorithm. But, we note that in three spatial dimensions, there is no quasi-conformal map, and so our approach is not applicable.

CGO solutions have also been used to solve the inverse obstacle problem under various mathematical models. For the isotropic case, see, for example, [27,28,31, 32,45,50,51,54]. For the anisotropic case in the three-dimensions, one can use the oscillating-decaying solutions to reconstruct unknown obstacles in a given medium, see [37,38,46,47]. It is worth mentioning that for the anisotropic case in the plane,

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one can also construct CGOs via the quasi-conformal map, and we refer readers to [53] for more detailed discussions.

The paper is organized as follows. Section 2 contains the mathematical formulation of the anisotropic inverse conductivity problem. In Section 3, we provide a rigorous mathematical analysis for the anisotropic elliptic problem. We prove that the linearization of the quadratic form is injective when evaluated at any positive definite 2 × 2 constant matrix. The tool is to use the quasi-conformal map in the plane. In Section 4, we provide a reconstruction methods based on Calder´on’s approach for anisotropic conductivities. In addition, we also provide a numerical implementation for the inverse anisotropic conductivity problem in Section 5.

2. Mathematical formulation

The mathematical model for the EIT problem with an anisotropic conductivity can be formulated as follows: Let Ω ⊂ R² be a simply connected domain with Lipschitz boundary ∂Ω. Assume that the following conditions hold.

Hypothesis 2.1. Let A(x) = A^jk(x)

1≤j,k≤2 be the anisotropic conductivity, which satisfies:

(a) Symmetry: A^jk(x) = A^kj(x) for x ∈ Ω.

(b) Ellipticity: There is a universal constant λ > 0 such that

2

X

j,k=1

A^jk(x)ξjξk ≥ λ|ξ|², for any x ∈ Ω and ξ = (ξ1, ξ2) ∈ R².

(c) Smoothness: The anisotropic conductivity A ∈ L^∞(Ω; R^2×2).

Let φ ∈ H^1/2(∂Ω) be the voltage given on the boundary. The electric field u arising from the applied voltage φ on ∂Ω is governed by the following second order elliptic equation

(∇ · (A∇u) =P2 j,k=1

∂

∂x^j A^jk(x)_∂x^∂ku = 0 in Ω,

u = φ on ∂Ω.

(2.1)

Under Hypothesis2.1for A, given any Dirichlet data φ on ∂Ω, by the Lax-Milgram theorem, it is known that (2.1) is well-posed (for example, see [21]). Therefore, the DN map is well-defined and is given by

Λ_A: H^1/2(∂Ω) → H^−1/2(∂Ω)

ΛA: φ 7→ ν · A∇u|∂Ω=

2

X

j,k=1

A^jk(x)∂u

∂x^jνk

_∂Ω

,

where u ∈ H¹(Ω) is the solution of (2.1), and ν = (ν₁, ν₂) is the unit outer normal vector on ∂Ω. It is natural to consider the quadratic form Q_A,Ω(φ) with respect to (2.1), which is defined by

(2.2) QA,Ω(φ) :=

Z

Ω 2

X

j,k=1

A^jk(x)∂u

∂x^j

∂u

∂x^k dx = Z

∂Ω

ΛA(φ)φ dS,

where dS denotes the arc length on ∂Ω and we have utilized the integration by parts in the last equality. The quantity QA,Ω represents the power needed to maintain the potential φ on ∂Ω. By the symmetry of the matrix A, knowledge of QA,Ω is equivalent to knowing ΛA.

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The inverse problem of anisotropic EIT is to ask whether A is uniquely determined by the quadratic form QA,Ω and if so, how to calculate the matrix-valued function A in terms of QA,Ω.

3. Injectivity of the inverse problem

Assume the conductivity A(x) satisfies Hypothesis 2.1and A(x) is of the form A = a(x)A0,

where a(x) is a scalar function to be determined and A₀ is a known constant 2 × 2 symmetric positive definite anisotropic tensor. Then by [41], the inverse anisotropic conductivity problem of determining a(x) has a unique solution.

Introduce the norms in the space of A and in the space of quadratic forms QA≡ QA,Ω(φ) as follows. Let

(3.1) kφk²:=

Z

Ω

|∇u|²dx,

where u ∈ H¹(Ω) is the solution of the following second order elliptic equation with constant matrix-valued coefficient,

(∇ · (A0∇u) = 0 in Ω,

u = φ on ∂Ω,

(3.2) and

(3.3) kQAk = sup

kφk≤1

|QA(φ)| .

