Computational inversion based on complex geometrical optics solutions

Transcription

1 Computational inversion based on complex geometrical optics solutions Samuli Siltanen Department of Mathematics and Statistics University of Helsinki, Finland International Conference on Inverse Problems and Applications in Honor of Gunther Uhlmann s 60th Birthday Hangzhou, China September 19, 2012

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3 This is a joint work with Takanori Ide, Tokyo Metropolitan University, Japan Masaru Ikehata, Gunma University, Japan David Isaacson, Rensselaer Polytechnic Institute, USA Hiroshi Isozaki, Tsukuba University, Japan Kim Knudsen, Technical University of Denmark Matti Lassas, University of Helsinki, Finland Susumu Nakata, Ritsumeikan University, Japan Jon Newell, Rensselaer Polytechnic Institute, USA Jennifer Mueller, Colorado State University, USA Gunther Uhlmann, University of Washington, USA

4 Outline Electrical impedance tomography The D-bar method for EIT Electrical inclusion detection

5 Outline Electrical impedance tomography The D-bar method for EIT Electrical inclusion detection

6 Electrical impedance tomography (EIT) is an emerging medical imaging technique Feed electric currents through electrodes. Measure the resulting voltages. Repeat the measurement for several current patterns. Reconstruct distribution of electric conductivity inside the patient. Different tissues have different conductivities, so EIT gives an image of the patient s inner structure. EIT is a harmless and painless imaging method suitable for long-term monitoring.

7 This talk concentrates on applications of EIT to chest imaging Applications: monitoring cardiac activity, lung function, and pulmonary perfusion. Also, electrocardiography (ECG) can be enhanced using knowledge about conductivity distribution.

8 The mathematical model of EIT is the inverse conductivity problem introduced by Calderón Let Ω R 2 be the unit disc and let conductivity σ : Ω R satisfy 0 < M 1 σ(z) M. Ω Applying voltage f at the boundary Ω leads to the elliptic PDE { σ u = 0 in Ω, u Ω = f. Boundary measurements are modelled by the Dirichlet-to-Neumann map Λ σ : f σ u n Ω. Calderón s problem is to recover σ from the knowledge of Λ σ. It is a nonlinear and ill-posed inverse problem.

9 Why is Calderón s problem nonlinear? Define a quadratic form P σ for functions f : Ω R by P σ (f ) = σ u 2 dz, (1) where u is the solution of the Dirichlet problem { σ u = 0 in Ω, u Ω = f. Ω Now the map σ P σ is nonlinear because u depends on σ in (1). Physically, P σ (f ) is the power needed for maintaining the voltage potential f on the boundary Ω. Integrate by parts in (1): P σ (f ) = Ω f (σ u n ) ds = Ω Thus the map σ Λ σ cannot be linear in σ. f (Λ σ f ) ds.

10 We illustrate the ill-posedness of Calderón s problem using a simulated example σ 1 σ 2

11 We apply the voltage distribution f (θ) = cos θ at the boundary of the two different phantoms σ 1 u 1 θ σ 2 u 2

12 The measurement is the distribution of current through the boundary σ 1 u 1 θ σ 1 u 1 n θ σ 2 u 2 σ 2 u 2 n θ

13 The measurements are very similar, although the conductivities are quite different σ 1 σ 1 u 1 n σ 2 u 2 n σ 2

14 Let us apply the more oscillatory distribution f (θ) = cos 2θ of voltage at the boundary σ 1 u 1 σ 2 u 2

15 The measurement is again the distribution of current through the boundary σ 1 u 1 σ 1 u 1 n θ σ 2 u 2 σ 2 u 2 n θ

16 The current distribution measurements are again very similar σ 1 σ 1 u 1 n σ 2 u 2 n σ 2

17 EIT is an ill-posed problem: big differences in conductivity cause only small effect in data σ 1 cos θ cos 4θ cos 2θ cos 5θ σ 2 cos 3θ cos 6θ

18 EIT is an ill-posed problem: noise in data causes serious difficulties in interpreting the data σ 1 cos θ cos 4θ cos 2θ cos 5θ σ 2 cos 3θ cos 6θ

