Solving the electrical impedance tomography inverse ...

Solving the electrical impedance tomography inverse problem for logarithmic conductivity Numerical sensitivity

Sergio de Paula Pellegrini Flávio Celso Trigo Raul Gonzalez Lima

Universidade de São Paulo Escola Politécnica

September 15th 2016 Pellegrini, Trigo, Lima (USP)

EIT inverse problem in log conductivity

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Outline

1 Introduction 2 Logarithmic conductivity improves convexity in EIT problem 3 Logarithmic conductivity might improve convergence rate

Pellegrini, Trigo, Lima (USP)


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Introduction

Electrical Impedance Tomography (EIT) Estimate distribution of electrical properties in a domain with imposition of currents and measurements of potentials at its boundary

• Inverse • Ill-posed • Nonlinear



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Introduction

EIT: methodology A typical approach: • Maxwell equations & material constitutive laws

• Finite Element Method • Iterative procedure: 1 Departing from an initial estimation of the property distribution (ˆ σ0 2

3 4

5

[e] )

−1 Estimate the potentials at the electrodes Vˆ[m] = T[m×n] Y[n×n] C[n] , k with Y[n×n] = f σ ˆ

An error function is evaluated: Φ1 σ ˆk =

1 2

V − Vˆ k

t

V − Vˆ k

A new distribution of electrical properties σ ˆ k+1 is determined in order to decrease the error function Go back to step 2 until stop criterion is matched



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Introduction

EIT: methodology A typical approach: • Maxwell equations & material constitutive laws

• Finite Element Method • Iterative procedure: 1 Departing from an initial estimation of the property distribution (ˆ σ0 2

3 4

5

[e] )

−1 Estimate the potentials at the electrodes Vˆ[m] = T[m×n] Y[n×n] C[n] , k with Y[n×n] = f σ ˆ direct problem

An error function is evaluated: Φ1 σ ˆk =

1 2

V − Vˆ k

t

V − Vˆ k

A new distribution of electrical properties σ ˆ k+1 is determined in order to decrease the error function inverse problem Go back to step 2 until stop criterion is matched



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Introduction

EIT: the negative conductivity problem Obtaining estimates in conductivity σ or resistivity ρ is particularly challenging when the actual values are between [0; 1], as the search algorithm (step 4) is prone to reach negative values

Actual values (1% uniform noise added to potential “measurements”) Pellegrini, Trigo, Lima (USP)

Converged solution using Gauss-Newton method, with σ ˆ0 = 2 σ ¯ and α = 40.0%


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Introduction

EIT: Literature solutions for negative conductivity problem

• Constrained search with backtracking: control the under relaxation

factor α to ensure positiveness

• Parametrization: using a different variable to describe the unknowns



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Introduction





Logarithmic conductivity ς, such that σ = σ0 exp ς with σ0 = 1



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Introduction





Logarithmic conductivity ς, such that σ = σ0 exp ς with σ0 = 1

S m

• Ensures positiveness – EIT numerical search is implicitly

constrained to respect Physics

• However, no study explored the comparative effect of

using logarithmic conductivity in EIT



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Logarithmic conductivity improves convexity in EIT problem

1 Introduction

2 Logarithmic conductivity improves convexity in EIT problem

3 Logarithmic conductivity might improve convergence rate



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Threshold α for monotone convergence Gauss-Newton method: there is a threshold under relaxation factor αthr that leads to monotone convergence. For the problem in analysis, with σ ˆ0 = 2 σ ¯:



search variable

αthr

σ

34.0%

ρ

17.7%

ς

63.2%

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Threshold α for monotone convergence Gauss-Newton method: there is a threshold under relaxation factor αthr that leads to monotone convergence. For the problem in analysis, with σ ˆ0 = 2 σ ¯:

search variable

αthr

σ

34.0%

ρ

17.7%

ς

63.2%

Converged solution Pellegrini, Trigo, Lima (USP)


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Threshold α for monotone convergence Gauss-Newton method: there is a threshold under relaxation factor αthr that leads to monotone convergence. For the problem in analysis, with σ ˆ0 = 2 σ ¯: 5

10

0

10

−5

10

−10

10

Φ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ∂Φ ¯¯ ¯¯ ∂Vc t ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ∂σ ¯¯ = ¯¯ ∂σ (Vm −Vc )¯¯ ¯¯ k−1 k ¯¯ ¯¯σ −σ ¯¯

10

20

30

40


50

60

70

80

search variable

αthr

σ

34.0%

ρ

17.7%

ς

63.2%

90

It. EIT inverse problem in log conductivity

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Threshold α for monotone convergence Gauss-Newton method: there is a threshold under relaxation factor αthr that leads to monotone convergence. For the problem in analysis, with σ ˆ0 = 2 σ ¯: 5

10

5.4 · 10−4 0

10

−5

10

−10

10


10

20

30

40


50

60

70

80

search variable

αthr

σ

34.0%

ρ

17.7%

ς

63.2%

90

It. EIT inverse problem in log conductivity

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Existence of local minimae Gauss-Newton in σ, with σ ˆ0 = 2 σ ¯ and α = 40.0%



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Existence of local minimae Gauss-Newton in σ, with σ ˆ0 = 2 σ ¯ and α = 40.0% 100