In the spirit of Calder´on [13], the next step is to show that the mapping from conductivity to power, namely

(3.4) Φ : A → QA,

is analytic, where the conductivity to power map QA is given by (2.2). The argu- ment outlined below establishes that Φ is analytic.

Consider A(x) as a perturbation from the constant matrix A0of the form A(x) := (1 + δ(x))A₀,

(3.5)

where δ(x) is regarded as a scalar perturbation function. We further assume that kδk_L^∞_(Ω)< 1 is sufficiently small so that the matrix A is also positive definite in order to derive the well-posedness of the following second order elliptic equation

(LA(w) := ∇ · (A(x)∇w) = 0 in Ω,

w = φ on ∂Ω.

(3.6)

As in Calder´on [13] for the isotropic case, we will use perturbation arguments. Let w := u + v, where w is the solution of (3.6), and u is the solution of (3.2) with the same boundary data w = u = φ on ∂Ω. Then

(3.7) L_A(w) = L_(1+δ(x))A₀(u + v) = L_A₀v + L_δA₀v + L_δA₀u = 0 in Ω, and v|∂Ω= 0. Then we have the following estimate for the function v.

Lemma 3.1. Let v be the function defined via the equation (3.7) in Ω with v|∂Ω= 0.

The operator LA₀ has a bounded inverse operator G and v has the following H¹ bound

(3.8) kvkH¹(Ω)≤ kGk_L(H−1;H¹)kδk_∞kA0kFkφk 1 − kGkkδk∞kA0k ,

where kGk_L(H⁻¹_;H1)denotes the operator norm from H⁻¹ to H¹, kA0kF stands for the Frobenius norm of the matrix A0 and kφk is given by (3.1).

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Recalling that the Frobenius norm of a 2 × 2 matrix A = (A_jk)_1≤j,k≤2is defined by

kAk²_F :=

2

X

j=1 2

X

k=1

|Ajk|².

Proof of Lemma 3.1. Since u ∈ H¹(Ω) is the unique solution of the boundary value problem LA₀u = 0 in Ω, u = φ on ∂Ω, the operator LA₀ has a bounded inverse G.

Then from (3.7) one sees,

(3.9) G (L_A₀v + L_δA₀v + L_δA₀u) = 0.

That is,

(3.10) (I + GL_δA₀) v = −GL_δA₀u.

Note that

(3.11) kLδA₀wk_H−1(Ω)= sup

ψ∈H¹₀(Ω)

R

Ω∇ · [(δ(x)A0)∇w]ψ dx kψk_H1

0(Ω)

, and

Z

Ω

∇ · [(δ(x)A0)∇w]ψ dx

= Z

Ω

∇ψ · [(δ(x)A0)∇w] dx

≤kδ(x)A₀k_L∞(Ω)k∇wk_L2(Ω)kψk_H1 0(Ω). (3.12)

Thus, from (3.11) and (3.12),

(3.13) kLδA₀wk_H⁻¹_(Ω)≤ kδ(x)A0k_L^∞_(Ω)k∇wk_L2(Ω). Next, consider the operator norm

kGLδA₀k_L(H1;H¹)= sup

w6=0

kGLδA0wkH¹(Ω)

kwk_H1(Ω)

≤ kGk_L(H⁻¹;H¹)kLδ(x)A₀wk_H−1(Ω)

kwk_H1(Ω)

. (3.14)

That is,

(3.15) kGLδA₀k_L(H1;H¹)≤ kGk_L(H−1;H¹)kδA0k_L∞(Ω),

where k · k_{L(X;Y )} stands for the operator norm from the Banach space X to Y . Moreover, when

kδ(x)A₀k_L∞(Ω)< 1 kGk_L(H⁻¹;H¹)

, the Neumann series





∞

X

j=0

(−1)^j(GLδA₀)^j



(GLδA₀u) , converges, and from (3.10), one has

(3.16) v = −





∞

X

j=0

(−1)^j(GLδA₀)^j



(GLδA₀u).

