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 ,
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
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.
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
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.
8.8
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.
8.10
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,
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
Since as a function of r′ is integrable over ∂Ω, it follows from the Lebesgue dominated convergence theorem and Lemma 8.1.2 that
From (8.13) and (8.11), we obtain
8.14
8.15
8.16
8.17
8.20
8.21
8.22
8.24
8.26
8.27
8.29
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
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 Br denotes the ball with radius r and center at the origin. We consider a weak solution for the elliptic equation
where A0 and AD are positive constant matrices.
Denote by F0(r) and FD(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).
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,
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.
8.38
8.41
8.43
8.45
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: = σD − σ0, where σD 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:
Throughout this section, , where u0 is the potential satisfying
8.48
8.51
8.52
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.
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 vt be the solution of the problem
The following lemma provides a way to compute |D|.
With the aid of the above lemma, Kwon and Seo (2001) developed the following method of finding the total size of multiple anomalies.
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.
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 + μχD 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 .
8.55
8.59
8.61
8.63
8.65
8.74
Now, we will prove Theorem 8.2.3. For p > 1, we have
We can cover D with internally non-overlapping closed cubes Qj, j = 1, …, J, with diameter ρ and ρ < d0/6. Then, (8.75) yields
We will take advantage of the fact that is a Muckenhoupt weight (Garofalo and Lin 1986):
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
where , C2 = C1Cρλ and Cρ 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).
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.
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
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.
8.82
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.
8.83
8.84
8.86
8.89
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.
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 PM( − 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 PM is drawn with a dotted line. The relative error of the estimated size was about 5.24%.
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) eiωt, 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.
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.
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 uω at ω satisfies the following mixed boundary value problem:
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 gω(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.
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 uω be the solution of (8.97). Denoting the real and imaginary parts of uω by and , the mixed boundary value problem (8.97) can be expressed as the following coupled system:
The measured Neumann data gω 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:
and
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 BL is a ball with radius L and its center at the origin.
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
The next observation explains why we should use a weighted difference g2 − αg1 instead of g2 − g1.
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 σj, n, σj, 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 vj and hj are H1-solutions of the following coupled system for j = 1, 2:
Let uj = vj + ihj. Then V0(1 − uj) 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
where C1 is a positive constant, Bδ 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
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).
8.112
8.114
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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