Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging

Kwon, Kiwoon

doi:https://doi.org/10.1155/2015/908251

Advances in Mathematical Physics

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Review Article | Open Access

Volume 2015 | Article ID 908251 | https://doi.org/10.1155/2015/908251

Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging

Kiwoon Kwon¹

Academic Editor: Hagen Neidhardt

Received12 Nov 2014

Accepted12 Jan 2015

Published31 Jan 2015

Abstract

Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known. However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.

1. Introduction

Let us consider the following second order elliptic partial differential equations with Diriclet boundary value in a Lipschitz domain compactly embedded in :where is a nonnegative number, are real functions satisfying , , and is a real matrix and is a nonnegative function such that for all and some positive constants and . It is known that (1a) and (1b) have a unique solution [1]. Therefore, we can define the Dirichlet-to-Neumann map as using the boundary trace operator. We will explain EIT, ISP, and DOT using (1a) and (1b) and Dirichlet-to-Neumann map or corresponding measurements map.

When , EIT is formulated as to find such that for given Dirichlet-to-Neumann map . In finite measurements case, EIT is to find the conductivity satisfying , , for given finite Dirichlet and Neumann boundary measurement pairs .

We discuss uniqueness of nonuniqueness of EIT by classifying into six cases for a Lipschitz domain compactly embedded in such that is connected: Case 1: , , Case 2: , Case 3: , , Case 4: , , Case 5: , Case 6: , ,where (i): a positive number,(ii): a positive function,(iii): a symmetric positive-definite matrix,(iv): a symmetric positive-definite matrix function,(v): the identity matrix.

For Cases , and 4, uniqueness of the coefficient is known for general conditions on regularity of the conductivity. For Case 5, cloaking is heavily studied recently. Invisibility and cloaking of are closely related with the nonuniqueness of coefficients not only in EIT [2–6], but also in acoustic scattering [7–9], electromagnetic scattering [10–12], and quantum scattering [13]. The idea of physical devices related to cloaking or invisibility is suggested more concretely than before such as wormhole and metamaterials and these studies draws much attention from physics and material engineering societies [14–16]. However, in Case 6, cloaking of the domain of anomaly is impossible but the illusion of the property is possible as in [11].

Acoustic wave propagation is described as follows: where is a positive-definite matrix function, is a positive function, and It is known that there exists a unique function satisfying (3) and (4) [17]. is called scattering wave. The scattering wave is approximated by far-field pattern by the following relation:

The inverse scattering problem (ISP) is defined as follows: given far-field patterns for all incident directions , , identify coefficients and .

In this paper, we classify and as the following cases: Case 7: , , , Case 8: , , , Case 9: , , Case 10: , , Case 11: , , , , where is a complex number and is a complex function. Note that from (2) we have for all .

ISP for (4) with given in Cases , and 11 is equivalently formulated as an inverse problem to find and from in (1a) and (1b) for some Lipschitz domain compactly embedded in and containing and . By Theorem in [18], we can take as a ball centered at the origin with radius being chosen such that is not a Dirichlet eigenvalue of (1a) with .

In this paper, DOT is explained as an inverse problem with respect to a forward problem formulated as an elliptic partial differential equation. Propagation of light in biological tissues is usually described by diffusion approximation equation in the frequency domain, the simplest but nontrivial approximation of the Boltzmann equation, as follows:where is photon density distribution, absorption coefficient, reduced scattering coefficient, a diffusion coefficient, and refractive index. Usually, we assume and is boundary reflection coefficient, the speed of light, modulation frequency of light, and outer unit normal vector.

DOT is to find the optical coefficients and from the measurement informations which is the value of the solution of (7a) and (7b) at when , . The and are usually called source and detector point, respectively.

Near infrared light is known to be deepest in the penetration depth to the tissue, compared to visible or near-visible lights. Therefore, near infrared light is used in DOT. DOT is known to be of low cost, portable, nonionized, and nonmagnetized. And DOT has higher temporal resolution and more functional information than conventionl structural medical imaging modalities such as magnetic resonance imaging (MRI) and computerized tomography (CT). For the comparison to other functional imaging modalities such as functional MRI (fMRI), photon emission tomography (PET), and electroencephalogram (EEG), see [19]. DOT is used in the area of breast imaging [20–22], functional neuroimaging [23, 24], brain computer interface (BCI) [25, 26], and the study about seizure [27, 28], new born infants [29, 30], osteoarthritis [31], and rat brain [32, 33].

We interpret DOT also as an inverse problem for (1a) and (1b) in isotropic coefficient and the uniqueness is discussed in the following cases: Case 12: is given and , space dimension , Case 13: is given and , space dimension , Case 14: or is to be determined.

