Chapter 8

Anomaly Estimation and Layer Potential Techniques

Layer potential techniques have been used widely to deal with the inverse problem of recovering anomalies in a homogeneous background. The reason is that the method provides a concrete expression connecting the anomalies with measured data.

For example, consider the inverse problem of detecting an electrical conductivity anomaly, occupying a region D, inside a three-dimensional region Ω bounded by its surface ∂Ω. Assume that the complex conductivity distribution at angular frequency ω changes abruptly across the boundary ∂D and in . With the aid of the fundamental solution F(r): = − 1/(4π|r|) of the Laplacian, we can provide a rigorous connection between the anomaly D and the boundary voltage–current data via the following integral equation (Kang and Seo 1996): for ,

8.1

where g represents Neumann data corresponding to the sinusoidal injection current with an angular frequency ω, u is the induced time-harmonic voltage inside Ω, , is the double-layer potential given by

8.2

and is the single-layer potential given by

8.3

When the Neumann data g and Dirichlet data f are available along the boundary ∂Ω, the inverse problem is to estimate the anomaly D from knowledge of the right-hand side of the identity (8.1). Owing to the expression in the sensitivity part in (8.1) containing location information of D, the formula provides useful information in estimating the anomaly D.

8.1 Harmonic Analysis and Potential Theory

8.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation

For simplicity, we will restrict ourselves to three-dimensional cases, although all the arguments in this chapter work for general dimensions with minor modifications. We also assume that both D and Ω are Lipschitz domains and . The boundary value problem of the Laplace equation can be solved by single- or double-layer potentials with a surface potential density. The reason is that a solution of the Laplace equation in Ω can be expressed as

8.4

where

Here, the relation between the Dirichlet data f and Neumann data g is dictated by

8.5

To see the relation more clearly, define a trace operator

8.6

The operator in (8.6) appears to be the same as in (8.2), but there exists a clear difference between them due to the singular kernel at r = r′. The following theorem explains how the double-layer potential jumps across ∂Ω due to its singular kernel.

Theorem 8.1.1

The trace operator is bounded on . (When ∂Ω is a C¹ domain, is a compact operator.) For , the following trace formulas hold almost everywhere on ∂Ω (and limit in sense):

8.7

8.8

8.9

where I is the identity operator on and is the dual operator of .

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Here, the term “almost everywhere” on ∂Ω means all points except a set of measure zero in ∂Ω. For detailed explanations on these issues in measure theory, see section 4.5. The proof of the boundedness of the trace operator in Theorem 8.1.1 for the Lipschitz domain Ω requires a deep knowledge on the harmonic analysis (Coifman et al. 1982, David and Journé 1984), while the proof in the C² domain Ω is a lot simpler (Folland 1976). To prove Theorem 8.1.1, we need to use the following lemma.

Proof.

For ease of explanation, we restrict ourselves to the case of the C¹ domain Ω. For , we have

If , there exists a ball (as shown in Figure 8.1) such that and application of the divergence theorem over the region leads to

8.10

Direct calculation over the sphere yields

Finally, let (see Figure 8.1b). Denoting , we have

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Figure 8.1 Diagrams showing (a) and and (b) . The ratio is about half for small

Now, we are ready to prove the trace formula in Theorem 8.1.1. We will only prove the trace formula (8.7) under the assumption that ∂Ω is C² and . For ,

From Lemma 8.1.2, and and, therefore,

8.11

Since (this property is not necessary for the proof of (8.7), and see Remark 8.1.3), ϕ(r) − ϕ(r′) = O(|r − r′|) and

8.12

Since as a function of r′ is integrable over ∂Ω, it follows from the Lebesgue dominated convergence theorem and Lemma 8.1.2 that

8.13

From (8.13) and (8.11), we obtain

Remark 8.1.3

The limit (8.13) via the Lebesgue dominated convergence theorem (LDCT) holds true even for the Lipschitz domain Ω and . To apply the LDCT, we use some estimate using the Hardy–Littlewood maximal function to replace the estimate of (8.12). To understand it simply, we assume that is the domain above the graph . For , there is a positive constant C depending only on so that

where ψ_r(r′) = ϕ(r′) − ϕ(r) and for .

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Proof.

We only prove that is invertible. Recall the definition of the null space N and the range R of an operator:

It is well known in functional analysis (see Rudin 1973, theorem 4.12) that

8.18

We will prove that implies ϕ = 0. From (8.9),

where and denote the limits of a function f from outside and inside of Ω, respectively. If , then satisfies

From the uniqueness (up to a constant) of the exterior Neumann boundary value problem, in and in because for |r| = ∞. Hence satisfies the interior Dirichlet boundary value problem

From the maximum principle, in Ω and, therefore, in . As a result, we have

and hence,

From (8.18), and, therefore, the closure of is .

Now we will prove by showing that the range of is closed. According to the Banach closed range theorem (Rudin 1973, p. 96), it suffices to show that has a closed range. We need to prove that, if is the limit of , then there exists such that .

