Post on 16-Feb-2021
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)
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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
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
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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
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]?
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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-
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-
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