Simultaneous reconstruction of outer boundary shape and conductivity distribution in electrical impedance tomography

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1 Simultaneous reconstruction of outer boundary shape and conductivity distribution in electrical impedance tomography Nuutti Hyvönen Aalto University joint work with J. Dardé, A. Seppänen and S. Staboulis.

2 Outline of the talk 1. Motivation. 2. Complete electrode model. 3. Differentiation with respect to boundary shape. 4. Reconstruction algorithm. 5. Numerical examples.

3 1. Motivation.

4 EIT reconstruction with correct boundary shape

5 EIT reconstruction with incorrect boundary shape

6 Methods for handling geometric uncertainties (i) [Kolehmainen et al. (2005)]: A method based on allowing slightly anisotropic conductivities. (ii) [Nissinen et al. (2011)]: An approximation error approach where statistics of the modeling error are approximated in advance via simulations and this information is then accounted for in Bayesian inversion. (iii) [Dardé et al. (2013)]: Proving Fréchet differentiability with respect to geometry and employing this information in a Gauss Newton scheme: Minimize a suitable least squares (MAP) functional where the geometric information is included as unknown parameters. (iv) [Mustonen et al. (2016)]: Stochastic collocation...

7 2. Complete electrode model.

8 Realistic electrode measurements U I

9 Complete electrode model [Cheng et al. 89] The admittance σ L (D) has a strictly positive definite real part. The boundary of D is partially covered with mutually disjoint and connected electrodes E m D, E := M m=1e m. The electrode net currents and voltages are denoted by {I m }, {U m } C, respectively. In electrode measurements, the contacts at the electrode-object interfaces are never perfect. This is characterized by the contact impedances z C M, with Re z m > 0, which are usually unknown.

10 The forward problem corresponding to the complete electrode model (CEM) is as follows: For the electrode net currents I C M, find (u, U) (H 1 (D) C M )/C that satisfies weakly σ u = 0 in D, ν σ u = 0 on D \ E, u + z m ν σ u = U m on E m, m = 1,..., M, ν σ u ds = I m, E m m = 1,..., M. These equations define the electromagnetic potential u and the electrode potentials U uniquely up to a common additive constant. In practice, (a noisy version of) the linear current-to-voltage operator R : I U, C M C M /C, is the data that can be obtained via measurements of EIT.

11 Regularity of the CEM solution Assuming that the admittance σ and the boundary D are regular enough (Lipschitz and C 1,1 ), the following result holds: Theorem. The interior potential u belongs to in H 2 ɛ (D)/C, ɛ > 0. (In particular, the singularities at the boundaries of the electrodes are not very severe.) Corollary. When z m goes to zero, the regularity is partially lost. In the limit, u H 3/2 ɛ (D)/C, ɛ > 0. In consequence, aiming for low contact impedances makes the CEM forward problem more difficult to solve numerically(!).

12 3. Differentiation with respect to boundary shape.

13 Perturbation of the object boundary Assume that σ and D are smooth enough. Let h C 1 ( D; R n ) be a small enough vector field and define the corresponding perturbed object boundary via D h = {y R n y = x + h(x) for some x D}. The electrodes are assumed to move/stretch accordingly. It is rather obvious that the current-to-voltage map of the CEM may be considered as an operator of two variables R : (h, I) U[h], B d C M C M /C, where B d is an origin-centered ball of radius d > 0 in the topology of C 1 ( D; R n ) and (u[h], U[h]) (H 1 (D h ) C M )/C is the solution of the CEM forward problem for the net current pattern I and the object D h.

14 Derivative with respect to the boundary perturbation The Fréchet derivative of R : B d C M C M /C with respect to the first variable at the origin is given by the bilinear map R : (h, I) U [h], C 1 ( D; R n ) C M C M /C, where (u [h], U [h]) (H 1 ɛ (D) C M )/C is the solution of the following derivative problem : σ u = 0 in D, M ν σ u 1 m (U u 1 )χ m = f 1 (f 2 χ m f 3 δ m ) z m=1 m z m=1 m on D, (U m u ) ds = E m f 2 ds E m f 3 ds, E m m = 1,..., M. Here, χ m is the characteristic function of E m, δ m is the delta functional on E m, and f 1 H 1/2 ɛ ( D), f 2 H 1/2 ɛ (E) and f 3 H 1 ɛ ( E).

