is measured (n is the unit outward normal) at the boundary, to define the Dirichlet-

Transcription

1 SIAM J MATH ANAL Vol 45, No 5, pp c 013 Society for Industrial and Applied Mathematics ON A CALDERÓN PROBLEM IN FREQUENCY DIFFERENTIAL ELECTRICAL IMPEDANCE TOMOGRAPHY SUNGWHAN KIM AND ALEXANDRU TAMASAN Abstract Recent research in electrical impedance tomography produced images of biological tissue from frequency differential boundary voltages and corresponding currents Physically one is to recover the electrical conductivity and permittivity ɛ from the frequency differential boundary data Let γ = +iωɛ denote the complex admittivity, Λ γ be the corresponding Dirichlet-to-Neumann map, and dλγ dω ω=0 be its frequency differential at ω =0 If C 1,1 () is constant near the boundary and ɛ C 1,1 dλγ 0 (), we show that dω ω=0 uniquely determines ( ɛ ɛ ln )/ inside In addition, if Λ γ ω=0 is also known, then ɛ and can be reconstructed inside The method of proof uses the complex geometrical optics solutions Key words Calderón problem, frequency differential electrical impedance tomography, complex geometrical optics AMS subject classifications 35R30, 35J5, 65N1 DOI / Introduction Electrical impedance tomography (EIT) aims to determine the electrical conductivity and permittivity distribution ɛ of a body from surface electrical measurements of voltages and corresponding currents One major application is in medical imaging, where the change of the electrical properties of biological tissues with their physiological and pathological conditions is used to provide diagnostic information Driven by its applications, considerable progress in both the engineering and mathematical facets of EIT has been achieved, and its development can be traced over the past two decades in the reviews [6, 3, 8,, 19] We consider a conducting bounded body R n, n 3, with C 1,1 -boundary Its conductivity distribution C 1,1 () is bounded away from zero, and its permittivity distribution ɛ C 1,1 () We assume that is constant near the boundary and ɛ is supported in (these assumptions may be relaxed as explained later) For a real valued function f H 1/ ( ) and an angular frequency ω the sinusoidal voltage f(x)cos(ωt) is imposed at the boundary Then a time harmonic complex voltage potential u ω distributes inside according to the problem (11) ( + iɛω) u ω =0inandu ω = f The problem (11) has a unique complex (voltage potential) solution u ω H 1 (); see also Theorem 31 below The exiting current (1) g ω := ( + iωɛ)n u ω is measured (n is the unit outward normal) at the boundary, to define the Dirichlet- Received by the editors January 4, 013; accepted for publication (in revised form) July 8, 013; published electronically September 1, Division of Liberal Arts, Hanbat National University, Daejeon , Korea (sungwhan@ hanbatackr) This author was supported by Hanbat National University and by the 007 Faculty Fund of Hanbat National University Department of Mathematics, University of Central Florida, Orlando, FL 3816 (tamasan@ mathucfedu) This author was supported in part by NSF grant DMS