It is easy to see that

kv + GLδA₀vk_H1(Ω)≥kvk_H1(Ω)− kGLδA₀vk_H1(Ω)

≥kvkH¹(Ω) 1 − kGk_L(H⁻¹_;H1)kδ(x)A0kL^∞(Ω) . (3.17)

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Finally, by plugging relations (3.9) and (3.13) into (3.17), one has 1 − kGk_L(H⁻¹_;H1)kδ(x)A0kL^∞(Ω) kvkH¹(Ω)

≤kv + G(−LδA₀u − LA₀v)k_H1(Ω)

=kv − GL_δA₀u − vk_H1(Ω)

≤kGk_L(H⁻¹;H¹)kLδA0ukH⁻¹(Ω)

≤kGk_L(H⁻¹_;H1)kδ(x)A0k_L^∞_(Ω)kφk, which is equivalent to

(3.18) kvkH¹(Ω)≤ kGk_L(H⁻¹;H¹)kδ(x)A0kL^∞(Ω)kφk 1 − kGk_L(H⁻¹_;H1)kδ(x)A0k_L^∞_(Ω).

Therefore, the above calculations allow us to conclude that the mapping Φ defined by (3.4) is analytic at A0, which completes the proof. Next, let us linearize the map Q_A(φ) around a positive definite constant matrix A(x) = A0 as follows:

Q_(1+δ(x))A₀(φ) = Z

Ω

[(1 + δ(x))A0∇w] · ∇w dx

= Z

Ω

[(A0+ δ(x)A0)∇u] · ∇u dx + 2 Z

Ω

(δ(x)A0∇u) · ∇v dx +

Z

Ω

[(A0+ δ(x)A0)∇v] · ∇v dx, (3.19)

where we have used that ∇ · (A0∇u) = 0 in Ω. We now show that Z

Ω

(δ(x)A0∇u) · ∇v dx and Z

Ω

[(A0+ δ(x)A0)∇v] · ∇v dx are of O(kδk). It is easy to see that

Z

Ω

(δ(x)A0)∇u · ∇v dx

≤ CA₀kδk_L^∞_(Ω)kφkkvk_H1(Ω), (3.20)

and

Z

Ω

[(A0+ δ(x)A0)∇v] · ∇v dx

≤ CA₀ 1 + kδk_L∞(Ω) kvk²_H1(Ω), (3.21)

for some constant C_A₀ > 0 independent of δ. By inserting (3.18) into (3.20), (3.21) and taking kδk_L∞(Ω) sufficiently small, one obtains the desired result. Thus, the Fr´echet derivative of the quadratic form Q_A(φ) at A(x) = A₀is given by

(3.22) dQ_A(φ)

_A=A

0

= Z

Ω

((δ(x)A₀) ∇u) · ∇u dx,

where u ∈ H¹(Ω) is a solution of ∇ · (A0∇u) = 0 in Ω with u = φ on ∂Ω.

Theorem 3.1. The Fr´echet derivative dQA(φ)|_A=A

0 is injective.

Proof. In order to prove that dQA(φ)|_A=A

0 is injective, we only need to show that Z

Ω

(δ(x)A₀∇u) · ∇u dx = 0 implies that δ ≡ 0,

where u ∈ H¹(Ω) is a solution of LA₀u = 0 in Ω with u = φ on ∂Ω. On the other hand, since the last integral in (3.22) vanishes for all such u, then it is equivalent to prove

(3.23)

Z

Ω

((δ(x)A0)∇u1) · ∇u2dx = 0,

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where u₁, u₂∈ H¹(Ω) are solutions of L_A₀u₁ = L_A₀u₂= 0 in Ω. Inspired by [13], we want to find special exponential solutions to prove it.

In order to achieve our aim, we utilize the celebrated quasi-conformal map in the plane. We first identify R² with the complex plane C. Recall that δ(x) ∈ C¹(Ω) since the conductivity A(x) is a C¹(Ω) matrix function. We now extend δ(x) to C (still denoted by δ(x)) with δ(x) ∈ C0¹(C) by considering δ(x) ≡ 0 for |x| > r, for some large constant r > 0 such that Ω b Br(0). Meanwhile, we also extend the constant matrix A₀ to C, denoted by A0(x), such that A₀(x) ∈ C¹(C) with A₀(x) = I₂ (a 2 × 2 identity matrix) for |x| > r.