Case (i). Consider the case where . Owing to the weak compactness (Evans 2010), there exists a subsequence weakly in . Then, we have

Hence, .

Case (ii). It remains to prove the case where . We may assume that . Denoting , we have

8.19

Since , there exists a subsequence such that weakly in .

We repeat the previous argument to get

Hence and weakly on ∂Ω. Verchota (1984) obtained the following remarkable contradiction, which indicates that case (ii) is not possible.

Here, we have used the estimate in Lemma 8.15 (see below), (8.19) and the fact that, if weakly on ∂Ω, then

From (i) and (ii), we obtain . Hence, to prove its invertibility, it remains to prove that is one-to-one. It is sufficient to prove that has a dense range on . For the Lipschitz domain Ω, we may use a sequence of C² domains such that . We know that for C² boundary , is a compact operator and the Fredholm alternative theorem can be applied to get . With the aid of this property, we can obtain that the closure of is . (Please refer to Verchota (1984) for the details of the proof.) This completes the proof of the invertibility of . Hence, we also get the invertibility of . To prove the invertibility of , we may repeat similar arguments.

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Lemma 8.1.5 (Verchota 1984)

Let Ω be a bounded Lipschitz domain in with a connected boundary. Then, for all , we have

8.20

8.21

where C depends only on the Lipschitz constant for Ω.

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Proof.

Let and denote

8.22

We decompose on ∂Ω into

The proof is based on the following Rellich identity.

(Rellich identity) Let be a vector field satisfying on ∂Ω. Then, we have the following identity:

8.23

where is

The identity (8.23) follows from

8.24

which can be obtained by applying the Gauss divergence to the following identity:

The Rellich identity (8.23) provides the following estimates:

where the constant C depends on and . Since , we obtain

8.25

8.26

8.27

where the constant C is independent of u and ϕ. Setting

(the average), the term can be estimated by

8.28

where the constant C is independent of u and ϕ. Here, we have used the Poincaré inequality

Combining all the estimates (8.25)–(8.28) with the Hölder inequality, we obtain

8.29

where the constant C is independent of u and ϕ. This completes the proof.

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We briefly summarize the layer potential method for the boundary value problems (BVP) of the Laplace equation on the Lipschitz domain Ω (Fabes et al. 1978). With the aid of the layer potential method, solving the Neumann BVP can be converted into the invertibility of . Since it is invertible, the potential

satisfies

8.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity

In this section, we study the regularity of solutions of elliptic equations on a Lipschitz interface based on the work by Escauriaza and Seo (1993). Throughout this section, let be the unit ball and let D be a Lipschitz domain with a connected boundary contained in B₁/2 where B_r denotes the ball with radius r and center at the origin. We consider a weak solution for the elliptic equation

8.30

where A₀ and A_D are positive constant matrices.

Denote by F₀(r) and F_D(r), respectively, the fundamental solutions of the constant-coefficient elliptic operators

We define the corresponding single-layer potentials and as

The corresponding trace operators are defined as, for r ∈ ∂D,

For ϕ ∈ H¹(B), we define the interior and exterior non-tangential maximal functions of ϕ at r ∈ ∂D, respectively, as

We write

We have the following regularity result for a solution of (8.30).

Theorem 8.1.6 (Escauriaza and Seo 1993)

Suppose A_D − A₀ is either a positive or negative definite matrix. If is a weak solution to (8.30), then

where the constant C depends only on the Lipschitz character of D, A₀, A_D and A_D − A₀.

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We will prove Theorem 8.1.6 assuming that Theorem 8.1.7 has already been proved. Take with ϕ = 1 on B₃/5, and ϕ = 0 outside B₄/5. Then, uϕ satisfies

Introducing the Newtonian potential

we have

Since , it is easy to see that . Let

According to Theorem 8.1.7,

8.31

Define

From (8.31), w satisfies

and we must have w = 0 in from the maximum principle, since w(r) = O(|r|⁻¹) at infinity. Hence, we have the following representation formula:

The theorem follows from the property of the singular integral on the Lipschitz domain.

The main step in the proof of Theorem 8.1.7 is the following estimate.

Proof.