2 ON A CALDERÓN PROBLEM IN fdeit 701 to-neumann map Λ +iɛω : f g ω H 1/ ( ) Originally formulated by Calderón [5] at the ω = 0 frequency, the goal in EIT is to determine and ɛ from knowledge of Λ +iɛω At zero frequency only is sought; in such a case the corresponding voltage potential v 0 is real valued and solves (13) ( v 0 )=0inandv 0 = f The Calderón problem (at ω = 0) has been mostly settled in the affirmative and we refer to [19] for a state of the art solution Of relevance to our work here, we mention the breakthrough result in [18], where Λ := Λ γ ω=0 is shown to uniquely determine in three or higher dimensions, and the reconstruction method in [14] which allows for the C 1,1 -regularity assumed here We note however that, while not explicitly stated, the results in [18, 14] extend to the complex admittivity γ = + iωɛ to show that Λ γ uniquely determines γ in three dimensions or higher The analogous results in two dimensions at ω = 0 were obtained in [15], with a nontrivial refinement in [1] Also in two dimensions, but at an arbitrary fixed frequency ω (not necessarily zero) Λ γ was shown recently to uniquely determine the complex admittivity γ in [4] (a previous result in [7] recovered γ for a sufficiently small imaginary part ωɛ) Recent research in [9, 11, 1, 16, 17] produce physiologically relevant images by using the frequency dependent behavior of the complex potential u ω These new methods are known as frequency differential electrical impedance tomography (fdeit) Physically one imposes boundary voltages at two distinct frequencies and measures a difference between corresponding boundary exit currents Despite the apparent usefulness in medical diagnostics, the quantities behind the images in fdeit are not so well understood In this paper we take a first step towards explaining what can be quantitatively obtained by fdeit We formulate the problem in terms of the frequency differential operator at the boundary: What can be obtained from knowledge of dλγ dω ω=0? To simplify notation, let D be defined for real valued functions f H 1/ ( ) by (14) D(f) := d dω Λ +iωɛ (f), ω=0 and then extended by complex linearity: D(f + ig) :=D(f)+iD(g) Our main result, Theorem, shows that D : H 1/ () H 1/ () is a well-defined bounded operator which uniquely determines the function ( ɛ ɛ ln )/ inside, in three or higher dimensions The method of proof uses the complex geometrical optics solutions in [18] We note that only the action of D on real valued functions is needed If, in addition, the Dirichlet-to-Neumann map Λ := Λ γ ω=0 is also known, we are able to recover separately the conductivity and the permittivity ɛ inside Moreover, in this case we do not need to assume constant near the boundary Specifically, if C 1,1 () then its boundary values and its normal derivative n can be recovered from Λ as shown in [14] for this regularity (and earlier in [13] and [18] for C -conductivity) Then canbeextendedwithpreservedc 1,1 -regularity

3 70 SUNGWHAN KIM AND ALEXANDRU TAMASAN to the whole space while making it constant near the boundary of a neighborhood of As shown in [15] the Dirichlet-to-Neumann map can be transferred from to (the fact that we deal with a complex valued coefficient does not change the proof from the real case) Therefore the assumption that is constant near the boundary does not restrict generality Similarly, the assumption that ɛ has compact support in can be replaced by the knowledge of the boundary values ɛ and its normal derivative ɛ n ; see also (44) With v ω := R(u ω )andh ω := I(u ω ) denoting the real, respectively, imaginary part of the voltage potential obtained for a real valued boundary data f, the problem (11) can be rewritten as a Dirichlet problem for the coupled system: [( )( )] ( ) ωɛ vω 0 (15) = in, ωɛ h ω 0 ( vω h ω ) = ( f 0 Not only do v ω and h ω have a nonlinear dependence on the conductivity, permittivity ɛ, and angular frequency ω, but also their intrinsic mutual relation makes this dependence difficult to investigate directly from (15) Key to this work, in Theorem 31 we identify the regime of frequencies (16) ω < ɛ 1 L () in which the family of operators ω Λ +iɛω is analytic in the strong operator topology (from H 1/ () to H 1/ ()) This analytic dependence allows for a recurrence type of decoupling Also the frequency differential boundary operator can be made more explicit: (17) D(f) =iɛ v 0 n + i d h ω dω ω=0 n = i ɛ Λ (f)+i d h ω dω ω=0 n Statement of results The main result is formulated in terms of the complex geometrical optics solutions of Sylvester and Uhlmann in [18] whose existence is recalled below both for convenience and to set notation The coefficients, andɛ assumed constant near the boundary, are extended by (the corresponding) constant on the complement of For δ R, the weighted norm f := f(x) (1 + x ) δ dx is used For L δ R n k,η,l R n with k η = k l = k η =0,and η = k 4 + l, consider the vectors ( ) k ξ 1 (η, k, l) :=η i + l, ( ) k (1) ξ (η, k, l) := η i l Note that ξ 1 ξ 1 = ξ ξ =0, ξ 1 = ξ =( k + l ), and ξ 1 + ξ = ik We restate their result in the variant below Theorem 1 (Theorem 3 in [18]) Let n 3 and C 1,1 () be constant near the boundary For 1 <δ<0 there are two constants R, C > 0 dependent only on δ, Δ / L (), and such that, for ξ j C n, j =1, as in (1) with l >R, )