From [7, Lemma 3.1] and [53, Theorem 2.1], given a C¹-smooth anisotropic conductivity A(x), it is known that there exists a C¹bijective map Φ^(A): R²→ R² with y = Φ^(A)(x) such that

Φ^(A)_∗ A =

det A ◦ (Φ^(A))⁻¹1/2

I2, (3.24)

is a scalar conductivity, where

Φ^(A)∗ A(y) = ∇Φ^(A)(x)A(x)∇(Φ^(A))^T(x) det(∇Φ^(A)(x))

_x=(Φ_(A)₎₋₁_(y) , (3.25)

with

(Φ^(A)∗ A)^i`(y)

:= 1

det(∇Φ^(A))

2

X

j,k=1

∂_xj(Φ^(A))ⁱ(x)∂_xk(Φ^(A))^`(x)A^jk(x)

_x=(Φ_(A)₎₋₁_(y). Moreover, Φ^(A) solves the following Beltrami equation in the complex plane C,

∂Φ^(A)= µA∂Φ^(A), (3.26)

where

µ_A= A²²− A¹¹− 2iA¹² A¹¹+ A²²+ 2√

det A, (3.27)

and

∂ = 1

2(∂x₁+ i∂x₂), ∂ = 1

2(∂x₁− i∂x₂).

Here we point out that the coefficient µA defined in (3.27) is supported in Br⊂ C, since δ(x) = 0 and A0 was extended C¹-smoothly to the identity matrix I2 for the same domain |x| > r.

Next, for the given anisotropic tensor A0, we can find a corresponding Beltrami equation for A0. We do the same extension of A0as before, such that A0(x) ∈ C¹(C) with A0(x) = I2 for |x| > r (for the same large number r > 0). Now, from the representation formulas (3.24) and (3.25), we know that there exists a quasi- conformal map Φ^(A⁰⁾ such that

Ae0:=Φ^(A_∗ ⁰⁾A0= ∇Φ^(A⁰⁾(x)A0∇(Φ^(A⁰⁾)^T(x) det(∇Φ^(A⁰⁾(x))

_x=(Φ_(A0)₎₋₁_(y)

=p

det A0I2

(3.28)

with det A0> 0. Furthermore, Φ^(A⁰⁾solves the Beltrami equation

∂Φ^(A⁰⁾= µ_A₀∂Φ^(A⁰⁾, where

µA0= A²²₀ − A¹¹− 2iA¹²₀ A¹¹+ A²²₀ + 2√

det A0

. (3.29)

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Similarly, we also have that µ_A₀ is supported in the same disc B_rsince A₀= I₂ for

|x| > r. Note that δ = δ(x) is a scalar function, then by using the formula (3.25) and (3.28), one can see that

Φ^(A_∗ ⁰⁾(δ(x)A₀) =∇Φ^(A⁰⁾(x) (δ(x)A₀) ∇(Φ^(A⁰⁾)^T(x) det(∇Φ^(A⁰⁾(x))

x=(Φ^(A0))⁻¹(y)

=δ(x)|_x=(Φ_(A0)₎−1(y)

pdet A0I2.

Let Φ ≡ Φ^(A⁰⁾, eΩ := Φ(Ω) and eu_j(y) := u_j◦ (Φ⁻¹(y) for j = 1, 2, by using (3.23) and change of variables y = Φ(x) via the quasi-conformal map, then we obtain that

Z

Ωe

Φ∗(δ(x)A0)∇yeu1· ∇yeu2dy = Z

Ω

((δ(x)A0)∇u1) · ∇u2dx = 0, (3.30)

where eu_j are solutions of L

Ae₀eu₁= L

Ae₀eu₂= 0 in eΩ.

(3.31)

In fact, (3.31) is equivalent to the Laplace equation ∆yuej = 0 in eΩ for j = 1, 2 because eA0 = √

det A0I2 with det A0 being a positive constant. Based on Calder´on’s constructions [13], we can consider two special exponential solutions in the transformed space as follows. Let ξ ∈ R² be an arbitrary vector and b ∈ R² such that ξ · b = 0 and |ξ| = |b|, then one can define

ue1(y) := eπi(ξ·y)+π(b·y) and ue2(y) := eπi(ξ·y)−π(b·y), (3.32)

and it is easy to check that ue1 and eu2 are solutions of Laplace’s equation. By substituting these exponential solutions (3.32) into (3.30), one has

2π|ξ|² Z

Ωe

δ ◦ Φ⁻¹(y) pdet A0e^2πiξ·ydy = 0, for any ξ ∈ R²,

which implies that δ = 0, due to the positivity of det A₀. This proves the assertion.