After a linear change of coordinates, we may assume that A₀ = I and A_D = A, where I denotes the identity matrix. From the assumptions, either A − I ≥ αI or I − A ≥ αI for some α > 0. We only consider the case where A − I ≥ αI for α > 0. Denote

and

8.32

Then, u satisfies

There exists C depending on the Lipschitz character of ∂D, A and α such that

8.33

To prove (8.33), we will derive Rellich-type identities as in the proof of Lemma 8.1.5. We choose a vector field with on ∂D, where C₁ depends only on the Lipschitz character of D. Then,

8.34

8.35

Integrating (8.34) over D and applying the divergence theorem, we get

8.36

Using the orthonormal basis {n, t₁, t₂}, we can decompose

and

Applying the above identities to (8.36), we obtain

8.37

Similarly, (8.35) for u⁻ leads to

8.38

We can rewrite the last equality as

From the transmission conditions (8.32), we can rewrite the last equality in terms of the gradient of u⁺, ϕ₁ and ϕ₂ on ∂D, obtaining

8.39

Subtracting (8.39) from (8.37), we have

8.40

From the orthonormality of the linear base {n, t₁, t₂}, we have

for a positive constant C depending only on A and α. From (8.40) with the above estimate, we have

The above estimate together with a similar estimate for give the estimate (8.33). The proof of Lemma 8.1.8 follows from the estimate (8.33) and Lemma 8.1.5.

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Theorem 8.1.9

Under the assumption of Theorem 8.1.7, there exists a positive constant C depending only on the Lipschitz character of D, A_D and A₀ such that, given , there exists a function u satisfying

8.41

8.42

8.43

8.44

8.45

Moreover, u can be represented as

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Proof.

With a linear change of the coordinate system, we may assume that A₀ = I. Denote A_D = A. From the Lax–Milgram theorem, for each r > 1, there is a unique such that

In particular,

Hence, we have

Here, means that there exists a constant C depending only on the Lipschitz character of D, A_D and A₀ such that Φ ≤ CΨ. From the trace theorem, we obtain

From the Sobolev inequality, we have

Using a small constant and Jenson's inequality, we get

and, therefore,

where the constant C is independent of r. Therefore, we can choose a sequence such that weakly. Clearly, this u satisfies (8.42)–(8.44). From the standard Schauder interior estimate, we have

From Theorem 8.1.4 by Verchota (1984), we can find f, g ∈ L²(∂D) such that

From the maximum principle, we have

Hence,

This completes the proof of Theorem 8.1.9.

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Proof of Theorem 8.1.7.

To prove that the operator is one-to-one, let f, g ∈ L²(∂D) satisfy and on ∂D. Then, the function u defined as in D and in lies in and is a weak solution of Lu = 0 in . Thus u must be identically zero by the maximum principle. Hence in and in . From the jump relations,

Therefore, f must be zero on ∂D. Similar arguments give g = 0 on ∂D.

Next, we will prove that has a closed range. As in Theorem 8.1.4, we first assume that

If , we may then assume that f_j → f weakly for some f ∈ L²(∂D) and g_j → g weakly for some g ∈ L²(∂D) and, therefore, we can easily conclude that and .

If , we may assume , and . Since is one-to-one, we may assume that f_j → 0 weakly in L²(∂D) and g_j → 0 weakly in L²(∂D). Since and are compact operators from L²(∂D) to L²(∂D), and converge strongly to zero in L²(∂D). But, since

there is a contradiction.

Now it remains to prove that has a dense range. As in Verchota (1984), approximation of D by smooth domains and the estimate in Lemma 8.1.8 will give that the operator in Theorem 8.1.7 has a dense range, if the range of this operator is dense when D is a smooth domain. Let ϕ₁ and ϕ₂ in be given. From Theorem 8.1.9, we can find f₁ and g in L²(∂D) such that

where . From Verchota (1984), we can find f ∈ L²(∂D) such that

Clearly, and has a dense range when D is smooth. This completes the proof of Theorem 8.1.7.

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8.2 Anomaly Estimation using EIT

In this section, we focus our attention on the estimation of the sizes of anomalies with different conductivity values compared with the background tissues. We describe how to estimate their size using the relationship between injection currents and measured boundary voltages. There are many potential applications where the locations and sizes of anomalies or changes in them with time or space are of primary concern. They include monitoring of impedance-related physiological events, breast cancer detection, bubble detection in two-phase flow and others in medicine and non-destructive testing.

Let denote an electrically conducting medium and let the anomalies occupy a region D contained in the homogeneous medium Ω. Then, the conductivity distribution σ can be written as

8.46

where σ₀ is a positive constant (which will be assumed to be 1 for simplicity) and μ is a constant such that − 1 < μ ≠ 0 < ∞. Physically, σ₀ is the conductivity of the homogeneous background and σ₀μ: = σ_D − σ₀, where σ_D is the conductivity of the anomaly D. A high contrast in conductivity occurs at the interface ∂D between the anomaly D and the background .

The goal is to develop an algorithm for extracting quantitative core information about D from the relationship between the applied Neumann data and the measured Dirichlet data . Here u is the induced potential due to the Neumann data g, and it is determined by solving the Neumann problem:

8.47

Throughout this section, , where u₀ is the potential satisfying

8.48

Theorem 8.2.1 (Kang and Seo 1996)

The function for can be expressed as

8.49

where F(r) = − 1/(4π|r|). The difference can be expressed as

8.50

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Proof.

