Model-aware Newton-type regularization in electrical impedance tomography

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Model-aware Newton-type regularization in electrical impedance tomography Andreas Rieder Robert Winkler FAKULTÄT FÜR MATHEMATIK INSTITUT FÜR ANGEWANDTE

1 Model-aware Newton-type regularization in electrical impedance tomography Andreas Rieder Robert Winkler FAKULTÄT FÜR MATHEMATIK INSTITUT FÜR ANGEWANDTE UND NUMERISCHE MATHEMATIK SFB 1173 KIT University of the State of Baden-Württemberg and National Research Center of the Helmholtz Association

2 Electrical impedance tomography Preprocessing Resolving the underdetermination Numerical experiments 2 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

3 Electrical impedance tomography Preprocessing Resolving the underdetermination Numerical experiments Electrical impedance tomography 3 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

4 The complete electrode model (CEM) current I i voltage U k σ(x) B p electrodes E with contact impedances z 1 E div(σ u) = 0 in B u+z σ n u = U on E σ n u = 0 on B \ E E σ n uds = I = f E 4 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

5 The complete electrode model (CEM) current I i voltage U k σ(x) B p electrodes E with contact impedances z 1 E div(σ u) = 0 in B u+z σ n u = U on E σ n u = 0 on B \ E E σ n uds = I = f E Given f E p := { span{χ E1,...,χ Ep }: B ds = 0} L 2 ( B) find (u,u) H 1 (B) E p : ( ) b σ (u,u),(w,w) = fwds (w,w) H 1 (B) E p (1) where b σ ( (v,v),(w,w) ) = B B σ v wdx + p =1 (Existence & Uniqueness: Cheney, Isaacson & Somersalo, 1992) 1 z E (v V)(w W)dS 4 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

6 The forward operator P = {B 1,...,B l }, partition of B Σ P = span{χ B1,...,χ Bl }, conductivity space Σ + P = {s Σ P : s c > 0 a.e.}, admissible conductivities Then, F : Σ + P L (B) L ( ) { } E p, σ f U, F(σ)f = U, where U is the second component of the solution of (1). F(σ) Current-to-Voltage or Neumann-to-Dirichlet map Note: We identify I, U E p with their coefficient vectors w.r.t. {χ E1,...,χ Ep } and σ Σ P with its coefficient vector w.r.t. {χ B1,...,χ Bl }. 5 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

7 The inverse conductivity problem (ICP) For a set of M currents I = (I 1 I M ) R p M we measure U v := F(σ)I+N v where N v k,l iid N(0,v). Given I and U v find a σ Σ + P s.t. F( σ)i U v. DOF provided by measurements (at most): (p 1)+(p 2)+...+(p M) = pm Thus, the ICP is typically under-determined as M(M +1) 2 p(p 1). 2 dimσ P p(p 1). 2 6 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

8 Generic Newton scheme Input: (I,U v ), σ (0), contact impedances z, noise level δ. Output: conductivity σ. k := 0; d (0) := U v F(σ (0) )I; while d (k) Fro > 1.1 δ do Find regularized solution s (k) of F (σ (k) )[s (k) ]I = d (k) ; σ (k+1) := σ (k) +s (k) ; d (k+1) := U v F(σ (k+1) )I; k := k +1; end while In a pre-processing step we determine σ (0) as best constant, estimate z and δ = U v U Fro. Further: Strategies for decreasing the nonlinearity and for resolving the underdetermination of ICP. 7 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

9 Electrical impedance tomography Preprocessing Resolving the underdetermination Numerical experiments Preprocessing 8 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

10 Initialization of σ (0) : shunt electrode model Here, z = 0, that is, u U on each electrode. Let u and v solve Laplace s equation with σ σ 0 and σ 1, resp., under identical Neumann data, that is i ν := σ 0 ν u = ν v. Then, ui ν ds = u vdx = 1 vi ν ds. σ 0 Ω Solving for σ 0 and averaging over M currents, Ω Ω σ 0 = 1 M M m=1 Ω v(m) i (m) ν ds Ω u(m) i (m) ν ds 1 M M m=1 p =1 V (m) I (m) p =1 U(m) I (m) =: σ shunt 0 (I, U) where U (m) and V (m) respect to the current I (m) are the measured and simulated voltages with = E i (m) ν ds. 9 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

11 Initialization of σ (0) : CEM On each electrode we have that u+z i ν = U and E i ν ds = I, = 1,...,p, while i ν = 0 on the gaps. Under the assumption u E const we get If we substitute this term for U (m) σ CEM 0 (I,U) := 1 M u = U z i ν U z E 1 I. M m=1 and V (m) p =1(V (m) p =1(U (m) Challenge. Unkown contact impedances z. in σ shunt 0, resp., then z E 1 I (m) z E 1 I (m) ) ) I (m) I (m). 10 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

12 Initialization of σ (0) and z simultaneously Assumption: z z > 0, = 1,...,p. Let V be the simulated data (σ 1) with contact impedance w > 0. We obtain a linear system for z and ρ 0 = 1/σ 0 : a m ρ 0 +b m z = c m, m = 1,...,M, where a m = p =1 ( V m w E 1 I m ) I m, b m = p E 1 (I m ) 2, c m = =1 p U m I m. =1 Then, σ CEM,z 0 := ρ 1 0 and z CEM,z 0 := ẑ where ρ 0 and ẑ are the least-squares solutions. Attention. The system might be under-determined. 11 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