Remark 3.1. It is worth mentioning that

(a) Due to the remarkable quasi-conformal mapping in the plane, one can re- duce the anisotropic conductivity equation into an isotropic one. This method helps us to develop the reconstruction algorithm for the anisotropic conductivity equation proposed by Calder´on [13].

(b) The method fails when the space dimension n ≥ 3, because there are no suitable exponential-type solutions for the anisotropic case. For the three- dimensional case, we do not have complex geometrical optics solutions but we have another exponential solution, which is called the oscillating- decaying solution (see [38]).

4. The linearized reconstruction method

Since the tensor of anisotropy A₀ is known a priori, we can now transform the problem to the isotropic case, reconstruct the transformed conductivity on the transformed domain using Calder´on’s method on an arbitrary domain as in [48], and then use the quasi-conformal map to transform the conductivity back to the original one. Recall that from the definitions of our choices of extensions for A0

and δ(x), the representation formulations of (3.27) and (3.29) yield that µA= µA0 in C, and µA= µA0= 0 for |x| > r > 0.

Thus, without loss of generality, we may assume that Φ^(A⁰⁾= Φ^(A)by using µA₀ = µA. In the rest of this article, we simply denote the quasi-conformal map by

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y = Φ(x) ≡ Φ^(A⁰⁾(x) = Φ^(A)(x). By utilizing the change of variables via the quasi-conformal map, we also have that

Z

∂Ω

u1(ΛAu2) dS = Z

∂ eΩeu1 Λ

Aeeu2 dS,

where eΩ = Φ(Ω), eA(y) = Φ^(A)_∗ A(y), defined in (3.25), and uej = uj ◦ Φ⁻¹(y) for j = 1, 2.

Since the DN data is preserved under the quasi-conformal map (i.e., change of variables), we can reconstruct the scalar conductivity ã(x) from Calderón’s method as in [48], and then map it back. Defining Calderón’s linearized bilinear form for the isotropic problem by

B(φ1, φ2) = Z

∂Ω

w1(ΛAw2) dS, (4.1)

w1|∂Ω= φ1= u1|∂Ω and w2|∂Ω= φ2= u2|∂Ω, this can be computed directly from our measured data. Calder´on proved [13] that the Fourier transform of an isotropic conductivity can be decomposed into two terms, one of which is negligible for small perturbations. Thus, denoting the Fourier transform of a function f by bf , by [13]

we can write

b˜

a(z) = bF (z) + R(z), (4.2)

where b˜

a(z) = − 1 2π²|z|²

Z

Ω˜

(1 + ˜δ(x))∇u₁· ∇u2dx

= − 1

2π²|z|² Z

Ω˜

˜

a(x)e^2πi(z·x)dx

F (z) = −b 1 2π²|z|²B

eiπ(z·x)+π(b·x), eiπ(z·x)−π(b·x) R(z) = 1

2π²|z|² Z

Ω˜

δ(∇u˜ 1· ∇v2+ ∇v1· ∇u2) + (1 + ˜δ)∇v1· ∇v2dx.

(4.3)

For small |z|, the term R(z) is small when the perturbation ˜δ is small in magnitude and is to be neglected in numerical implementation. Thus, the isotropic conductivity can be approximated by the inverse Fourier transform of bF (z). Since the deformed domain ˜Ω is not circular, we adopt the algorithm introduced in [48], in order to compute the function bF (z). We will first invert bF (z) numerically to obtain a reconstruction of the scalar isotropic conductivity eA = Φ_∗A(x) on eΩ = Φ^(A⁰⁾Ω.

Next, we compute the mapping Φ^(A⁰⁾by solving the Beltrami equation (3.26). We then pull back the scalar conductivity from the deformed coordinates to the original coordinates by applying (Φ^(A⁰⁾)⁻¹ to ˜a to obtain a(x). Finally, we obtain the anisotropic conductivity A(x) by multiplying a(x) by the known matrix A₀.

5. Numerical implementation

5.1. Numerical solution of the forward problem for data simulation. A finite element method (FEM) implementation of the complete electrode model (CEM) [52] for EIT was developed for data simulation. We first provide the equations of the CEM. Assume the anisotropic conductivity A satisfies Hypothesis2.1.