Using the fact that in the distribution sense, we have

for all . We now use the refraction condition of u along ∂D to get

8.51

These prove (8.49). The expression (8.50) follows from (8.49) and

8.52

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This theorem plays an important role in extracting location information about D. This section considers non-iterative anomaly estimation algorithms for searching its location and estimating its size.

8.2.1 Size Estimation Method

Kang et al. (1997) derived that, with the special Neumann data g = a · n where a is a unit constant vector, the volume of D can be estimated by

8.53

where u₀ is the corresponding solution of (8.47) with the homogeneous conductivity distribution. Alessandrini et al. (2000) provided a careful analysis on the bound of the size of inclusions for a quite general g and conductivity distribution. In this section, we will explain the results of Alessandrini et al. (2000) in detail.

Before presenting an analysis of size estimation, we begin by explaining the algorithm to estimate the total size of anomalies proposed by Kwon and Seo (2001). The total size estimation of anomalies uses the projection current g = a · n, where a is a unit constant vector. We may assume that Ω contains the origin. Define the scaled domain for a scaling factor t > 0. Let v_t be the solution of the problem

The following lemma provides a way to compute |D|.

Proof.

Let as a function of t defined in the interval (0, 1). If t₁ < t₂, it follows from integration by parts that

These identities give the monotonicity of η(t):

Since , a similar monotonicity argument leads to the following inequalities;

Since η(t) is continuous, there exists a unique t₀ so that .

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With the aid of the above lemma, Kwon and Seo (2001) developed the following method of finding the total size of multiple anomalies.

• Suppose that . With an applied current g = a · n, where a is a unit constant vector, choose the unique t₀, 0 < t₀ < R, so that

• Then, the size of the ball is a good approximation of the total size of .

Various numerical experiments indicate that the above algorithm gives a nearly exact estimate for arbitrary multiple anomalies with a quite general conductivity distribution, as shown in Figure 8.2.

Figure 8.2 Numerical simulation for size estimation using Lemma 8.2.2 and phantom experiments for size estimation and location search. From Kwon et al. (2003). Reproduced with permission from IEEE

Next, we will provide an explanation on the background idea of the size estimation for the case where μ is small. Integrating by parts yields

where σ = 1 + μχ_D and . By adding the above two identities, we obtain

According to the choice of t₀,

If μ ≈ 0, then and are approximately constant, and the above identity is possible when the volume of is close to the total volume .

Theorem 8.2.3 (Alessandrini et al. 2000)

Let Ω be a bounded domain in with a connected C¹ boundary such that is connected for all d > 0. Let for a fixed d₀ > 0. Assume that the conductivity distribution is

where λI ≤ σ ≤ (1/λ)I for some λ > 0, and A_D and A₀ are constant matrices such that αI ≤ A_D − A₀ and A_D ≤ μA₀ for some positive constants α, μ. Then, we have

8.54

where p > 1, C₁ and C₂ are positive constants depending on d₀, λ, Ω and

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Lemma 8.2.6

Under the assumption of Theorem 8.2.3, we have the following estimate for every ρ > 0:

8.60

where C_ρ depends only on ρ, d₀, λ, Ω and

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Proof.

From the three sphere inequality given by Garofalo and Lin (1986), there exist C ≥ 1 and δ ∈ (0, 1) depending on λ and d₀ such that

where

Applying the Caccioppoli and Poincaré inequalities, we have

8.61

which leads to

8.62

Here, the constant C differs on each occurrence but all of the C are independent of u₀ and depend only on ρ, d₀, λ, Ω and .

Let be any two points. Note that the two points and r′ can be joined by a polygonal arc with node points such that |r_j − r_j+1| = 2ρ for j = 1, 2, …, L − 1 and . From the estimate (8.62), we have

8.63

and by induction

8.64

By covering with balls of radius ρ using the estimate (8.64), we have

8.65

and, therefore,

8.66

It remains to derive an appropriate lower bound for W to estimate (8.60). To be precise, it suffices to prove that there exists ρ_* > 0 depending only on λ, Ω, and such that

8.67

If (8.67) holds true, then for ρ < ρ_*, and (8.66) and (8.67) yield

8.68

If (8.68) holds true, it is obvious that the estimate on the right-hand side of (8.68) holds true for ρ > ρ_*.

Hence, it remains to prove the estimates (8.67). It can be proven by showing that there exists a ρ_* such that

8.69

The proof of (8.69) follows from a careful adaptation of results by Kenig et al. (2007):

8.70

where C > 0 depends on λ and the Lipschitz character of Ω only. The proof of the estimate (8.70) requires a deep knowledge of harmonic analysis when ∂Ω is only Lipschitz, while it is a lot simpler when ∂Ω is C². For ease of explanation, we will give the proof under the assumption that ∂Ω is in C². By the Hölder inequality and Sobolev inequality,

8.71

where C depends on Ω only. Moreover, we have

8.72

where C depends on λ and Ω only. From (8.71) and (8.72), we have

8.73

Using the standard estimate and (8.73), we have

8.74

where C depends on λ and Ω. Hence, we obtain (8.69) by choosing a sufficiently small ρ_*. This completes the proof.