19 Many different types of reconstruction methods have been suggested for EIT in the literature Linearization: Barber, Bikowski, Brown, Calderón, Cheney, Isaacson, Mueller, Newell Iterative regularization: Dobson, Gehre, Hua, Jin, Kaipio, Kindermann, Kluth, Leitão, Lechleiter, Lipponen, Maass, Neubauer, Rieder, Rondi, Santosa, Seppänen, Tompkins, Webster, Woo Bayesian inversion: Fox, Kaipio, Kolehmainen, Nicholls, Pikkarainen, Ronkanen, Somersalo, Vauhkonen, Voutilainen Resistor network methods: Borcea, Druskin, Mamonov, Vasquez Layer stripping: Cheney, Isaacson, Isaacson, Somersalo D-bar methods: Astala, Bikowski, Bowerman, Delbary, Hansen, Isaacson, Kao, Knudsen, Lassas, Mueller, Murphy, Nachman, Newell, Päivärinta, Perämäki, Saulnier, S, Tamasan Teichmüller space methods: Kolehmainen, Lassas, Ola, S Methods for partial information: Alessandrini, Ammari, Bilotta, Brühl, Eckel, Erhard, Gebauer, Hanke, Harrach, Hyvönen, Ide, Ikehata, Isozaki, Kang, Kim, Kress, Kwon, Lechleiter, Lim, Morassi, Nakamura, Nakata, Potthast, Rossetand, Seo, Sheen, S, Turco, Uhlmann, Wang, and others

20 The forward map F : X D(F ) Y of an ill-posed problem does not have a continuous inverse Model space X = L (Ω) Data space Y σ F Λ σ δ Λ σ δ D(F ) F (D(F ))

21 Regularization means constructing a continuous map Γ α : Y X that inverts F approximately Model space X = L (Ω) Data space Y F σ Λ σ Λδ δ σ Γ α D(F ) F (D(F ))

22 A regularization strategy needs to be constructed so that the assumptions below are satisfied A set Γ α : Y X of continuous maps parameterized by α > 0 is a regularization strategy for F if for each fixed σ D(F ) we have lim Γ α(λ σ ) σ X = 0. α 0 Further, a regularization strategy with a choice α = α(δ) of regularization parameter is called admissible if α(δ) 0 as δ 0, and for any fixed σ D(F ) the following holds: { sup Γα(δ) (Λ δ σ) σ X : Λ δ σ Λ σ Y δ } 0 as δ 0. Λ δ σ

23 Outline Electrical impedance tomography The D-bar method for EIT Electrical inclusion detection

24 History of CGO-based methods for real 2D EIT Infinite-precision data Practical data 1980 Calderón 2008 Bikowski-Mueller 1987 Sylvester-Uhlmann (d 3) 2008 Boverman-Isaacson-Kao-Saulnier-Newell 1988 Nachman 2010 Bikowski-Knudsen-Mueller 1988 R. G. Novikov 2011 Delbary-Hansen-Knudsen 1996 Nachman (σ C 2 (Ω)) 2000 S-Mueller-Isaacson 1997 Liu 2003 Mueller-S 2004 Isaacson-Mueller-Newell-S 2006 Isaacson-Mueller-Newell-S 2007 Murphy-Mueller 2008 Knudsen-Lassas-Mueller-S 2009 Knudsen-Lassas-Mueller-S 2012 S-Tamminen 1997 Brown-Uhlmann (σ C 1 (Ω)) 2001 Knudsen-Tamasan 2001 Barceló-Barceló-Ruiz 2003 Knudsen 2003 Astala-Päivärinta (σ L (Ω)) 2009 Astala-Mueller-Päivärinta-S 2005 Astala-Lassas-Päivärinta 2011 Astala-Mueller-Päivärinta Barceló-Faraco-Ruiz Perämäki-S 2008 Clop-Faraco-Ruiz

25 Nachman s 1996 uniqueness proof in 2D uses complex geometric optics (CGO) solutions Define a potential q by setting q(z) 0 for z outside Ω and q(z) = σ(z) σ(z) for z Ω. Then q C 0 (Ω). We look for solutions of the Schrödinger equation ( + q)ψ(, k) = 0 in R 2 parametrized by k C \ 0 and satisfying the asymptotic condition e ikz ψ(z, k) 1 W 1, p (R 2 ), where p > 2 and ikz = i(k 1 + ik 2 )(x + iy).