4 ON A CALDERÓN PROBLEM IN fdeit 703 there exist w(,ξ j ) H 1 loc (Rn ) solutions of w(,ξ j )=0in R n, of the form () w(x, ξ j )=e x ξj 1/ (1 + ψ(x, ξ j )) with (3) ψ(,ξ j ) L δ C ξ j The main result which will be proven in section 4 is the following Theorem Let R n, n 3 be an open domain with C 1,1 -boundary, C 1,1 () be constant near the boundary, and ɛ C 1,1 0 () Recall the frequency differential map D defined in (14) For each k R n,letf j := w(,ξ j ), j =1, be the traces of the complex geometrical optics solutions in () Then [ ] ( ɛ ɛ ln ) (4) F (k) = lim i D(f 1 )f ds, l where F denotes the Fourier transform We stress here that only the action of D on real valued functions is needed in Theorem above If the Dirichlet-to-Neumann map Λ (at frequency ω = 0) is also known, then can be recovered inside as shown in [14] As a corollary to Theorem one is also able to reconstruct ɛ inside Corollary 1 Let R n, n 3 be an open domain with smooth C 1,1 - boundary Assume C 1,1 () constant near the boundary and ɛ C 1,1 0 () Then and ɛ inside can be reconstructed from knowledge of Λ and D 3 Analytic dependence in frequency In this section we prove the analytic dependence in the frequency of ω Λ +iωɛ in the strong operator topology from H 1/ () H 1/ () The assumptions on the coefficients are slightly relaxed By C 0,1 () we denote the space of Lipschitz continuous maps Theorem 31 Let L () bounded away from zero and ɛ C 0,1 () Assume that ω lies in the frequency range (16) Then the Dirichlet problem (11) has a unique solution u ω H 1 () with the following series representation: (31) v ω (x) :=R(u ω (x)) = v k (x)ω k and k=0 h ω (x) :=I(u ω (x)) = h k 1 (x)ω k 1, k=1 where the summation is convergent in the H 1 (), v 0 is the solution to (13), and for k =1,, the following recurrence holds: (3) ( h k 1 )= (ɛ v (k 1) ) in, ( v k )= (ɛ h k 1 ) in, h k 1 = v k =0 Assuming the recurrences in (3) hold we establish first some basic estimates

5 704 SUNGWHAN KIM AND ALEXANDRU TAMASAN Lemma 31 Let v k and h k 1, k = 1,, be defined in (3) Then, for k =1,,, we have (33) and (34) [ v k ɛ dx [ h k 1 ɛ dx k L () k 1 L () [ [ v 0 dx v 0 dx Proof Let us fix a positive natural number k From (3) the divergence theorem implies that h k 1 dx = ɛ v k h k 1 dx Cauchy s inequality applied to the right-hand side above yields h k 1 dx ɛ [ [ v k dx h k 1 dx, L () and thus [ ɛ [ (35) Similarly we obtain (36) h k 1 dx [ v k ɛ dx L () L () v k dx [ h k 1 dx By induction, the estimates (33) and (34) follow ProofofTheorem31 The coupled system (15) is equivalent to the two elliptic equations (37) ( v ω )= (ωɛ h ω ) in, and (38) ( h ω )= (ωɛ v ω ) in We seek solutions in the ansatz (39) v(x, ω) := v k (x)ω k and h(x, ω) := h k (x)ω k k=0 Let us assume first that the series representations in (39) are convergent in H 1 () If (37) and (38) are satisfied, then 0= ( v 0 )+ k=0 ( v k+1 ɛ h k )ω k+1 k=0