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Now, we will prove Theorem 8.2.3. For p > 1, we have

8.75

We can cover D with internally non-overlapping closed cubes Q_j, j = 1, …, J, with diameter ρ and ρ < d₀/6. Then, (8.75) yields

8.76

We will take advantage of the fact that is a Muckenhoupt weight (Garofalo and Lin 1986):

8.77

where M and p depend only on ρ, d₀, λ, Ω and . The left term in (8.77) can be estimated by

From (8.76) and the above estimate, we have

8.78

By (8.60) and (8.78), we have

8.79

where , C₂ = C₁C_ρλ and C_ρ is the constant in (8.60). From (8.79) and the left-hand side of (8.56), we have

This completes the proof of the right-hand side of (8.54).

8.2.2 Location Search Method

Kwon et al. (2002) developed a location search method to detect an anomaly using a pattern injection current g and boundary voltage f. We assume that the object contains a single anomaly D that is small compared with the object itself and is located away from the boundary ∂Ω. The location search algorithm is based on simple aspects of the function H(r) outside the domain Ω, which can be computed directly from the data g and f.

We choose for some fixed constant vector a.

1. Take two observation regions Σ₁ and Σ₂ contained in given by

2. Find two points P_i ∈ Σ_i(i = 1, 2) so that

and

3. Draw the corresponding plane Π₁(P₁) and the line Π₂(P₂) given by

4. Find the intersecting point P of the plane Π₁(P₁) and the line Π₂(P₂). Then this point P can be viewed as the location of the anomaly D.

Figure 8.3 illustrates how the location search method works. The above location search method is based on the assumption that for some constant α. Noting that with the injection current g = a · n, it follows from (8.50) that

8.80

where

We can express w as a single-layer potential and, therefore, w is harmonic both in D and in . Hence, we can get the following observation by the mean value property of harmonic functions in , and the uniqueness of the interior Dirichlet problem for the Laplace equation in D by considering the limit value of w(r) to the boundary ∂D.

Figure 8.3 Relations between the location of the anomaly and the pattern of H(r) in the case of μ > 0

Observation 8.2.7 (Kwon et al. 2002)

The function w(r) in (8.80) is harmonic in . If D = B_r(r_*) is a ball, w(r) satisfies

8.81

Assuming H ≈ (μα/σ₀)w outside Ω and examining the sign of w in (8.81) as depicted in Figure 8.3, we have the following:

8.82

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We should note that this nice observation is made under the assumption that , that is, inside the anomaly D is a fixed constant vector . Hence, we need to check whether the current generates for some scalar α. In the special case where Ω and D are concentric balls, we can compute u explicitly via Fourier expansion.

Observation 8.2.9

For ease of explanation, let σ₀ = 1 and . Let and D = B_ρ(r_*), the ball centered at r_* of radius ρ. Assume that for a positive number d and that μ ≠ 0 satisfies − 1 + 1/μ₀ ≤ μ ≤ μ₀ for a number μ₀ ≥ 1. Take an observation line Σ₁: = a line parallel to a with . Then, there exists with as ρ → 0, so that for the point P₁ ∈ Σ₁ satisfying H(P₁) = 0, the plane or the line Π₁(P₁): = {r:a · (r − P₁) = 0} hits the -neighborhood of the center r_* of D.

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Proof.

We will compare u with a new function v (which makes quantitative analysis a lot easier) defined by

8.84

where the constant c is chosen so that . Here, the term 3μ/(3 + μ) is chosen so that v satisfies in , which will be explained below.

From Observation 8.2.7, we have

8.85

A straightforward calculation of (8.85) using parametric spherical coordinates centered at r_* yields

where v⁺ = v|_D and . The above identities lead to the transmission condition

and, therefore, v satisfies the conductivity equation in . The Neumann data of v on is given by

where E(r, r_*) is defined as

Hence, η: = u − v solves the following Neumann problem:

8.86

We will estimate that

8.87

By simple arithmetic, we can decompose E(r, r_*) into the following two terms:

Since R = |r|, we get the following by a simple Cauchy–Schwarz inequality:

In order to estimate U(r, r_*), consider the spherical coordinates for and r_* with θ and α as latitudes measured from the xz plane to the y axis, respectively:

for − π/2 ≤ θ, α ≤ π/2 and 0 ≤ η, β ≤ 2π. Denoting t: = τ/R and r: = cos(η − β), U(r, r_*) is expressed as

8.88

By a tedious calculation with (8.88), we get

Thus, U(r;t, α, β, θ): = U(r, r_*) is monotonic for − 1 ≤ r ≤ 1 for fixed t, α, β (that is, for fixed r_*) and θ. To check the extremes of U(r, r_*), we only need to check

for 0 ≤ t < 1 and − π/2 ≤ θ, α ≤ π/2. Letting s: = cos(θ ± α), we get

Then, it is easy to get |U(r, r_*)| ≤ 1. Hence, we obtain the desired assertion (8.87):

