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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
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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
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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?
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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)
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a two-dimensional half-space model
hmax = 0.1 m deℓ = 0.2 m zc = 0.02 Ω·m σ = 1 S/m
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perturbation
Naïve Perturbation Naïve Perturbation
Jm =δUδx
all electrode nodes are perturbed
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perturbation
Perturbation Perturbation
Jm =δUδx
electrode boundary nodes are perturbed
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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
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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
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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
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fréchet derivative for tangential movement
Boyle*
What if the contact impedance zc units are wrong:[Ω·m] rather than [Ω]?
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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+
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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+
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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+
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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?)
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Questions?
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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-
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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-
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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-
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