Electrical impedance tomography with punctual electrodes Fabrice - - PowerPoint PPT Presentation

Electrical impedance tomography with punctual electrodes

Fabrice Delbary1, Rainer Kreß1

1Universit¨

at G¨

• ttingen

AIP 2009 Vienna

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 1 / 26

Introduction

• Electrical impedance tomography

Applications : medicine, geophysics, nondestructive control Difficulties : nonlinear ill-posed inverse problem

• One of the recent progress for the electrical impedance tomography :

Nonlinear integral equations for the complete electrode model in inverse impedance tomography (H. Eckel, R. Kreß) Each electrode covers a part of the boundary No current outside the electrodes, current-voltage known on the electrodes Contact impedance has to be taken into account = ⇒ Possible simplification : if the electrodes are small enough, consider they are punctual

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 2 / 26

Introduction

• Electrical impedance tomography

Applications : medicine, geophysics, nondestructive control Difficulties : nonlinear ill-posed inverse problem

• One of the recent progress for the electrical impedance tomography :

Nonlinear integral equations for the complete electrode model in inverse impedance tomography (H. Eckel, R. Kreß) Each electrode covers a part of the boundary No current outside the electrodes, current-voltage known on the electrodes Contact impedance has to be taken into account = ⇒ Possible simplification : if the electrodes are small enough, consider they are punctual

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 2 / 26

Introduction

• Electrical impedance tomography

Applications : medicine, geophysics, nondestructive control Difficulties : nonlinear ill-posed inverse problem

• One of the recent progress for the electrical impedance tomography :

Nonlinear integral equations for the complete electrode model in inverse impedance tomography (H. Eckel, R. Kreß) Each electrode covers a part of the boundary No current outside the electrodes, current-voltage known on the electrodes Contact impedance has to be taken into account = ⇒ Possible simplification : if the electrodes are small enough, consider they are punctual

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 2 / 26

Plan of the talk

Overview of imaging methods Description of the problem and the method Numerical results Conclusion and future work

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 3 / 26

Imaging methods

Two important sets of reconstruction methods

• Sampling methods (A. Kirsch, D. Colton, F. Cakoni, H. Haddar, M.

Hanke-Bourgeois, M. Br¨ uhl, B. Gebauer, S. Schmitt, M. Piana, R. Aramini, G. Bozza, M. Brignone...) = ⇒ + Fast methods (compared with iterative methods) + Faster when using a ”no sampling” approach (find a ”global regularization parameter” on the grid of sampling points : R. Aramini, G. Bozza, M. Brignone, M. Piana)

• Usually less precise than iterative methods
• Qualitative methods (informations on the shape of the inclusion but

not on the physical parameters)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 4 / 26

Imaging methods

Two important sets of reconstruction methods

• Sampling methods (A. Kirsch, D. Colton, F. Cakoni, H. Haddar, M.

Hanke-Bourgeois, M. Br¨ uhl, B. Gebauer, S. Schmitt, M. Piana, R. Aramini, G. Bozza, M. Brignone...) = ⇒ + Fast methods (compared with iterative methods) + Faster when using a ”no sampling” approach (find a ”global regularization parameter” on the grid of sampling points : R. Aramini, G. Bozza, M. Brignone, M. Piana)

• Usually less precise than iterative methods
• Qualitative methods (informations on the shape of the inclusion but

not on the physical parameters)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 4 / 26

Imaging methods

Two important sets of reconstruction methods

• Sampling methods (A. Kirsch, D. Colton, F. Cakoni, H. Haddar, M.

Hanke-Bourgeois, M. Br¨ uhl, B. Gebauer, S. Schmitt, M. Piana, R. Aramini, G. Bozza, M. Brignone...) = ⇒ + Fast methods (compared with iterative methods) + Faster when using a ”no sampling” approach (find a ”global regularization parameter” on the grid of sampling points : R. Aramini, G. Bozza, M. Brignone, M. Piana)

• Usually less precise than iterative methods
• Qualitative methods (informations on the shape of the inclusion but

not on the physical parameters)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 4 / 26

Imaging methods

Two important sets of reconstruction methods

• Newton-like methods (iterative methods) (W. Rundell, R. Kreß, O.

Ivanyshyn, R. Potthast, H. Eckel, E. Heinemeyer,...) = ⇒ + Good accuracy (compared with sampling methods) + Quantitative methods

• Quite slow (compared with sampling methods, specially when a ”no

sampling” approach is used

• Need of an initial guess =

⇒ possibility to use a sampling method

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 5 / 26

Imaging methods

Two important sets of reconstruction methods

• Newton-like methods (iterative methods) (W. Rundell, R. Kreß, O.