6 ON A CALDERÓN PROBLEM IN fdeit 705 and 0= ( h 0 )+ ( h k+1 + ɛ v k )ω k+1, k=0 where the divergence is taken in the weak sense In particular we obtain (310) ( v 0 )=0, ( h 0 )=0 in, and, for k =0, 1,,, (311) ( v k+1 )= (ɛ h k ), ( h k+1 )= (ɛ v k ) in By our assumption, both series are convergent in H 1 (), and their sums have well-defined traces in H 1/ ( ), which are the corresponding sums of the traces of the terms Now v(x, ω) =f(x) andh(x, ω) =0forx yield v 0 = f and h 0 = 0, and, for k =1,,, h k 1 = v k =0 on Note that v 0 is the solution of (13) and h 0 0 Conversely, for f H 1/ ( ), let v 0 be the solution of (13) and define two sequences of functions {v k } 0 and {h k } 1 via the recurrence (3) From Lemma 31 it follows that for any k =1,,, v k L ()ω k 1 m [ v k dx ω k 1 ωɛ [ k m L () M m v 0 L () ωɛ v 0 dx k L (), and h k 1 L ()ω k 1 1 m [ h k 1 dx ω k 1 1 ωɛ [ k 1 m L () M m v 0 L () ωɛ v 0 dx k 1 L () If ω satisfies (16), the H 1 ()-convergence of the series well defines v(x, ω) and h(x, ω) in (39) to be the unique solutions of (15) The result below further clarifies the frequency differential operator D in (14) Corollary 3 Let ɛ C 0,1 (), and L () with essinf > 0 For f,g H 1/ ( ) real valued we have (31) D(f + ig) =iɛ v 0 n + i h 1 n,

7 706 SUNGWHAN KIM AND ALEXANDRU TAMASAN where v 0 solves ( v 0 )=0in, and v 0 = f + ig, and h 1 solves ( h 1 )= ɛ v 0 in and h 1 =0 Proof Ifg = 0 the corollary follows directly from the definition (14) and the fact that d h ω dω n ω=0 = h1 n, with h 1 defined in the recurrence (3) If g is arbitrary, the result follows from the complex linearity of the two terms in the right-hand side of (31) From Theorem 31 it follows that v 0 is the zeroth order term of the series expansion of the real part v ω and h 1 is the first order term of the series expansion of the imaginary part h ω Moreover, when ω 1 is small, and ω 1,ω = O(ω), we have v ω1 v ω = v (ω 1 + ω )+O(ω 3 h ω1 h ω (313) )and = h 1 + O(ω ), ω 1 ω ω 1 ω where h 1 and v satisfy the Poisson s problems ( h 1 )= (ɛ v 0 ) in, (314) ( v )= (ɛ h 1 ) in, h 1 = v =0 4 Proof of Theorem and its corollary For j =1, letw j (x) :=w(x; ξ j ) be the complex geometrical optics (CGO) solutions corresponding to a fix vector k R n as provided by Theorem 1 For w 1 in the CGO above, let h 1 be the solution of the Poisson equation ( h 1 )= (ɛ w 1 ) in and h 1 =0 First we carry out the calculation without the assumption that ɛ =0nearthe boundary to emphasize the fact that knowledge of ɛ and its normal derivative at the boundary suffices Following the explicit formula for D in Corollary 3 we obtain (41) D(w 1 )w ds = i = i ( ɛ w 1 n + h 1 n ) w ds h 1 w + ɛ w 1 w dx Since h 1 =0on andw solves the conductivity equation w =0,the first integral on the right-hand side of (41) is zero so that (4) D(w 1 )w ds = i ɛ w 1 w dx Now use w 1 w =[Δ(w 1 w ) (w 1 Δw + w Δw 1 )] and the fact that the w j s also solve Δw j + ln w j = 0 in, to obtain ɛ w 1 w dx = 1 ( ɛ (w 1w ) ɛ ) n n (w 1w ) ds + 1 [(Δɛ)(w 1 w )+ɛ ln (w 1 w )] dx