26 The CGO solutions are constructed using a generalized Lippmann-Schwinger equation Define µ(z, k) = e ikz ψ(z, k). Then ( + q)ψ = 0 implies ( 4ik z + q)µ(, k) = 0, (2) where the D-bar operator is defined by z = 1 2 ( x + i y ). A solution of (2) satisfying µ(z, k) 1 W 1, p (R 2 ) can be constructed using the Lippmann-Schwinger type equation µ = 1 g k (qµ), where g k satisfies ( 4ik z )g k = δ and is defined by g k (z) = 1 4π 2 R2 e iz ξ ξ 2 + 2k(ξ 1 + iξ 2 ) dξ 1dξ 2.

27 The Faddeev fundamental solution g 1 (z) has a logarithmic singularity at z = 0 Real part of g 1 (z) Imaginary part of g 1 (z) It is enough to know g 1 (z) because of the relation g k (z) = g 1 (kz).

28 The conductivity σ can be recovered from the functions µ(z, k) at k = 0 Recall that with the asymptotics Substituting k = 0 gives ( 4ik z + q)µ(, k) = 0 µ(z, k) 1 W 1, p (R 2 ). ( + σ σ )µ(, 0) = 0, (3) and setting µ(z, 0) = σ(z) gives the unique solution of (3) satisfying µ(z, 0) 1 W 1, p (R 2 ).

29 The crucial intermediate object in the proof is the non-physical scattering transform t(k) We denote z = x + iy C or z = (x, y) R 2 whenever needed. The scattering transform t : C C is defined by t(k) := e ikz q(z)ψ(z, k) dxdy. (4) R 2 Sometimes (4) is called the nonlinear Fourier transform of q. This is because asymptotically ψ(z, k) e ikz as z, and substituting e ikz in place of ψ(z, k) into (4) results in e i(kz+kz) q(z)dxdy R 2 = e i( 2k 1,2k 2 ) (x,y) q(z)dxdy R 2 = q( 2k 1, 2k 2 ).

30 Infinite-precision data: Solve boundary integral equation ψ(, k) Ω = e ikz S k (Λ σ Λ 1 )ψ for every complex number k C \ 0. Practical data: Solve boundary integral equation ψ δ (, k) Ω = e ikz S k (Λ δ σ Λ 1 )ψ δ for all 0 < k < R = 1 10 log δ. Evaluate the scattering transform: t(k) = e ikz (Λ σ Λ 1 )ψ(, k) ds. Ω Fix z Ω. Solve D-bar equation k µ(z, k) = t(k) 4πk e i(kz+kz) µ(z, k) with µ(z, ) 1 L r L (C). For k R set t δ R (k) = 0. For k < R t δ R(k) = e ikz (Λ δ σ Λ 1 )ψ δ (, k) ds. Ω Fix z Ω. Solve D-bar equation k µδ R(z, k) = tδ R (k) 4πk e i(kz+kz) µ δ R (z, k) with µ δ R (z, ) 1 Lr L (C). Reconstruct: σ(z) = (µ(z, 0)) 2. Set Γ 1/R(δ) (Λ δ σ) := (µ δ R (z, 0))2.

31 We define spaces for our regularization strategy = L (Ω) Let M > 0 and 0 < ρ < 1. The domain D(F ) consists of functions σ : Ω R with σ C 2 (Ω) M, σ(z) M 1, σ(z) 1 for ρ < z < 1. Bounded linear operators A : H 1/2 ( Ω) H 1/2 ( Ω) satisfying A(1) = 0, A(f ) ds = 0. Ω

32 Main result: nonlinear low-pass filtering yields a regularization strategy with convergence speed Theorem (Knudsen, Lassas, Mueller & S 2009) There exists a constant 0 < δ 0 < 1, depending only on M and ρ, with the following properties. Let σ D(F ) be arbitrary and assume given noisy data Λ δ σ satisfying Then Γ α with the choice R(δ) = 1 10 Λ δ σ Λ σ Y δ < δ 0. log δ, α(δ) = 1 R(δ), is well-defined, admissible and satisfies the estimate Γ α(δ) (Λ δ σ) σ L (Ω) C( log δ) 1/14.

33 Let us analyze how the regularization works using a simulated heart-and-lungs phantom

34 This is how the actual scattering transform looks like in the disc k < 10, computed by knowing σ Real part of t(k) Imaginary part

35 Scattering transform in the disc k < 10, here computed from noisy measurement Λ δ σ Real part of t(k) Imaginary part

36 Regularized reconstructions from simulated data with noise amplitude δ = Λ δ σ Λ σ Y δ 10 6 δ 10 5 δ 10 4 δ 10 3 δ 10 2 The percentages are the relative square norm errors in the reconstructions.