Then it satisfies the anisotropic generalized Laplace equation

(5.1) ∇ · (A∇u) =

2

X

j,k=1

∂

∂x^j

A^jk(x)∂u

∂x^k

= 0 in Ω.

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The boundary conditions for the CEM with L electrodes are defined as follows.

The current Ilon the l the electrode is given by (5.2)

Z

el

2

X

j,k=1

A^jk(x) ∂u

∂x^k

dS = Il, l = 1, 2, ..., L,

2

X

j,k=1

A^jk(x) ∂u

∂x^k

= 0 off

L

[

l=1

el,

where el is the region covered by the l-th electrode, and ν is the outward normal to the surface of the body. The voltage on the boundary is given by

(5.3) u + z_l

2

X

j,k=1

A^jk(x) ∂u

∂x^k

= U_l on e_l for l = 1, 2, ..., L, where z_lis the contact impedance corresponding to the l-th electrode. For a unique solution to the forward problem, one must specify the choice of ground,

(5.4)

L

X

l=1

U_l= 0, and the current must satisfy Kirchhoff’s Law:

(5.5)

L

X

l=1

I_l= 0.

Denote the potential inside the domain Ω by u or v and the voltages on the boundary by U or V . The variational formulation of the complete electrode model is given by

(5.6) B_s((u, U ), (v, V )) =

L

X

l=1

I_lV¯_l,

where v ∈ H¹(Ω) and V ∈ C^L, and the sesquilinear form B_s: H × H → C is given by

(5.7) Bs((u, U ), (v, V )) = Z

Ω

A∇u · ∇¯v dx dy +

L

X

l=1

1 zl

Z

e_l

(u − Ul)(v − ¯Vl) dS.

Discretizing the variational problem leads to the finite element formulation. The domain Ω is discretized into small triangular elements with N nodes in the mesh.

Suppose (u, U ) is a solution to the complete electrode model with an orthonormal basis of current patterns ϕ_k. Then a finite dimensional approximation to the voltage distribution inside Ω is given by:

(5.8) u^h(z) =

N

X

k=1

α_kϕ_k(z), and on the electrodes by

(5.9) U^h(z) =

N +(L−1)

X

k=N +1

β_{(k−N )}~n_(k−n),

where discrete approximation is indicated by h and the basis functions for the finite dimensional space H ⊂ H¹(Ω) is given by ϕ_k, and α_kand β_{(k−N )}are the coefficients to be determined. Let

(5.10) ~nj= (1, 0, ..., 0, −1, 0, ...0)^T ∈ R^L×1,

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where −1 is in the (j − 1)st position. The choice of ~n_{(k−N )} satisfies the condition for a choice of ground in (5.4), since ~n_{(k−N )}in (5.9) results in

(5.11) U^h(z) =

L−1

X

k=1

β_k, −β₁, ..., −β_L−1

!^T .

In order to implement the FEM computationally we need to expand (5.6) using approximating functions (5.8) and (5.9) with v = ϕ_j for j = 1, 2, ...N and V = ~n_j for j = N + 1, N + 2, ...N + (L − 1) to get a linear system

(5.12) M~b = ~f ,

where −→

b = (−→α ,−→

β )^T ∈ C^{N +L−1} with the vector −→α = (α1, α2, ..., αN) and the vector−→

β = (β1, β2, ...βL−1), and the matrix G ∈ C^{(N +L−1)} is of the form

(5.13) M =

B C

C˜ D

The right-hand-side vector is given by

(5.14) −→

f = (0, ˜I)^T,

where 0 ∈ C^1×N and ˜I = (I₁− I₂, I₁− I₃, ....I₁− I_L) ∈ C^1×(L−1). The entries of

−

→α represent the voltages throughout the domain, while those of−→

β are used to find the voltages on the elctrodes by

(5.15) U^h= C−→

β where C is the L × (L − 1) matrix

(5.16) C =







1 1 1 . . . 1

−1 0 0 . . . 0 0 −1 0 . . . 0

. ..

0 0 0 . . . −1





 .

The entries of the block matrix B are determined in each of the following cases:

• Case (i) 1 ≤ k, j ≤ N.