In Sections 2, 3, and 4, uniqueness and nonuniqueness of EIT, ISP, and DOT are treated, respectively. The relation between nonuniqueness and cloaking or illusion is also explained.

2. EIT

Consider for the Dirichlet elliptic problem (1a) and (1b). EIT is formulated as follows:(i)find such that for given or measured Dirichlet-to-Neumann map .

2.1. Uniqueness in Cases 1, 2, 3, and 4

The uniqueness studies of EIT in Cases , and 4 are summarized in the following way.

Case 1. One or two measurements uniqueness results are known when is assumed to be known [34–36]. The finite measurement uniqueness is known only for Case 1. In this case, the number of measurements, the geometry of the obstacle , and the choice of suitable Dirichlet or Neumann data minimizing the number of measurements are interesting issues.

Case 2. Many mathematicians conduct extensive works in this case and thus we are able to understand the unique determination of electrical conductivity when for two dimensional case [37–39] and for the case in dimensions higher than two [40–43].

Case 3. In [44], an orthogonality relation between the two solutions of (7a) and (7b) for arbitary obstacles and is derived. Based on this orthogonal relation and the Hahn-Banach theorem, the uniqueness for Case 3 is derived.

Case 4. The uniqueness in Case 4 is also proved in [44] with additional condition that is positive-definite: this additional condition is generalized and removed by [45, 46].

2.2. Nonuniqueness in Case 5

The nonuniqueness of EIT in Case 5 is observed early in [47]: if is a boundary fixing diffeomorphism on and push-forward map is defined by then we have

The proof of (9) is summarized well in [6]: Knowing the Dirichlet-to-Neumann map is equivalent to knowing the quadratic form by the polarization identity. Then, (9) follows from the change of variable method for the quadratic form

Many researchers raised questions whether the change of variable (9) is a unique obstruction to the uniqueness. And it is proved that for some boundary fixing diffeomorphism in the following cases: (i) and [48],(ii) and (Lipschitz functions) [49],(iii) and [50],(iv): and are analytic [51, 52].

2.3. Near-Cloaking in Case 5

Let be some domain and contained in , , , and Define for a boundary-fixing diffeomorphism on transforming into arbitrary small domain and on . Then, is nearly cloaked into , since by (9). If goes to , is called perfectly cloaked. Otherwise, is called nearly cloaked into . Conversely, if is nearly cloaked into , there is a diffeomorphism such that by (12).

For example, if and are two-dimensional disks of radii 2, 1, and , respectively, centered at , we could take the diffeomorphism on mapping into as follows:

2.4. Illusion in Case 6

In Case 6, the uniqueness of is solved in [44, 45, 53] and the nonuniqueness of inside is shown using (12) [53]. This is called illusion of material property inside the uniquely determined domain . Note that is not cloaked into any more.

In more detail, if then we have And there is a boundary fixing diffeomorphism on such that

Therefore, in Case 6, the domain is uniquely determined by the Dirchlet-to-Neumann map; however, the property is nonunique up to the change of variables inside . Specifically, the domain of anomaly cannot be cloaked into a much smaller domain, but the property could be illuded into other property . We will call (18) property illusion or just illusion.

2.5. Cloaking in Case 5 versus Illusion in Case 6

Define three kinds of conductivities as follows:

Cloaking and illusion for three conductivities (19a), (19b), and (19c) are summarized as follows:(i) implies that and, with some more assumptions [53], there exists a diffeomorphism on such that . Therefore, only property illusion between and is possible and near cloaking of domain is not possible;(ii) implies that there exists a diffeomorphism on such that . Therefore, near-cloaking of domain is possible.

Comparing the above two results, we can conclude that if background conductivity is known only on the boundary, near-cloaking of the domain is possible to any smaller domain . But if the background conductivity is known in the neighborhood of the boundary, near-cloaking of the domain is not possible and only property illusion between and is possible.

3. ISP

ISP is formulated as follows:(i)given far-field patterns for all incident directions , identify coefficients and for (4). Let be a ball centered at the origin with radius being chosen such that is not a Dirichlet eigenvalue of (1a) with and compactly embedded in and containing . Then ISP is also formulated as follows (Theorem in [54]): (ii)find and from in (1a) and (1b) with for .

3.1. Uniqueness in Cases 7, 8, and 9

Case 7. The case for positive constant can be understood as special cases of Case 8. The limiting cases and could be considered as (3) in and the boundary condition on as (sound-soft case) and (sound-hard case) on . The uniqueness for sound-soft and sound-hard obstacle is considered in [17, 55].

Case 8. This case is called “inverse transmission problem” and the uniqueness is solved in [17].

Case 9. Uniqueness with a few subcases is solved in [18].

