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

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.

We discuss uniqueness of nonuniqueness of EIT by classifying  into six cases for a Lipschitz domain  compactly embedded in Ω such that Ω \  is connected: For Cases 1, 2, 3, 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][3][4][5][6], but also in acoustic scattering [7][8][9], electromagnetic scattering [10][11][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][15][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  ∈  −1 ,  = 2, 3, . .., identify coefficients  and .
In this paper, we classify  and  as the following cases: for all  ∈ R  \ {O}.
ISP for (4) with ,  given in Cases 7, 8, 9, 10, and 11 is equivalently formulated as an inverse problem to find  and  from Λ , in (1a) and (1b) for some Lipschitz domain Ω compactly embedded in R  and containing  and  = 0.By Theorem 6.1.3 in [18], we can take Ω as a ball centered at the origin with radius  being chosen such that  is not a Dirichlet eigenvalue of (1a) with  = 0.
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 Φ is photon density distribution,   absorption coefficient,    reduced scattering coefficient,  = 1/3(  +    ) a diffusion coefficient, and  refractive index.Usually, we assume  = 1 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 Φ , which is the value of the solution of (7a) and (7b) at   ∈ Ω when () = (,   ),   ∈ Ω.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 nearvisible 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][21][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:  =  0 is given and  ̸ = 0, space dimension  ≥ 3, Case 13:  =  0 is given and  ̸ = 0, space dimension  = 2, Case 14:  = 0 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.

Uniqueness in
Case 1.One or two measurements uniqueness results are known when  is assumed to be known [34][35][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 3. In [44], an orthogonality relation between the two solutions of (7a) and (7b) for arbitary obstacles  1 and  2 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].

Nonuniqueness in Case 5.
The nonuniqueness of EIT in Case 5 is observed early in [47]: if  is a boundary fixing diffeomorphism on Ω 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)  = 2 and  ∈  3 (Ω) [48], (ii)  = 2 and  ∈  0,1 (Ω) (Lipschitz functions) [49], (iii)  = 2 and  ∈  ∞ (Ω) [50], (iv)  ≥ 3:  and  are analytic [51,52].
Then,  is nearly cloaked into , since Λ  = Λ   by (9).If  goes to 0,  is called perfectly cloaked.Otherwise,  is called nearly cloaked into .Conversely, if  is nearly cloaked into , there is a diffeomorphism  such that Λ  = Λ   by (12).For example, if Ω,  and  are two-dimensional disks of radii 2, 1, and  > 0, respectively, centered at O, we could take the diffeomorphism  on Ω 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  1 =  2 .Specifically, the domain of anomaly  cannot be cloaked into a much smaller domain, but the property  1 could be illuded into other property  2 .We will call (18) property illusion or just illusion.

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) Λ  1 = Λ  3 implies that  = 1 and, with some more assumptions [53], there exists a diffeomorphism  on  such that  1 =  *  3 .Therefore, only property illusion between  1 and  3 is possible and near cloaking of domain is not possible; (ii) Λ  2 = Λ  3 implies that there exists a diffeomorphism  on Ω such that  2 =  *  3 .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  1 and  2 is possible.
Let   be a ball centered at the origin with radius  being chosen such that  is not a Dirichlet eigenvalue of (1a) with  = 0 and compactly embedded in R  and containing .Then ISP is also formulated as follows (Theorem 6.1.3 in [54]): (ii) find  and  from Λ , in (1a) and (1b) with  = 0 for   .

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  = 0 could be considered as (3) in Ω \  and the boundary condition on  as Φ = 0 (sound-soft case) and Φ/] = 0 (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].

Nonuniqueness in Case 10
. Let  : R  → R  be a diffeomorphism such that | R  \Ω is an identity map on R  \Ω.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)  = 2 [56], (ii)  ≥ 3 [57].

Near-Cloaking in Case 10.
Let  be some domain and contained in Ω,  + = | Ω\ ,  − = |  , and Define for a diffeomorphism  on R  transforming  into arbitrary small domain ,  *  + =   on Ω\, and fixing R  \Ω.Then,  is nearly cloaked into , since   (⋅, ) =  ∞   (⋅, ) for all  ∈  −1 by (20).If  goes to 0,  is called perfectly cloaked.Otherwise,  is called nearly cloaked into .Conversely, if  is nearly cloaked into , there is a diffeomorphism  such that   (⋅, ) =  ∞   (⋅, ) for all  ∈  −1 by (23).For example, if Ω,  and  are two-dimensional disks of radii 2, 1, and  > 0, respectively, centered at O, we could take the diffeomorphism  on Ω mapping  into  as follows: Advances in Mathematical Physics 5 3.4.Illusion in Case 11.In case 11,  is uniquely determined and is studied in [18,53,[58][59][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 R  .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.Case 11.Define three pairs of coefficients as follows:
Also as in EIT, we can conclude that if background identity coefficient is known only in R  \ Ω, near-cloaking of the domain is possible to any smaller domain .But if the background coefficient is known and fixed in R  \ , near-cloaking of the domain is not possible and only property illusion between  1 and  2 is possible.

DOT
DOT is formulated using source-to-detector map as follows: (i) find the optical coefficients   and    from the measurement informations Φ , which is the value of the solution of (7a) and (7b) at   ∈ Ω when () = (,   ),   ∈ Ω.  and   are usually called source and detector point, respectively.We assumed that   ,    are isotropic in this paper.
Boundary value problem (7a) and (7b) with () = (,   ) is equivalent to boundary value problem with (7a) for  = 0 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  −1/2 (Ω) to  1/2 (Ω).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.

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  ̸ = 0.
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  ∞ (Ω) (Theorem 5.2.2 in [54]) and   ,    is determined by comparing the real and imaginary part of .
Case 14.Even though  is determined, if refractive index  is not known or modulation frequency  = 0, 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  = 0, 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  = 0 is used, we cannot uniquely determine the optical coefficients and if refractive index is known and frequency domain light source  ̸ = 0 is used, we can uniquely determine optical coefficients.