In this case u^h 6= 0, U^h = 0, v 6= 0, but V = 0. The sesquilinear form can be simplified to

(5.17) Bs((u^h, U^h), (v, V )) :=

Z

Ω

A∇u^h· ∇¯v dx +

L

X

l=1

1 zl

Z

e^l

u^h¯v dS = 0.

Thus, the (k, j) entry of the block matrix B becomes,

(5.18) Bkj=

Z

Ω

A∇φk· ∇φ_jdx +

L

X

l=1

1 zl

Z

e_l

φkφ_jdS.

The entries of the block matrix C are determined as follows:

• Case (ii) 1 ≤ k ≤ N, N + 1 ≤ j ≤ N + (L − 1).

In this case u^h 6= 0, U^h= 0, v = 0, and V 6= 0. The sesquilinear form simplifies to

(5.19) B_s((u^h, 0), (0, V )) := −

L

X

l=1

1 z_l

Z

e^l

u^hV¯_ldS = I₁− I_j+1

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Therefore, entries of C matrix becomes,

(5.20) C_kj= − 1

zl

Z

e_l

ϕ_k(−→n_j)_ldS

• Case (iii) The entries of the block matrix ˜C are determined as follows:

For N ≤ k ≤ N + (L − 1), 1 ≤ j ≤ N. Here u^h = 0, U^h6= 0, v 6= 0, V = 0. The expression for sesquilinear is

(5.21) Bs((0, U^h), (v, 0)) := −

L

X

l=1

1 zl

Z

e^l

U^h¯vldS = 0

Thus the kj entry of the ˜C is

(5.22) C = −˜

^L X

l=1

Z

e_l

ϕ_jdS − 1 zl+ 1

Z

e_j+1

ϕ_j+1dS

• Case (iv) The entries of the block matrix D are determined as follows:

For N ≤ k, j ≤ N + (L − 1). Here u^h= 0, U^h6= 0, v = 0, V 6= 0 The sequilinear form is given by

(5.23) Bs((0, U^h), (0, V )) :=

L

X

l=1

1 zl

Z

e^l

U^hV¯ldS = I1− Ij+1

Thus the entries of matrix D is given by

(5.24) D_kj=

(_|e

1|

z1 +^|e_z^j+1^|

j+1 , j = k − N

|e1|

z1 , j 6= k − N.

Solving (5.12) gives us the coefficients β(k−N ) required for the voltages U^h on the electrodes.

5.2. Computing the quasi-conformal map. A discrete approximation to the quasi-conformal map Φ^(A⁰⁾was computed by solving the Beltrami equation (3.26) by the method referred to as Scheme 1 in [20]. Let T [h] denote the Hilbert transform of a function h

(5.25) T [h] = i

2π lim

→0

Z Z

C\B(z,)

h(ξ)

(ξ − z)²d ¯ξ ∧ dξ, and P [h] denote the Cauchy transform of h

(5.26) P [h] = i

2π Z Z

C

h(ξ) ξ − z

h(ξ)

ξ − d ¯ξ ∧ dξ.

It is shown in [20] that a solution to equation (3.26) can be computed as follows.

1. Begin with an initial guess h⁰ to the solution of the equation h^∗= T [µh^∗] + T [µ]

2. Compute the iterates

hⁿ⁺¹= T [µhⁿ] + T [µ]

until the method converges to within a specified tolerance, and denote the solution by h^∗.

3. Compute the solution f (z) from

f (z) = P [µ(h^∗+ 1)](z) + z

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The iterates converge since the map h 7→ T [µh] is a contraction in L_p(C), p > 2 by the extended version of the Ahlfors-Bers-Boyarskii theorem [4,20].

5.3. Reconstruction. Once we compute w = f (z) for all discretized values of z ∈ Ω, by using the reconstruction algorithm in [48] and the data from the forward modeling in subsection5.1, we compute the isotropic conductivity ˜σ(w) = ˜σ(f (z)).

Note that Calder´on’s method is a pointwise reconstruction algorithm. Then we get σ(z) = ˜σ(f (z)).

6. Examples

In this section we illustrate the method on a radially symmetric discontinuous conductivity on the unit disk with high and low contrast. This simple example was chosen to illuminate the features of the reconstruction and facilitate comparison with the reconstructions by the D-bar method and Calder´on’s method in [35] of an isotropic conductivity of the same nature.