13 Initialization of σ (0) and z : numerical experiment unit disk with σ 0.25, 16 equally spaced electrodes covering 50% of the boundary, w = variance z % ±10% z CEM,z σ CEM,z z CEM,z σ CEM,z Bottom: contact impedances varying by ±10%, i.e. z = z(1+0.1n ), = 1,...,p, where n [ 1,1] are independent uniformly distributed numbers. 12 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

14 Estimation of the noise level I spanning set for E p = UI + = F(σ) symmetric Symmetry error of measured data: e v := U v I + ( U v I + ) 2 Fro. We have as well as Ee v = 2(p 1) I + 2 Fro v Eδ = E N v Fro = 2v Replacing Ee v by e v we estimate δ CEM (I,U v ) := Γ ( Mp+1 2 ( Mp 2 Γ M pe v 2(p 1) ) ) Mpv. I + 1 Fro Eδ. 13 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

15 Testing the noise level estimator δ δ CEM (p = 16) δ CEM (p = 64) 2.0e e e-3 2.0e e e-2 2.0e e e-1 1 δ CEM /δ 4.5% 6.5% unit disk with σ 1, equally spaced electrodes covering 50% of the boundary, adacent currents, similar results for non-constant conductivities. 14 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

16 Conductivity transformation Avoiding the positivity constraint. Reducing the nonlinearity. Consider inective C 1 transformations t : (0, ) R, σ t (σ) =: t, and their corresponding transformed forward operators F (t (σ)) = F(σ), that is, F (t) = F(t 1 (t)) =: F(σ (t)). Well-known: t (σ) = log(σ) Our alternative: t α (σ) := (1 α) 1 σ ασ where α = Remark. The choice α = 0.25 minimizes a certain nonlinearity measure under all transformation t α, 0 < α < / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

17 Electrical impedance tomography Preprocessing Resolving the underdetermination Numerical experiments Resolving the underdetermination 16 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

18 Reconstruction prior Computing the Newton update s R l from Ss = d where S := F (σ)[ ]I R pm l is highly under-determined (and ill-posed). Let S := F (σ)[χ B ]I be the th column of S. If S = βs k for some β R\{0} then e βe k N(S). = We have one DOF in choosing s versus s k. Reconstruction prior. In case of S = βs k choose s and s k such that s k = sgn(β) s. σ k σ Consequence. Identical conductivity values are updated by the same amount. 17 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

19 Weighted minimum norm solution Define a weighted minimum norm solution s + W := argmin s N(S) W Ss d 2 where W denotes orthogonality w.r.t. W, for W > 0. Theorem: If W = W S,σ := diag then s + W satisfies the reconstruction prior. Remark. If S βs k and W = W S,σ then ( S1 2,..., S ) l 2 σ 1 σ l s k σ k sgn(β) s σ. Consequence. No spurious oscillations are introduced. 18 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

20 Electrical impedance tomography Preprocessing Resolving the underdetermination Numerical experiments What you should take home from this talk Numerical experiments 19 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

21 MANTIS Input: (I,U v ), domain/electrode geometries, t ; Output: conductivity σ; Intialize partition P of B; Estimate σ (0), z, δ; k := 0; d (0) := U v F(σ (0) )I; while d (k) Fro > 1.1 δ do t (k) := t (σ (k) ); S (k) := F (t (k) )[ ]I; s (k) := cg ( S (k),d (k) ),W (k) S,σ ; %adaptive stop criterion provided (k) σ (k+1) := σ ( t (k) +s (k) k := k +1; end while ) ; d (k+1) := U v F(σ (k+1) )I; 20 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

22 Test conductivity / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

23 Settings Exp. p z 1,...,p δ z CEM,z 0 δ CEM σ CEM,z 0 A e-1 1.5e e e B e-2 1.0e e e P = l (no. of triangles for reconstructing σ) 22 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

24 tlog, W = Id tα, W = Id tlog, W = WS,σ tα, W = WS,σ ǫ = 56.6%, ncg = 98 ǫ = 59.7%, ncg = 135 ǫ = 46.0%, ncg = 41 ǫ = 44.9%, ncg = 45 ǫ = 46.4%, ncg = 366 ǫ = 46.4%, ncg = 1100 ǫ = 41.5%, ncg = 48 ǫ = 39.8%, ncg = 59 1 ǫ = Z P σrec (x) P σ(x) dx, P σ(x) ncg overall number of cg iterations

25 A glimpse on Hilbert vs. Banach space 10 weighted L 2 weighted L ǫ = 45.5%, n cg = 120 ǫ = 34.1%, n LW = p z 1,...,p δ z CEM,z 0 δ CEM σ CEM,z e-1 1.0e e e / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

26 What you should take home from this talk MANTIS is a parameter-free Newton-type inversion scheme which extracts estimates for the contact impedance, background conductivity, and the noise level from the measured data, relies on a novel conductivity transformation to reduce the nonlinearity, uses weighted inner products to resolve the underdetermination and to suppress spurious oscillations. R. Winkler, A. Rieder: Model-aware Newton-type inversion scheme for electrical impedance tomography, Inverse Problems 31 (2015) (includes reconstructions from measured data) 25 / 25 c Andreas Rieder Model-aware Newton-type regularization in EIT AIP2015, Helsinki, May 2015

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