Since , we see that |r − r_*| ≥ d for all . In addition, the fact that |μ|/(3 + μ) ≤ 1 for all − 1 < μ < ∞ gives the estimate

8.89

Under the assumption that − 1 < − 1 + 1/μ₀ ≤ μ ≤ μ₀ < ∞, we are ready to compare H(r) with

in the region . From (8.49), we have

8.90

Therefore, by the observation that

we obtain the following estimate by the standard Hölder estimate of (8.90):

8.91

We claim that there exists a positive constant so that

8.92

The proof of (8.92) is a bit technical and lengthy, so we omit its here. Conditions (8.91) and (8.92) lead to the following estimate:

8.93

Note that by using the mean value theorem for harmonic functions, we easily get

8.94

From (8.93) and (8.94), we obtain the final estimate

8.95

where is given by

8.96

Now let be defined as follows, which clearly satisfies as ρ → 0:

where , independent of ρ, is given by (8.96). Because there is nothing to be proved if , we may assume that ρ < 1/C from now on.

For r ∈ Σ₁ satisfying , we have

Hence, by the positiveness of (r − r_*) · a and the definition of , we obtain

On the other hand, for r ∈ Σ₁ satisfying , we have

Being careful of the negativeness of (r − r_*) · a, we similarly obtain

In the end, by the estimate (8.95) we have, if μ > 0,

Note that the sign of H(r) is exchanged if μ < 0. Therefore, the zero point P₁ ∈ Σ₁ of H(r) satisfies , which completes the proof.

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Although the basic idea of the algorithm is simple, several technical arguments are needed for its proof. Combining this location search algorithm with the size estimation algorithm proposed in Kwon and Seo (2001), one can select an appropriate initial guess. Figure 8.4 explains this algorithm.

Figure 8.4 Location detection by finding an intersecting point of two lines Π₁(P₁) and Π₂(P₂)

In order to test the feasibility of the location search and size estimation methods, Kwon et al. (2003) carried out phantom experiments. They used a circular phantom with 290 mm diameter as a container and filled it with NaCl solution of conductivity 0.69 S m⁻¹. Anomalies with different conductivity values, shapes and sizes were placed inside the phantom. A total of 32 equally spaced electrodes were attached on the surface of the phantom. Using a 32-channel EIT system, they applied the algorithms to measured boundary voltage data. The circular phantom can be regarded as a unit disk by normalizing the length scale. To demonstrate how the location search and size estimation algorithm work, they placed four insulators into the phantom:

They injected a projection current with and measured the boundary voltage f. For the location search, they chose two observation lines:

They evaluated the two-dimensional version of H(r) with F replaced by . In Figure 8.3, the left-hand plot is the graph of H(r) on Σ₁ and the right-hand plot is the graph of H(r) on Σ₂. They found the zero point of H(r) on Σ₁ and the maximum point of |H(r)| on Σ₂ as denoted by the black dots in Figure 8.3. The intersecting points were calculated as P( − 0.1620, − 0.0980), which was close to the center of mass P_M( − 0.1184, − 0.0358). For the case of a single anomaly or a cluster of multiple anomalies, the intersecting point furnished meaningful location information.

For the size estimation, the estimated total size was 0.4537 compared with the true total size of 0.4311. In Figure 8.2, the corresponding disk with the size of 0.4537 centered at P( − 0.1620, − 0.0980) is drawn with a solid line and the corresponding disk with the true size centered at P_M is drawn with a dotted line. The relative error of the estimated size was about 5.24%.

8.3 Anomaly Estimation using Planar Probe

In this section, we describe anomaly estimation techniques using a local measurement from a planar probe placed on a portion of an electrically conducting object. The detection of breast cancer using a trans-admittance scanner (TAS) is a typical example of this setting. TAS is based on the experimental findings showing that the complex conductivity values of breast tumors differ significantly from those of surrounding normal tissues (Assenheimer et al. 2001; Hartov et al. 2005; Jossinet and Schmitt 1999; Silva et al. 2000). In TAS, with one hand a patient holds a reference electrode through which a sinusoidal voltage V₀sinωt is applied, while a scanning probe at the ground potential is placed on the surface of the breast. The voltage difference V₀sinωt produces an electric current flowing through the breast region (see Figure 8.5). The resulting electric potential at position r = (x, y, z) and time t can be expressed as the real part of u(r) e^iωt, where the complex potential u(r) is governed by the equation in the object, where σ and denote the conductivity and permittivity, respectively. The scanning probe is equipped with a planar array of electrodes and we measure exit currents (Neumann data) , which reflect the electrical properties of tissues under the scanning probe.