Ivanyshyn, R. Potthast, H. Eckel, E. Heinemeyer,...) = ⇒ + Good accuracy (compared with sampling methods) + Quantitative methods

• Quite slow (compared with sampling methods, specially when a ”no

sampling” approach is used

• Need of an initial guess =

⇒ possibility to use a sampling method

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 5 / 26

Imaging methods

Two important sets of reconstruction methods

• Newton-like methods (iterative methods) (W. Rundell, R. Kreß, O.

Ivanyshyn, R. Potthast, H. Eckel, E. Heinemeyer,...) = ⇒ + Good accuracy (compared with sampling methods) + Quantitative methods

• Quite slow (compared with sampling methods, specially when a ”no

sampling” approach is used

• Need of an initial guess =

⇒ possibility to use a sampling method

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 5 / 26

Problem and method

D σ1 Ω σ0 ∂D

b b b b b b b b b b b b b b b b

Γ = ∂Ω electrodes

Problem : Find the location, the shape and the conductivity σ1 of an inclusion D in a circular domain Ω using electrical tomography Method : Impose a current between each adjacent pair of electrodes and measure the resulting voltage between all other adjacent pairs (containing none of the emitting electrodes)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 6 / 26

Problem and method

D σ1 Ω σ0 ∂D

b b b b b b b b b b b b b b b b

Γ = ∂Ω electrodes

Problem : Find the location, the shape and the conductivity σ1 of an inclusion D in a circular domain Ω using electrical tomography Method : Impose a current between each adjacent pair of electrodes and measure the resulting voltage between all other adjacent pairs (containing none of the emitting electrodes)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 6 / 26

Problem and method

D σ1 Ω σ0 ∂D

b b b b b b b b b b b b b b b b

Γ = ∂Ω I V U

Problem : Find the location, the shape and the conductivity σ1 of an inclusion D in a circular domain Ω using electrical tomography Method : Impose a current between each adjacent pair of electrodes and measure the resulting voltage between all other adjacent pairs (containing none of the emitting electrodes)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 6 / 26

Previous works on the problem

• B. Gebauer (talk of Monday 3:15 pm)

Relative data measurements for moving objects Measurements at a time t + 1 ”compared” to measurements at a time t Factorization Method (qualitative) = ⇒ Reconstruction of the moving object

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 7 / 26

Previous works on the problem

• S. Schmitt (talk of Friday 3:30 pm)

Imaging in the half plane Absolute data measurements (conductivity of the background medium supposed to be known) Factorization Method (qualitative) = ⇒ Reconstruction of the inclusion

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 8 / 26

Equations of the potential

Notations

R : radius of Ω Piecewise constant conductivity in Ω σ =

• σ0

in Ω \ D σ1 in D Fundamental solution of the Laplace equation ∀x, y ∈ R2 , x = y , Φ(x, y) = − 1 2π ln |x − y| Imposed current Positive electrode located at x+ ∈ Γ Negative electrode located at x− ∈ Γ Algebraic value of the current : I Generic notation for the outward normal vector to a domain : ν

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 9 / 26

Equations of the potential

Notations

R : radius of Ω Piecewise constant conductivity in Ω σ =

• σ0

in Ω \ D σ1 in D Fundamental solution of the Laplace equation ∀x, y ∈ R2 , x = y , Φ(x, y) = − 1 2π ln |x − y| Imposed current Positive electrode located at x+ ∈ Γ Negative electrode located at x− ∈ Γ Algebraic value of the current : I Generic notation for the outward normal vector to a domain : ν

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 9 / 26

Equations of the potential

Notations

R : radius of Ω Piecewise constant conductivity in Ω σ =

• σ0

in Ω \ D σ1 in D Fundamental solution of the Laplace equation ∀x, y ∈ R2 , x = y , Φ(x, y) = − 1 2π ln |x − y| Imposed current Positive electrode located at x+ ∈ Γ Negative electrode located at x− ∈ Γ Algebraic value of the current : I Generic notation for the outward normal vector to a domain : ν

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 9 / 26

Equations of the potential

Notations

R : radius of Ω Piecewise constant conductivity in Ω σ =

• σ0

in Ω \ D σ1 in D Fundamental solution of the Laplace equation ∀x, y ∈ R2 , x = y , Φ(x, y) = − 1 2π ln |x − y| Imposed current Positive electrode located at x+ ∈ Γ Negative electrode located at x− ∈ Γ Algebraic value of the current : I Generic notation for the outward normal vector to a domain : ν