We consider four different conductivity tensors A0 for the background on the unit disk Ω:

A¹₀=

1 0 0 1.3

, A²₀=

1.3 0

0 1

, A³₀=

1 0 0 4

, A⁴₀=

4 0 0 1

. The tensor A¹₀ corresponds to µ = 0.0655, A²₀ corresponds to µ = −0.0655, A³₀ corresponds to µ = 0.3333, and A⁴₀ corresponds to µ = −0.3333, rounded to four digits after the decimal. The images of Ω under the quasi-conformal mapping computed by the method in Section 5.2 with initial guess h⁰ = µA₀ are found in Figure 1.

Figure 1. The unit disk Ω and its image ˜Ω (in blue) under the computed quasi-conformal mapping Φ^(A⁰⁾ for A¹₀ (upper left), A²₀ (upper right), A³₀ (lower left), A⁴₀ (lower right).

The discontinuous radially symmetric isotropic conductivity σ is defined by σM(x) =

1, 0.5 < |x| ≤ 1 M, 0 ≤ |x| < 0.5

Four anisotropic conductivity distributions were then constructed by defining A1(x) ≡ σ1.3(x)A¹₀, A2(x) ≡ σ1.3(x)A²₀, A3(x) ≡ σ4(x)A³₀, and A4(x) ≡ σ4(x)A⁴₀.

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Voltage data was simulated by the method described in Section5with trigonometric current patterns defined by

(6.1) T_`^k =

cos(kθ`), k = 1, ...,^L₂

sin (k −^L₂)θ`, k = ^L₂ + 1, ..., L − 1,

where T_`^k specifies the current amplitude injected on electrode ` located at angular position θ_` for the kth pattern.

Reconstructions of σ_1.3(x) with background tensor A¹₀and A²₀are found in Figure 2, and reconstructions of σ₄(x) with background tensor A³₀ and A⁴₀ are found in Figure 3.

Figure 2. Top row: Cross sections along the x-axis of reconstructions of σ1.3(x) where A0 = A¹₀ computed by Calder´on’s method for anisotropic conductivities with truncation radius R = 1.8 (left) and R = 2.0 (right). Bottom row: Cross sections along the x- axis of reconstructions of σ1.3(x) where A0 = A²₀ computed by Calder´on’s method for anisotropic conductivities with truncation radius R = 1.8 (left) and R = 2.0 (right).

7. Conclusions

A direct reconstruction algorithm for reconstructing the multiplicative scalar field for 2-D anisotropic conductivities with known entries for the background anisotropic tensors was presented based on Calder´on’s linearized method for isotropic conductivities. The quasi-conformal map was used to prove injectivity of the linearized problem in the plane. The map facilitates the reduction of the anisotropic problem to an isotropic problem that can then be solved by Calder´on’s method on the image of the original domain under the mapping, and pulled back to obtain the multiplicative scalar field. The method is demonstrated on simple radially symmetric conductivities with jump discontinuity of high and low contrast. Further work is needed to determine the method’s practicality for more complicated conductivity distributions and experimental data. Also, the method presented here is not applicable to the three-dimensional case.

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Figure 3. Top row: Cross sections along the x-axis of reconstructions of σ4(x) where A0 = A³₀ computed by Calder´on’s method for anisotropic conductivities with truncation radius R = 2.0 (left) and R = 2.3 (right). Bottom row: Cross sections along the x- axis of reconstructions of σ4(x) where A0 = A⁴₀ computed by Calder´on’s method for anisotropic conductivities with truncation radius R = 2.0 (left) and R = 2.3 (right).

Acknowledgment

The project was supported by Award Number R21EB024683 from the National Institute Of Biomedical Imaging And Bioengineering. The content is solely the responsibility of the authors and does not necessarily represent the official view of the National Institute Of Biomedical Imaging And Bioengineering or the National Institutes of Health. Y.-H. L. is partially supported by the Ministry of Science and Technology Taiwan, under the Columbus Program: MOST-109-2636-M-009- 006, 2020-2025. The authors would like to thanks Prof. Ikehata for some useful suggestions and fruitful discussions.

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Department of Mathematics, University of Helsinki, Finland Email address: [email protected]

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Tai- wan

Email address: [email protected]

Department of Mathematics, Colorado State University, USA Email address: [email protected]

Department of Mathematics and School of Biomedical Engineering, Colorado State University, USA

Email address: [email protected]