Monotony based imaging in EIT

Transkript

1 Monotony based imaging in EIT Bastian von Harrach Department of Mathematics - M, Technische Universität München, Germany Joint work with Marcel Ullrich, Universität Mainz, Germany ICNAAM 2, Rhodes, Greece, 9-25 September 2.

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(L2 ( 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: Reconstruct supp(σ σ ), σ : reference conductivity.

5 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 )

6 Monotony based imaging True conductivity: σ = +χ D, D: unknown inclusion Λ(σ): measured data Test conductivity: κ = +χ B, B: small ball Λ(κ) can be simulated for different balls B Monotony: B D = Λ(σ) Λ(κ) Monotony based reconstruction algo. for EIT Tamburrino/Rubinacci (22) For all balls B, calculate Λ(κ) and test whether Λ(σ) Λ(κ) Result: upper bound of D.

7 Monotony based imaging Monotony based reconstruction (up to now...) Simple theory, simple implementation Regularization seems straight-forward Only reconstructs upper bound Expensive, requires one forward solution for each test ball Needs definiteness assumption Comparison: Factorization Method Kirsch 998, Hanke/Brühl 2 Complicated theory and implementation No known convergent regularization strategies Reconstructs exact shape (if Ω\D connected) Cheap, requires only one homogeneous forward solution Needs definiteness assumption

8 Converse montony relation Theorem (H./Ullrich, 2) Ω\D connected. σ = +χ D, κ = +χ B. B D = Λ(κ) 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 Implementation/Definiteness Computation costs: Using linear approx. of Λ(κ) still fulfills monotony relation (still exact, no linearization error) Fast implementation, requires only homogeneous forward solution Comp. cost equivalent to linearized methods or FM Indefinite inclusions (larger and smaller than background conductivity) can be treated by step-wise shrinking of larger test domains.

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

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

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

14 Summary New results on the monotonicity method of Tamburrino and Rubinacci Method yields the exact shape not just an upper bound Method can be efficiently implemented by linearization (while still reconstructing the exact shape) Possible advantages Rigorous treatment of indefinite inclusions seems possible Convergent implementation of testing criteria seems possible Goal Enhance linearized/iterative methods by exact shape reconstruction (H./Seo SIMA 2, H./Seo/Woo IEEE TMI 2)