15 To be more precise, f 1 = Div(h ν (σ u D ) τ ), f 2 Em = h ν ( (n 1)H(U m u) + u ν ) Em, f 3 Em = (h ν E )(U m u) Em, where Div is the surface divergence, h ν is the normal component of h, H : D R is the mean curvature, ν E is the unit normal of the electrodes in the tangent bundle of D, and the pair (u, U) (H 2 ɛ (D) C M )/C is the solution of the standard CEM forward problem with the electrode net currents I. NB: There exists a similar characterization for the derivatives with respect to perturbations of the electrode boundaries. This allows one to build a reconstruction algorithm that estimates the locations/ sizes/shapes of the electrodes, as well.

16 A dual sampling formula Due to the following sampling formula, the derivative problem need not to be solved numerically in practice: Let (ũ, Ũ) be the CEM forward solution for some current pattern Ĩ C M. Then for any (h, I) C 1 ( D, R n ) C M it holds that R (h, I) Ĩ = h ν (σ u) τ ( ũ) τ ds D M m=1 M m=1 1 z m 1 z m E m h ν ( (n 1)(U m u)h u ν E m (h ν E )(U m u)(ũm ũ) ds. ) (Ũm ũ) ds

17 4. Reconstruction algorithm.

18 Parametrization of the object boundary We search for the unknown boundary in the star-shaped form γ α (θ) = r α (θ)e iθ, where and α j R. r α (θ) = α 0 + N (α j cos jθ + α j+n sin jθ) j=1 The derivatives of the CEM measurement map with respect to α j are computed with the help of the above sampling formula.

19 The algorithm The actual reconstruction method is a modified version of the (3D) output least squares algorithm used by Prof. Jari Kaipio s inverse problems group at the University of Eastern Finland (Kuopio). Regularization with respect to the conductivity distribution is achieved with a smoothness prior. (The initial guess is constant.) Regularization with respect to the boundary shape is achieved using suitable Sobolev norms. (The initial guess is a disk.) The actual minimization algorithm is a combination of the Gauss Newton method and the golden section line search. The degree of regularization is chosen within the Bayesian paradigm. The length of the boundary curve and the sizes of the electrodes are assumed to be known, but the locations of the electrodes and the contact impedances are estimated. (The initial guess is equiangular.)

20 The to-be-minimized least-squares/map functional looks something like this: Φ(σ, z, α, θ) = V U(σ, z, α, θ) 2 Γ 1 + σ 2 Σ 1 + β 1 z z β 2 θ θ β 3 γ α γ α0 H s.

21 4. Examples.

22 Simulated data

23 Experimental data

27 Open problems Characterization of the (severe) nonuniqueness. Purely three-dimensional geometries. More flexible parametrizations for the boundary (splines etc.). Proper handling of the discretization of the Fréchet derivative (the effect of contact impedance). Testing with diagnostic data.

28 Some relevant publications J. Dardé, H. Hakula, N. Hyvönen and S. Staboulis, Fine-tuning electrode information in electrical impedance tomography, Inverse Problems and Imaging, 6, (2012). J. Dardé, N. Hyvönen, A. Seppänen and S. Staboulis, Simultaneous reconstruction of outer boundary shape and admittance distribution in electrical impedance tomography, SIAM Journal on Imaging Sciences, 6, (2013). J. Dardé, N. Hyvönen, A. Seppänen and S. Staboulis, Simultaneous recovery of admittivity and body shape in electrical impedance tomography: An experimental evaluation, Inverse Problems, 29, (2013).

arxiv: v2 [math.na] 8 Sep 2016

arxiv: v2 [math.na] 8 Sep 2016 COMPENSATION FOR GEOMETRIC MODELING ERRORS BY ELECTRODE MOVEMENT IN ELECTRICAL IMPEDANCE TOMOGRAPHY N. HYVÖNEN, H. MAJANDER, AND S. STABOULIS arxiv:1605.07823v2 [math.na] 8 Sep 2016 Abstract. Electrical