estimating electrode movementin Two Dimensions

Alistair Boyle1, Markus Jehl2, Michael Crabb3, Andy Adler1EIT2015, Neuchâtel, Switzerland1Carleton University, Ottawa, Canada2University College London (UCL), London, UK3University of Manchester, Manchester, UK

electrode movement corrections

Jacobian Methods Jm,x = δUδx1. Naïve perturbation2. perturbation3. rank-one update1

4. Fréchet derivative2 VδU

δxElectrode

1Gómez-Laberge C, Adler A. Physiol Meas 29(6):S89–S99, 20082Dardé J, Hakula H, Hyvönen N, et al. Inverse Problems and Imaging 6(3):399–421, 2012

1

motivation

∙ explore sensitivity of the direct perturbation method

∙ understand Fréchet derivative and build an implementation

∙ ask the Engineering question:

Do these methods give the same answers for a simplified model?

2

movement jacobian

Movement In our case, two-dimensional tangential electrodemovement on a fixed boundary.

Jacobian An estimate of the local slope.

For Gauss-Newton updates, the Jacobian, in part, determines searchdirection δx, followed by a line search for distance α. If sufficientlylinear, then α = 1 can work well.

δx = (JTJ+ λ2RTR)−1 JTU (1)xn+1 = xn + α δx (2)

3

a two-dimensional half-space model

hmax = 0.1 m deℓ = 0.2 m zc = 0.02 Ω·m σ = 1 S/m

4

perturbation

Naïve Perturbation Naïve Perturbation

Jm =δUδx

all electrode nodes are perturbed

5

perturbation

Perturbation Perturbation

Jm =δUδx

electrode boundary nodes are perturbed

5

perturbation

Rank-One Update Gómez-Laberge

Jm =δUδx

“the discrete form of the Lie derivative of the conductivity tensor”;the matrix rank-one update to the conductivity Jacobian due to FEM

node movement

5

fréchet derivative for tangential movement

Crabb, Boyle

accounts for contact impedance zc effectsin the Complete Electrode Model

δU = Jmh+ O(h2)

Jm h =1zc

∫∂E(h · υ∂E)(Um − u)(Vm − v)ds

Um

u

Vm

v

zcυ∂E

Dardé J, Hakula H, Hyvönen N, et al. Inverse Problems and Imaging 6(3):399–421, 20126

fréchet derivative for tangential movement

Crabb, Boyle

accounts for contact impedance zc effectsin the Complete Electrode Model

δU = Jmh+ O(h2)

Jm h =1zc

∫∂E(h · υ∂E)(Um − u)(Vm − v)ds

Um

u

Vm

vzc

υ∂E

Dardé J, Hakula H, Hyvönen N, et al. Inverse Problems and Imaging 6(3):399–421, 20126

fréchet derivative for tangential movement

Boyle*

What if the contact impedance zc units are wrong:[Ω·m] rather than [Ω]?

7

contact impedance zc for the m+ electrode

10−14 10−10 10−6 10−2 102 106 1010−1

−0.5

0

0.5

1

zc [Ω·m]

J m,x

Naïve perturbationperturbationGómez-LabergeBoyle, CrabbBoyle*

hmax = 0.1 m deℓ = 0.2 m plot for m+

8

electrode diameter deℓ for the m+ electrode

10−2 10−1−1

−0.5

0

0.5

1

deℓ [m]

J m,x

Naïve perturbationperturbationGómez-LabergeBoyle, CrabbBoyle*

hmax = 0.1 m zc = 0.02 Ω·m plot for m+

9

mesh density hmax for the m+ electrode

10−0.8 10−0.6 10−0.4 10−0.2 100−1

−0.5

0

0.5

1

hmax [m]

J m,x

Naïve perturbationperturbationGómez-LabergeBoyle, CrabbBoyle*

deℓ = 0.2 m zc = 0.02 Ω·m plot for m+

10

discussion

What does this mean for movement solutions?

∙ Fréchet deriv. is fast(forward solutions reused from adjoint method/conductivity)

∙ Fréchet deriv. magnitudes differ from perturbation methods(sign is correct/stable;hyper-parameter λ and/or line search α values will be different)(contact impedance units?)

∙ Fréchet deriv. results collapse for small contact impedancezc ≪ Ωσ , and more sensitive to mesh density hmax thanconductivity solutions(use “rank-one update” method when necessary?)

11

Questions?

12

contact impedance zc for the s- electrode

10−14 10−10 10−6 10−2 102 106 1010−1

−0.5

0

0.5

1

zc [Ω·m]

J m,x

Naïve perturbationperturbationGómez-LabergeBoyle, CrabbBoyle*

hmax = 0.1 m deℓ = 0.2 m plot for s-

13

electrode diameter deℓ for the s- electrode

10−2 10−1−1

−0.5

0

0.5

1

deℓ [m]

J m,x

Naïve perturbationperturbationGómez-LabergeBoyle, CrabbBoyle*

hmax = 0.1 m zc = 0.02 Ω·m plot for s-

14

mesh density hmax for the s- electrode

10−0.8 10−0.6 10−0.4 10−0.2 100−1

−0.5

0

0.5

1

hmax [m]

J m,x

Naïve perturbationperturbationGómez-LabergeBoyle, CrabbBoyle*

deℓ = 0.2 m zc = 0.02 Ω·m plot for s-

15