10


50

10

0

10

−50

10


2

4

6

8 It.


10

12

14

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10


50

10

0

10

−50

10

2

4

6

8 It.

10

12

2.6 · 10−9

14

Solution is local minimum Pellegrini, Trigo, Lima (USP)


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10


50

10

2.8 · 100 0

10

−50

10

2

4

6

8 It.

10

12

2.6 · 10−9

14

Solution is local minimum This minimum has higher “energy” than physically plausible solution Pellegrini, Trigo, Lima (USP)


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Negative values may lead to other minimae Gauss-Newton in σ, with σ ˆ0 = 2 σ ¯ α = αthr = 34.0%

α = 34.1%

12

60

10

50

8

40

6 σ

σ

30

4

20

2 10

0 0

−2 −4

−10

10

20

30

40

50

60

70

80

90

2

It.



4

6

8

10

12

It.

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Negative values may lead to other minimae Gauss-Newton in σ, with σ ˆ0 = 2 σ ¯ α = αthr = 34.0%

α = 34.1%

12

60

10

50

8

40

6 σ

σ

30

4

20

2 10

0 0

−2 −4

−10

10

20

30

40

50

60

70

80

90

2

It.

4

6

8

10

12

It.

Negative conductivities inject “energy” into the numerical equations Pellegrini, Trigo, Lima (USP)


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Using ς allows faster algorithm 0

10

σ ρ ς

−1

α thr

10

−2

10

−3

10

−2

10

−1

10

0

10

1

10

2

10

3

10

4

10

σ ˆ 0/ σ ¯ Pellegrini, Trigo, Lima (USP)


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Hypersurface Φ is modified Local minimae with negative values for σ or ρ are not reachable using ς Using Nonlinear Conjugate Gradient: Estimating in σ

Estimating in ς

(Polak-Ribière with exact line search) Pellegrini, Trigo, Lima (USP)


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Logarithmic conductivity might improve convergence rate

1 Introduction

2 Logarithmic conductivity improves convexity in EIT problem

3 Logarithmic conductivity might improve convergence rate



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Convergence rate Gauss-Newton, with σ ˆ0 = 2 σ ¯ and α = 20.0% Estimating in ς

12

12

10

10

8

8

6

6

σ

σ

Estimating in σ

4

4

2

2

0

10

20

30

40 It.


50

60

70

0

10


20

30

40 It.

50

60

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Convergence rate Gauss-Newton, with σ ˆ0 = 2 σ ¯ and α = 20.0% Estimating in ς

12

12

10

10

8

8

6

6

σ

σ

Estimating in σ

4

4

2

2

0

10

20

30

40 It.

50

60

70

0

10

20

30

40 It.

50

60

70

Convergence rate more independent of actual property distribution Pellegrini, Trigo, Lima (USP)


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Assymetry in direct problem I Varying the conductivity of a inhomogeneity, electric potentials are measured (no noise added) for all bipolar electric current inputs



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Assymetry in direct problem II The measured potentials are compared with the ones of a reference homogeneous case, and plotted as a function of the inhomogeneity value 0.35

0.25 ¯¯ ¯¯ ¯¯ ˆ ˆ ¯¯ ¯¯V −Vref ¯¯

2

2

0.3

0.25 ¯¯ ¯¯ ¯¯ ˆ ˆ ¯¯ ¯¯V −Vref ¯¯

0.35

0.3

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0.1

0.2

0.3

0.4 σ (S/m)

0.5

0.6

0.7

0

−2

−1.5 ς (−)

−1

−0.5

• This analysis shows the aggregate influence of the inhomogeneous

elements

• The plots, based on the direct problem, show tendencies for

convergence rate of the inverse problem (formally, at initial guess)

• A more symmetric behavior implies in a convergence rate more

independent of under- or overestimation



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Assymetry in direct problem – large phantom I Same test for a numerical model of 385 601 elements and 32 electrodes

Varying conductivity of inhomogeneity from 3.38 · 10−3 to 6.17 · 107 Pellegrini, Trigo, Lima (USP)


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Assymetry in direct problem – large phantom II −3

−8

x 10

4.5

5

x 10

4 3.5

4 ¯¯ ¯¯ ¯¯ ˆ ˆ ¯¯ ¯¯V −Vref ¯¯

¯¯ ¯¯ ¯¯ ˆ ˆ ¯¯ ¯¯V −Vref ¯¯

2

2

3 2.5

3

2

2

1.5 1

1

0.5

0

1

2

3 σ (S/m)

4

5

6 7 x 10

0

−5

0

5

10

15

ς (−)

• Similar conclusions are drawn for symmetry near the homogeneous

setup • For the contrasting inhomogeneity values, the numerical scheme executes a large step when estimating in σ – which might lead to negative conductivities – but a smaller one when estimating for ς



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Core references

• T. Murai, Y. Kagawa. Electrical impedance computed tomography

based on a finite element model. IEEE Transactions on Biomedical Engineering, BME-32(3), 1985, pp. 177-184.

• P. J. Vauhkonen. Image Reconstruction in Three-Dimensional

Electrical Impedance Tomography. PhD thesis, University of Kuopio, 2004.

• A. Tarantola. Elements for Physics: quantities, qualities and intrinsic

theories. Springer, 2006.



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