Fast shape-reconstruction in electrical impedance tomography

Transcription

1 Fast shape-reconstruction in electrical impedance tomography Bastian von Harrach (joint work with Marcel Ullrich) Institut für Mathematik - IX, Universität Würzburg The Sixth International Conference Inverse Problems: Modeling and Simulation, Antalya, Turkey, May 2 26, 22.

2 Mathematical Model Forward operator of EIT: Λ : σ Λ(σ), conductivity measurements Conductivity: σ L + (Ω) Continuum model: Λ(σ): Neumann-Dirichlet-operator Λ(σ) : g u Ω, applied current measured voltage (σ u) = in Ω, σ ν u Ω = g on Ω. Linear elliptic PDE theory: Λ(σ) : L 2 ( Ω) L 2 ( Ω) linear, compact, self-adjoint

3 Inverse problem Non-linear forward operator of EIT Λ : σ Λ(σ), L +(Ω) L(L 2 ( Ω)) Inverse problem of EIT: Λ(σ) σ? Uniqueness ( Calderón problem ): Is Λ injective? Convergent numerical methods to reconstruct σ?

4 Reconstruction Convergent numerical methods to reconstruct σ? Newton iteration: almost no theory Dobson (992): (Local) convergence for regularized EIT equation. Lechleiter/Rieder(28): (Local) convergence for discretized setting. D-bar method: convergent 2D-implementation for σ C 2 Knudsen, Lassas, Mueller, Siltanen (28) In practice: large jumps in conductivity large interest in detecting shapes / inclusions / anomalies Inclusion/shape detection problem: Λ(σ) supp(σ σ )?, σ : reference conductivity.

5 Shape detection Promising approach: Factorization method (Kirsch 998) FM for EIT (999 ): Brühl, Hakula, Hanke, H., Hyvönen, Kirsch, Lechleiter, Nachman, Päivärinta, Pursiainen, Schappel, Schmitt, Seo, Teirilä Typical result: z supp(σ σ ) iff lim I α (z) =. α (I α(z): indicator function) Unsolved problems since 998: Convergent regularization strategies for infinity test? Theory needs definiteness assumption, e.g., σ σ everywhere In this talk: A monotonicity based sampling method.

6 Monotony Ω (σ σ 2 ) u 2 dx (g,(λ(σ 2 ) Λ(σ ))g) u solution corresponding to σ and boundary current g. Simple consequence: σ σ 2 = Λ(σ ) Λ(σ 2 )

7 Monotony based imaging True conductivity: σ = +χ D, D: unknown inclusion Λ(σ): measured data Test conductivity: +χ B, B: small ball Λ(+χ B ) can be simulated for different balls B Monotony: B D = +χ B +χ D = σ = Λ(+χ B ) Λ(σ) Monotony based reconstruction algo. for EIT (Tamburrino/Rubinacci 2) For all B, calculate Λ(+χ B ) & test whether Λ(+χ B ) Λ(σ) Result: upper bound of D. Only an upper bound? Converse monotony relation?

8 Converse montony relation Theorem (H./Ullrich) Ω\D connected. σ = +χ D. B D = Λ(+χ B ) Λ(σ). Monotony method detects exact shape. (Extensions possible for non-connected complement, inhomogeneous inclusions or background, continuous transitions between inclusion and background,...)

9 Converse montony relation Proof (σ = +χ D, κ = +χ B ) (κ σ) u κ 2 dx (g,(λ(σ) Λ(κ))g) Ω Apply localized potentials (H 28) to control power term u κ 2. D small power B large power g : (g,(λ(σ) Λ(κ))g) = Λ(σ) Λ(κ)

10 Fast implementation Testing Λ(+χ B ) Λ(σ) is expensive. One forw. prob. per B. Theorem (H./Seo, SIAM J. Math. Anal. 2) Let κ, σ, σ piecewise analytic and Λ (σ )κ = Λ(σ) Λ(σ ). Then supp Ω κ = supp Ω (σ σ ). supp Ω : outer support ( = support, if support is compact and has conn. complement) Replacing Λ(+χ B ) by its linear approx. should still recover the exact shape (linearization error does not affect the shape!)

11 Fast implementation Theorem (H./Ullrich) Let Ω\D connected, < k /2. B D Λ(½)+kΛ (½)χ B Λ(σ). Fast, requires only homogeneous forward solution Comp. cost equivalent to linearized methods or FM (Again, extensions possible for non-connected complement, inhomogeneous inclusions or background, continuous transitions between inclusion and background,...)

12 Indefinite inclusions Indefinite inclusions: σ = 2+χ D + χ D. With κ + = 2+χ B. κ = 2 χ B : D + D B Λ(κ ) Λ(σ) Λ(κ + ). Indefinite inclusions can be treated by step-wise shrinking of larger test domains. Result can be linearized. Linearized version yields exact shape (no linearization error!) Result extends to general pcw. anal. conductivities (with some technical effort)

13 Numerical results y Achse x Achse y Achse x Achse Reconstructions with exact data and with.% noise.

14 Numerical results.5.5 z Achse z Achse y Achse x Achse y Achse x Achse Reconstructions with exact data and with.% noise.

15 Numerical results y Achse x Achse.5.5 Reconstructions for smooth transitions between inclusion and background and for the indefinite case.

16 Outlook Goal: Enhance linearized methods Standard linearized method vs. heuristic combination with FM for frequency-difference EIT without ref. measurements (H./Seo/Woo, IEEE Trans. Med. Imaging 2)

17 Summary New monotony based shape reconstruction method yields the exact shape not just an upper bound can be efficiently implemented by linearization (while still reconstructing the exact shape) Advantages Rigorous treatment of indefinite inclusions Convergent regularization implementation of testing criteria seems possible For practical applications: Enhance linearized/iterative methods by exact shape reconstruction (H./Seo SIMA 2, H./Seo/Woo IEEE TMI 2)