3.2. Nonuniqueness in Case 10

Let be a diffeomorphism such that is an identity map on . Then where push-forward map is defined by This is just change of variable in the weak formulation of the direct problem such that

We have similar question for EIT. Is the change of variable unique obstruction to the uniqueness of anisotropic ISP? And it is proved that for some boundary fixing diffeomorphism in the following references: (i) [56],(ii) [57].

3.3. Near-Cloaking in Case 10

Let be some domain and contained in , , , and Define for a diffeomorphism on transforming into arbitrary small domain , on , and fixing . Then, is nearly cloaked into , since for all by (20). If goes to , is called perfectly cloaked. Otherwise, is called nearly cloaked into . Conversely, if is nearly cloaked into , there is a diffeomorphism such that for all by (23).

For example, if and are two-dimensional disks of radii 2, 1, and , respectively, centered at , we could take the diffeomorphism on mapping into as follows:

3.4. Illusion in Case 11

In case 11, is uniquely determined and is studied in [18, 53, 58–60] but is not uniquely determined. That is to say, is uniquely determined and not cloaked into any smaller domain but is illuded into another property by some diffeomorphism on . In more detail, if where then we have and there is exterior fixing diffeomorphism on such that

In [11], a few interesting property illusions are considered; the optical transformation of an object into another object with different property is considered. In the paper, stereoscopic image of a man is transformed into an illusion image of a woman and dielectric spoon of the electric permeability 2 into an illusion image of metallic cup of electric permeability −1 in an electromagnetic scattering problem.

3.5. Cloaking in Case 10 and Illusion in Case 11

Define three pairs of coefficients as follows:where is compactly imbedded in , which is also compactly imbedded in .

Let us denote . Cloaking and illusion for three coefficients in (31a), (31b), and (31c) are summarized as follows:(i) for all implies and, with additional assumptions in [53], there is a exterior fixing diffeomorphism on such that [56, 57]. Therefore, only property illusion between and is possible and near-cloaking of domain is not possible: (ii) for all implies that there is an exterior fixing diffeomorphism on such that

Therefore, near-cloaking of the domain is possible in this case.

Also as in EIT, we can conclude that if background identity coefficient is known only in , near-cloaking of the domain is possible to any smaller domain . But if the background coefficient is known and fixed in , near-cloaking of the domain is not possible and only property illusion between and is possible.

4. DOT

DOT is formulated using source-to-detector map as follows:(i)find the optical coefficients and from the measurement informations which is the value of the solution of (7a) and (7b) at when . and are usually called source and detector point, respectively. We assumed that are isotropic in this paper.

Suppose that on . Then DOT is also formulated using Dirichlet-to-Neumann map as follows:(ii)find complex valued from in (1a) and (1b) with and given .

4.1. Source-to-Detector Map, Dirichlet-to-Neumann Map, and Far-Field Map

We summarize the relation between source-to-detector map and Dirichlet-to-Neumann map following the approach used in [61].

By setting and with , we have

If and for some source point , we have the following solution of (34a) as follows: For the fundamental solution with nonconstant function , see [62].

When has upper and lower bound and is contained in or a Dirac delta function, (7a), (7b), (34a), and (34b) have a unique solution and contained in , respectively [1, 63].

Boundary value problem (7a) and (7b) with is equivalent to boundary value problem with (7a) for and nonzero Robin boundary condition replacing (7b). This argument can be proved using the function in [62]. Therefore, DOT is redescribed as to find the optical coefficients from Robin-to-Dirichlet map defined as a map from to . Using unique solvability of (7a) with Dirichlet or Neumann boundary condition replacing (7b), Robin-to-Dirichlet map is equivalent to Neumann-to-Dirichlet map and to Dirichlet-to-Neumann map.

4.2. Uniqueness and Nonuniqueness of DOT

The research about unique determination of the optical coefficients in DOT is rare except [61], but it is a very important issue for DOT as an inverse problem. The determination of optical coefficients in (7a) and (7b) is equivalent to the determination of in (34a) and (34b) when .

Using the relation between source-to-detector map for (7a) and (7b) and Dirichlet-to-Neumamm map for (34a), we add a comment on the result of [42, 54, 61] and reference therein.

Case 12. is determined in (Theorem in [54]) and is determined by comparing the real and imaginary part of .

Case 13. is “almost” determined in (Theorem in [54]) and is determined in a similar way for .

Case 14. Even though is determined, if refractive index is not known or modulation frequency , we cannot determine and , simultaneously. If is not known, we should have at least three equations and we only know at most two equations for real and imaginary part of . And if , we have no information on the imaginary part and we only have one equation. The detailed nonuniqueness example is given in [61].

In summary, if refractive index is not known or continuous light source case is used, we cannot uniquely determine the optical coefficients and if refractive index is known and frequency domain light source is used, we can uniquely determine optical coefficients.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2010624).

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Copyright © 2015 Kiwoon Kwon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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