8 ON A CALDERÓN PROBLEM IN fdeit 707 Using the Green s formula in the last integral and the assumption that is constant near the boundary we obtain ɛ w 1 w dx = 1 ( ɛ (w 1w ) ɛ ) n n (w 1w ) ds + 1 (43) ( ɛ ɛ ln )(w 1 w ) dx From (4) and (43) we have that ( ɛ ɛ ln )w 1 w dx = i (44) D(w 1 )w ds ( ɛ (w 1w ) ɛ n n (w 1w ) If we now use the assumption of ɛ being support in, (44) further simplifies to (45) ( ɛ ɛ ln )w 1 w dx = i D(w 1 )w ds By the choice of ξ j s in the complex geometrical optics w j s, we have ( ɛ ɛ ln ) e ix k (1 + ψ(x, ξ 1 ))(1 + ψ(x, ξ )) dx = i D(w 1 )w ds Since the integrant in the left-hand side above is supported in, the integral can be taken over the entire space R n The decay estimates (3) then yield [ ] ( ɛ ɛ ln ) F (k) = lim i D(w 1 )w ds, l where F is the Fourier transform in R n This completes our proof of Theorem The proof of Corollary 1 relies on the results in [14] which show that the traces of the geometrical optics solutions used in Theorem () can been recovered from a singular integral equation at the boundary: Since C 1,1 () is recovered inside from Λ, we may assume without loss of generality that = 1 near the boundary Then Lemmas 7 and 1(b) in [14] yield that the traces f j := w(,ξ j ) at the boundary are the unique solutions to the equation (46) f j = e ix ξj (S ξj Λ γ B ξj 1 ) I f j, j =1,, where the boundary operators S ξ and B ξ are the single and double layer potentials G ξ S ξ f(x) = G ξ (x, y)f(y)ds and B ξ f(x) =pv (x, y)f(y)ds n associated with the Fadeev Green kernel eix ξ Rn e ix η G ξ (x) = (π) n η +ξ η dη Once the traces of w 1, and w are determined, the right-hand side of (4) is determined By Fourier inversion, we then determine the essentially bounded function (47) Q[, ɛ] := ( ɛ ɛ ln ) ) ds

9 708 SUNGWHAN KIM AND ALEXANDRU TAMASAN With and Q known, the permittivity ɛ is the unique solution of the Dirichlet problem (48) Δɛ ɛ ln() ɛδln() =Q in, ɛ =0 This finishes the proof of Corollary 1 5 Concluding remarks We formulated a Calderón-type problem using frequency differential D := dλγ dω ω=0 of the Dirichlet-to-Neumann map at ω =0 Provided that is (an unknown) constant near the boundary and ɛ issupportedin, we showed that the frequency differential uniquely determines Q in (47) relating the conductivity to the permittivity ɛ However, if the Dirichlet-to-Neumann map at ω = 0 is also available, then and ɛ can be recovered inside We note here that ɛ need not be supported in, since the quantity Q in (47) can still be recovered if ɛ and its normal derivative ɛ n are known at the boundary, according to (44) Our results yield the following effect of the admittivity + iωɛ on the complex voltage potential u ω The real part R(u ω ) is influenced mainly by the conductivity, whereas the imaginary part I(u ω ) is influenced by the combination Q[, ɛ] in (47) There are infinitely many pairs, ɛ which yield the same quantity Q in (47) More precisely, let Q L () be in the range of the combination in (47) For an arbitrary f C 1,1 (), let f be any solution of the transport equation ln f f = Q Δf Then the pair ( f,ɛ) with ɛ = f f yields the same Q, independent of f If the actual value of at the boundary is not known, since Q[λ, λɛ] =Q[, ɛ] for any λ>0, the recovered quantity is not sensitive to the contrast in the pair of coefficients However, the boundary data D can distinguish the difference in scale between the conductivity and the permittivity ɛ since Q[, λɛ] =λq[, ɛ] In practice the angular frequency ω is not arbitrarily small However, due the scaling γ = + i(tω) ɛ t, we apply the results above to tω, with t small This scaling is meaningful at angular frequencies of up to a few khz, where the scaling factor is the permittivity of the vacuum ɛ 0 = F/m, since then ωɛ 0 is still numerically small From a numerical perspective, (313) shows a difference in scale (of order 1) between the real and imaginary parts of the complex voltage potential at small frequency (or ɛ 0 ω as explained above) They imply that D is approximated at O(ω )bythe difference quotient at two small frequencies without a need to distinguish the real from the imaginary part of the voltage potential ɛ Following from (17), in fdeit it is the quotient at the boundary which scales the boundary information about the admitivitty inside In particular, when ɛ = 0 it is only the imaginary part of the voltage potential which carries the information about the coefficients from inside to the frequency differential data at the boundary In such a case we can still expect to recover the quantity Q in (47) While the formulated problem is still severely ill-posed, these theoretical results are expected to help in understanding the quantitative feature of fdeit Acknowledgments This work was done during the first author s sabbatical visit to A Tamasan at the University of Central Florida The first author would like to thank them both for the hospitality