Figure 8.5 Configuration for breast cancer detection using TAS

The inverse problem of TAS is to detect a suspicious abnormality in a breast region underneath the probe from measured Neumann data g. One may utilize the difference g − g₀, where g₀ is reference Neumann data measured beforehand without any anomaly inside the breast region (Ammari et al. 2004; Kim et al. 2008; Seo et al. 2004). This difference g − g₀ can be viewed as a kind of background subtraction, so that it makes the anomaly apparently visible. However, it may not be available in practice, and calculating g₀ is not possible since the inhomogeneous complex conductivity of a specific normal breast is unknown. In such a case, we should use a frequency-difference TAS method.

8.3.1 Mathematical Formulation

Let the human body occupy a three-dimensional domain Ω with a smooth boundary ∂Ω. Let Γ and γ be portions of ∂Ω, denoting the probe plane placed on the breast and the surface of the metallic reference electrode, respectively. Through γ, we apply a sinusoidal voltage of V₀sinωt with frequency f = ω/2π in the range of 50 Hz to 500 kHz. Then the corresponding complex potential u_ω at ω satisfies the following mixed boundary value problem:

8.97

where n is the unit outward normal vector to the boundary ∂Ω. Note that both and depend on ω. The scan probe Γ consists of a planar array of electrodes and we measure the exit current g_ω(j) through each electrode :

In the frequency-difference TAS, we apply voltage at two different frequencies, f₁ = ω₁/2π and f₂ = ω₂/2π, with 50 Hz ≤ f₁ < f₂ ≤ 500 kHz, and measure two sets of corresponding Neumann data and through Γ at the same time. We assume that there exists a region of breast tumor D beneath the probe Γ so that changes abruptly across ∂D. The inverse problem of frequency-difference TAS is to detect the anomaly D beneath Γ from the difference between and .

In order for any detection algorithm to be practicable, we must take account of the following limitations.

a. Since Ω differs for each subject, the algorithm should be robust against any change in the geometry of Ω and also any change in the complex conductivity distribution outside the breast region.

b. The Neumann data g_ω are available only on a small surface Γ instead of the whole surface ∂Ω.

c. Since the inhomogeneous complex conductivity of the normal breast without D is unknown, it is difficult to obtain the reference Neumann data in the absence of D.

These limitations are indispensable to a TAS model in practical situations. In the frequency-difference TAS model, we use a weighted frequency difference of Neumann data instead of . The weight α is approximately

and it is a crucial factor in anomaly detection. We should note that the simple difference may fail to extract the anomaly owing to the complicated structure of the solution of the complex conductivity equation. We need to understand how reflects a contrast in complex conductivity values between the anomaly D and surrounding normal tissues.

We assume that σ and are isotropic, positive and piecewise smooth functions in . Let u_ω be the solution of (8.97). Denoting the real and imaginary parts of u_ω by and , the mixed boundary value problem (8.97) can be expressed as the following coupled system:

8.98

The measured Neumann data g_ω can be decomposed into

The solution of the coupled system (8.98) is a kind of saddle point (Borcea 2002; Cherkaev and Gibiansky 1994), and we have the following relations:

8.99

and

8.100

where we have and .

In order to detect a lesion D underneath the scanning probe Γ, we define a local region of interest under the probe plane Γ as shown in Figure 8.6. For simplicity, we let z be the axis normal to Γ and let the center of Γ be the origin. Hence, the probe region Γ can be approximated as a two-dimensional region , where L is the radius of the scan probe. We set the region of interest inside the breast as a half-ball , as shown in Figure 8.6, where B_L is a ball with radius L and its center at the origin.

Figure 8.6 (a) Simplified model of the breast region with a cancerous lesion D under the scanning probe. (b) Schematic of the scanning probe in the (x, y) plane

Remark 8.3.1

In Table 8.1 we summarize the conductivity and permittivity values of normal and tumor tissues in the breast. Both σ and have a unit of S m⁻¹ and , where is the permittivity of free space and is the relative permittivity. Note that for a frequency f = ω/2π ≥ 50 kHz (Surowiec et al. 1988).

Table 8.1 Conductivity and permittivity values of normal and tumor breast tissues

---

For successful anomaly detection, we should carefully choose the two frequencies ω₁ and ω₂. One may choose f₁ = ω₁/2π and f₂ = ω₂/2π such that

8.101

We denote by u₁ = v₁ + ih₁ and u₂ = v₂ + ih₂ the complex potentials satisfying (8.98) at ω₁ and ω₂, respectively, and let and . The frequency-difference TAS aims to detect D from a weighted difference between g₁ and g₂.