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 9 / 26

Equations of the potential

Notations

R : radius of Ω Piecewise constant conductivity in Ω σ =

• σ0

in Ω \ D σ1 in D Fundamental solution of the Laplace equation ∀x, y ∈ R2 , x = y , Φ(x, y) = − 1 2π ln |x − y| Imposed current Positive electrode located at x+ ∈ Γ Negative electrode located at x− ∈ Γ Algebraic value of the current : I Generic notation for the outward normal vector to a domain : ν

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 9 / 26

Equations of the potential ut in Ω

∇ · (σ∇ut) = in Ω σ0∂νut = Iδx+ − Iδx−

• n

Γ In other terms... ∆ut = in Ω \ ∂D ∂νut = Iδx+ − Iδx−

• n

Γ u+

t

= u−

t

• n

∂D σ0∂νu+

t

= σ1∂νu−

t

• n

∂D

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 10 / 26

Equations of the potential ut in Ω

∇ · (σ∇ut) = in Ω σ0∂νut = Iδx+ − Iδx−

• n

Γ In other terms... ∆ut = in Ω \ ∂D ∂νut = Iδx+ − Iδx−

• n

Γ u+

t

= u−

t

• n

∂D σ0∂νu+

t

= σ1∂νu−

t

• n

∂D

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 10 / 26

Equations of the potential

Imposed potential and induced potential

• ut = Λ + u

∀x ∈ R2 , x = x± , Λ(x, x+, x−) = 2I σ0 Φ(x, x+) − 2I σ0 Φ(x, x−) ∆u = in Ω \ ∂D ∂νu =

• n

Γ u+ − u− =

• n

∂D σ0∂νu+ − σ1∂νu− = −(σ0 − σ1)∂νΛ

• n

∂D

• Γ

u = (fix the constant) (1)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 11 / 26

Equations of the potential

Imposed potential and induced potential

• ut = Λ + u

∀x ∈ R2 , x = x± , Λ(x, x+, x−) = 2I σ0 Φ(x, x+) − 2I σ0 Φ(x, x−) ∆u = in Ω \ ∂D ∂νu =

• n

Γ u+ − u− =

• n

∂D σ0∂νu+ − σ1∂νu− = −(σ0 − σ1)∂νΛ

• n

∂D

• Γ

u = (fix the constant) (1)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 11 / 26

Equations of the potential

Imposed potential and induced potential

• ut = Λ + u

∀x ∈ R2 , x = x± , Λ(x, x+, x−) = 2I σ0 Φ(x, x+) − 2I σ0 Φ(x, x−) ∆u = in Ω \ ∂D ∂νu =

• n

Γ u+ − u− =

• n

∂D σ0∂νu+ − σ1∂νu− = −(σ0 − σ1)∂νΛ

• n

∂D

• Γ

u = (fix the constant) (1)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 11 / 26

Equations of the potential

Imposed potential and induced potential

• ut = Λ + u

∀x ∈ R2 , x = x± , Λ(x, x+, x−) = 2I σ0 Φ(x, x+) − 2I σ0 Φ(x, x−) ∆u = in Ω \ ∂D ∂νu =

• n

Γ u+ − u− =

• n

∂D σ0∂νu+ − σ1∂νu− = −(σ0 − σ1)∂νΛ

• n

∂D

• Γ

u = (fix the constant) (1)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 11 / 26

Equations of the potential

Boundary integral formulation

• Solution to problem (1) searched as a ”modified” single layer potential.

Φc(x, y) = − 1 2π ln   |y|

• x − R2y

|y|2

• R

  x, y ∈ Ω = ⇒ u(x) = S(∂D, ψ, x) + Sc(∂D, ψ, x) x ∈ Ω S(∂D, ψ, x) =

• ∂D

Φ(x, y)ψ(y)ds(y) x ∈ Ω Sc(∂D, ψ, x) =

• ∂D

Φc(x, y)ψ(y)ds(y) x ∈ Ω

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 12 / 26

Equations of the potential

Boundary integral formulation

• Solution to problem (1) searched as a ”modified” single layer potential.

Φc(x, y) = − 1 2π ln   |y|

• x − R2y

|y|2

• R

  x, y ∈ Ω = ⇒ u(x) = S(∂D, ψ, x) + Sc(∂D, ψ, x) x ∈ Ω S(∂D, ψ, x) =

• ∂D

Φ(x, y)ψ(y)ds(y) x ∈ Ω Sc(∂D, ψ, x) =

• ∂D

Φc(x, y)ψ(y)ds(y) x ∈ Ω

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 12 / 26

Equations of the potential

Boundary integral formulation

• Solution to problem (1) searched as a ”modified” single layer potential.