10 ON A CALDERÓN PROBLEM IN fdeit 709 REFERENCES [1] K Astala and L Päivärinta, Calderón s inverse conductivity problem in the plane, Ann of Math (), 163 (006), pp [] RH Bayford, Bioimpedance tomography (electrical impedance tomography), Annu Rev Biomed Eng, 8 (006), pp [3] L Borcea, EIT Electrical impedance tomography, Inverse Problems, 18 (00), pp R99 R136 [4] AL Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J Inverse Ill-Posed Problems, 16 (008), pp [5] AP Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc Brasiliera de Matematica, Rio de Janeiro, 1980, pp [6] M Cheney, D Isaacson, and J C Newell, Electrical impedance tomography, SIAMRev, 41 (1999), pp [7] E Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to- Neumann map, Inverse Problems, 16 (000), p 107 [8] D Holder, Electrical Impedance Tomography: Methods, History and Applications, IOP, Bristol, UK, 005 [9] S Kim, J Lee, JK Seo, E J Woo, and H Zribi, Multifrequency trans-admittance scanner: Mathematical framework and feasibility, SIAM J Appl Math, 69 (008), pp 36 [10] S Kim, JK Seo, and T Ha, A nondestructive evaluation method for concrete voids: Frequency differential electrical impedance scanning, SIAM J Appl Math, 69 (009), pp [11] S Kim, Assessment of breast tumor size in electrical impedance scanning, InverseProblems, 8 (01), [1] S Kim, EJ Lee, EJ Woo, and JK Seo, Asymptotic analysis of the membrane structure to sensitivity of frequency-difference electrical impedance tomography, Inverse Problems, 8 (01), [13] RV Kohn and M Vogelius, Determining conductivity by boundary measurements, Comm Pure Appl Math, 37 (1984), pp [14] A Nachman, Reconstructions from boundary measurements, Ann of Math (), 18 (1988), pp [15] A Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann of Math (), 143 (1996), pp [16] JK Seo, J Lee, SW Kim, H Zribi, and EJ Woo, Frequency-difference electrical impedance tomography (fdeit): Algorithm development and feasibility study, InverseProblems, 9 (008), pp [17] JK Seo, B Harrach, and EJ Woo, Recent progress on frequency difference electrical impedance tomography, ESAIM Proc, 6 (009), pp [18] J Sylvester and G Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann of Math (), 15 (1987), pp [19] G Uhlmann, Electrical impedance tomography and Calderón s problem, InverseProblems, 5 (009), pp 1 39