Now, let us investigate the connection between u₁ and u₂ and whether the frequency-difference Neumann data g₂ − αg₁ contain any information about D. Since both σ and depend on ω and , σ(r, ω₁) ≠ σ(r, ω₂) and . For simplicity, we denote

Suppose there is a cancerous lesion D inside and the complex conductivity changes abruptly across ∂D as in Table 8.1. To distinguish them, we denote

8.102

With the use of this notation, u₁ and u₂ satisfy

8.103

Remark 8.3.2

Owing to the complicated structure of (8.99) and (8.100) for the solution u_ω, it is quite difficult to analyze the interrelation between the complex conductivity contrast and the Neumann data g_ω. The simple frequency-difference data g₂ − g₁ on Γ may fail to extract the anomaly for more general cases of complex conductivity distributions in Ω because of the complicated structure of the solution of (8.98). To be precise, the use of the weighted difference is essential when the background comprises biological materials with non-negligible frequency-dependent complex conductivity values. To explain this clearly, consider a homogeneous complex conductivity distribution in Ω, where depends only on ω. Owing to the frequency dependence, the simple difference g₂ − g₁ is not zero, while g₂ − αg₁ = 0. Hence, any reconstruction method using g₂ − g₁ always produces artifacts because g₂ − g₁ does not eliminate modeling errors. See (8.113) for an approximation of g₂ − g₁ in the presence of an anomaly D.

---

The next observation explains why we should use a weighted difference g₂ − αg₁ instead of g₂ − g₁.

Observation 8.3.4

Denoting

it follows from a direct computation that u₂ − u₁ satisfies

8.104

For the detection of D, we use the following weighted difference:

where α = η|_Γ. If in (8.104), then u₁ = u₂ in Ω and g₂ − αg₁ = 0 on Γ. In other words, if in , it is impossible to detect D from g₂ − αg₁ = 0 regardless of the contrasts in σ and across ∂D. Any useful information on D could be found from non-zero g₂ − αg₁ on Γ when is large along ∂D.

For chosen frequencies ω₁ and ω₂, we can assume that σ and are approximately constant in the normal breast region and also in the cancerous region D. Hence, if η changes abruptly across ∂D, we roughly have

and therefore the term in (8.104) is supported on ∂D in the breast region . This explains why the difference g₂ − αg₁ on Γ can provide information of ∂D. We note that the inner product is to be interpreted in a suitable distributional sense if the coefficients jump at ∂D.

---

8.3.2 Representation Formula

Observation 8.3.4 in the previous section roughly explains how D is related to g₂ − αg₁. In this section, the observation will be justified rigorously in a simplified model. We assume that σ_{j, n}, σ_{j, c}, and are constants. According to Table 8.1, the change in conductivity due to the change in frequency is small, so we assume that

8.105

Since the breast region of interest is relatively small compared with the entire body Ω, we may assume that Ω is the lower half-space and γ = ∞.

Suppose that v_j and h_j are H¹-solutions of the following coupled system for j = 1, 2:

8.106

Let u_j = v_j + ih_j. Then V₀(1 − u_j) can be viewed as a solution of (8.103) with and γ = ∞.

Let us introduce a key representation formula explaining the relationship between D and the weighted difference g₂ − αg₁. For each , we define

where is the reflection point of r′ with respect to the plane {z = 0} and φ(r, · ) is the -solution of the following PDE:

The following theorem explains an explicit relation between D and .

Now, let us derive a constructive formula extracting D from the representation formula (8.107) under some reasonable assumptions. We assume that

8.108

where C₁ is a positive constant, B_δ is a ball with radius δ and center , and . Suppose we choose ω₁/2π ≈ 50 Hz and ω₂/2π ≈ 100 kHz. Then the experimental data in Remark 8.3.1 shows that

Hence, in practice, we can assume that

8.109

Based on the experimental data in Remark 8.3.1, we assume that

8.110

where κ₁ and κ₂ are positive constants less than and κ₃ is a positive constant less than 10. Taking advantage of these, we can simplify the representation formula (8.107).

Theorem 8.3.6

Under the assumptions (8.108) and (8.110), the imaginary part of the weighted frequency difference g₂ − αg₁ can be expressed as

8.111

where

and the error term Error(r) is estimated by

8.112

Here, is a polynomial function of order n such that and its coefficients depend only on κ_j, j = 1, 2, 3.

---

Remark 8.3.7

According to Theorem 8.3.6,

Hence, even if and are quite different, we cannot extract any information on D when

On the other hand, even if , we can extract information on D whenever

---

Remark 8.3.8

Based on (8.111), we can derive the following simple approximate formula for the reconstruction of D:

8.113

where U is the solution of (8.106) in the absence of any anomaly at . Note that the difference g₂ − g₁ can be approximated by

8.114

and therefore any detection algorithm using the above approximation will be disturbed by the term .

---

Remark 8.3.9

The reconstruction algorithm is based on the approximation formula (8.113). In practice, we may not have a priori knowledge of the background conductivities. In that case, α is unknown. But α can be evaluated approximately by the ratio of the measured Neumann data as follows:

8.115

Hence, we may choose .

---

We can prove the identity (8.115) for a bounded domain Ω. Using u₁|_γ = u₂|_γ = V₀, we have

The identity (8.115) follows from the fact that

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