37 The method works for real data as well, including laboratory phantoms and in vivo human data Saline and agar phantom Reconstruction (R = 4) [Isaacson, Mueller, Newell & S 2006] [Montoya & Mueller 2012]

38 Unknown boundary shape can be estimated from EIT data using Teichmüller space methods [Kolehmainen, Lassas, Ola & S 2012]

39 Outline Electrical impedance tomography The D-bar method for EIT Electrical inclusion detection

40 This is a brief history of the enclosure method 2000 Ikehata Enclosure method for EIT } 2000 Ikehata & S Implementation for 2D EIT 2000 Brühl & Hanke 2002 Ikehata Regularization analysis 2002 Ikehata & Ohe Polygonal cavities in 2D EIT 2004 Ikehata & S Regularized cone probing in 2D EIT 2007 Ide, Isozaki, Nakata, S & Uhlmann Spherical probing in 2D EIT 2008 Ikehata & Ohe Locating cracks in 2D EIT 2008 Uhlmann & Wang General 2D probing 2010 Ide, Isozaki, Nakata & S Spherical probing in 3D EIT 2010 Zhou Maxwell s equations 2012 Ikehata, Niemi & S Inverse obstacle scattering in 2D

41 Different probing strategies Half-space Cone Spherical

42 Main theorem of spherical probing, part 1 (Ide, Isozaki, Nakata, S and Uhlmann 2007) We denote the inclusion set by Ω 1. The background conductivity is σ 0 1. Assume that σ(x) = σ 0 (x) for x Ω 1 and σ(x) > σ 0 (x) on Ω 1. Ω σ σ 0 Take x 0 from the outside of the convex hull of Ω, and let R > 0. Then there exists a smooth solution u τ (x) of (σ 0 (x) u τ (x)) = 0 on Ω with the following special properties. Ω 1 σ > σ 0 x 0 R

43 Main theorem of spherical probing, part 2 Hyperbolic transformation: y 1 = x x 2 2 r 2 (x 1 + r) 2 + x2 2, y 2 = 2x 2 r (x 1 + r) 2 + x 2 2 (x 1, x 2 )-plane: u τ (x) (y 1, y 2 )-plane: exp( τy 1 + iτy 2 ) Decay Growth Decay Growth

44 Main theorem of spherical probing, part 3 Let f τ = u τ Ω. There is a 0 < δ = δ(ω 1, x 0, R) such that (Λ σ Λ σ0 )f τ, f τ < Ce δτ (Λ σ Λ σ0 )f τ, f τ > Ce δτ

45 Practical scheme for detecting inclusions Take 0 < τ 1 < τ 2. (Λ σ Λ σ0 )f τ, f τ < Ce δτ (Λ σ Λ σ0 )f τ, f τ > Ce δτ (Λ σ Λ σ0 )f τ1, f τ1 > (Λ σ Λ σ0 )f τ2, f τ2 (Λ σ Λ σ0 )f τ1, f τ1 < (Λ σ Λ σ0 )f τ2, f τ2

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194 Spherical probing from ideal and noisy EIT data Conductivity Maximal region Reconstruction (ideal data) Reconstruction (noisy data)

195 Spherical probing from ideal and noisy EIT data in 3D (Ide, Isozaki, Nakata & S 2010)

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197 T. Ide, H. Isozaki, S. Nakata, and S. Siltanen, Local detection of three-dimensional inclusions in electrical impedance tomography, Inverse problems, 26 (2010), p T. Ide, H. Isozaki, S. Nakata, S. Siltanen, and G. Uhlmann, Probing for electrical inclusions with complex spherical waves, Communications on pure and applied mathematics, 60 (2007), pp M. Ikehata, E. Niemi, and S. Siltanen, Inverse obstacle scattering with limited-aperture data, Inverse Problems and Imaging, 6 (2012), pp M. Ikehata and S. Siltanen, Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems, 16 (2000), pp M. Ikehata and S. Siltanen, Electrical impedance tomography and Mittag-Leffler s function, Inverse Problems, 20 (2004), pp D. Isaacson, J. Mueller, J. Newell, and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiological Measurement, 27 (2006), pp. S43 S50. D. Isaacson, J. L. Mueller, J. C. Newell, and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography, IEEE Transactions on Medical Imaging, 23 (2004), pp K. Knudsen, M. Lassas, J. Mueller, and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), pp J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), pp A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), pp S. Siltanen, J. Mueller, and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem, Inverse Problems, 16 (2000), pp