Φc(x, y) = − 1 2π ln   |y|

• x − R2y

|y|2

• R

  x, y ∈ Ω = ⇒ u(x) = S(∂D, ψ, x) + Sc(∂D, ψ, x) x ∈ Ω S(∂D, ψ, x) =

• ∂D

Φ(x, y)ψ(y)ds(y) x ∈ Ω Sc(∂D, ψ, x) =

• ∂D

Φc(x, y)ψ(y)ds(y) x ∈ Ω

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 12 / 26

Equations of the potential

Boundary integral formulation

• ψ : density function (minimal regularity H− 1

2(∂D)), solution of a

boundary integral equation on ∂D. ψ − 2µK′

∂Dψ − 2µKc′ ∂Dψ = 2µ∂νΛ

• µ = σ0 − σ1

σ0 + σ1

• K′

∂D, Kc′ ∂D : H− 1

2 (∂D) → H− 1 2 (∂D) boundary integral operators.

• K′

∂Dψ

• (x)

=

• ∂D

∂νxΦ(x, y)ψ(y)ds(y) , x ∈ ∂D

• Kc′

∂Dψ

• (x)

=

• ∂D

∂νxΦc(x, y)ψ(y)ds(y) , x ∈ ∂D

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 13 / 26

Equations of the potential

Boundary integral formulation

• ψ : density function (minimal regularity H− 1

2(∂D)), solution of a

boundary integral equation on ∂D. ψ − 2µK′

∂Dψ − 2µKc′ ∂Dψ = 2µ∂νΛ

• µ = σ0 − σ1

σ0 + σ1

• K′

∂D, Kc′ ∂D : H− 1

2 (∂D) → H− 1 2 (∂D) boundary integral operators.

• K′

∂Dψ

• (x)

=

• ∂D

∂νxΦ(x, y)ψ(y)ds(y) , x ∈ ∂D

• Kc′

∂Dψ

• (x)

=

• ∂D

∂νxΦc(x, y)ψ(y)ds(y) , x ∈ ∂D

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 13 / 26

Equations of the potential

Boundary integral formulation

• ψ : density function (minimal regularity H− 1

2(∂D)), solution of a

boundary integral equation on ∂D. ψ − 2µK′

∂Dψ − 2µKc′ ∂Dψ = 2µ∂νΛ

• µ = σ0 − σ1

σ0 + σ1

• K′

∂D, Kc′ ∂D : H− 1

2 (∂D) → H− 1 2 (∂D) boundary integral operators.

• K′

∂Dψ

• (x)

=

• ∂D

∂νxΦ(x, y)ψ(y)ds(y) , x ∈ ∂D

• Kc′

∂Dψ

• (x)

=

• ∂D

∂νxΦc(x, y)ψ(y)ds(y) , x ∈ ∂D

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 13 / 26

Equations of the potential

Boundary integral formulation

• ψ : density function (minimal regularity H− 1

2(∂D)), solution of a

boundary integral equation on ∂D. ψ − 2µK′

∂Dψ − 2µKc′ ∂Dψ = 2µ∂νΛ

• µ = σ0 − σ1

σ0 + σ1

• K′

∂D, Kc′ ∂D : H− 1

2 (∂D) → H− 1 2 (∂D) boundary integral operators.

• K′

∂Dψ

• (x)

=

• ∂D

∂νxΦ(x, y)ψ(y)ds(y) , x ∈ ∂D

• Kc′

∂Dψ

• (x)

=

• ∂D

∂νxΦc(x, y)ψ(y)ds(y) , x ∈ ∂D

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 13 / 26

Equations of the inverse scheme

Conductivity σ1 of D supposed to be known.

• Explanations when considering one emitting pair (xe

+, xe −) and one

measuring pair (xm

+, xm −) of electrodes

• Measured voltage between xm

+ and xm − for the emitting pair (xe +, xe −) :

Uem

• (xe

+, xe −) =

⇒ Induced potential ue = ⇒ Density ψe

• ∂D supposed to have a regular parametrization z(t) , t ∈ [0, 2π] =

⇒ density ψe expressed as a function ϕe of t ∈ [0, 2π] (ϕe(t) = ψe(z(t)))

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 14 / 26

Equations of the inverse scheme

Conductivity σ1 of D supposed to be known.

• Explanations when considering one emitting pair (xe

+, xe −) and one

measuring pair (xm

+, xm −) of electrodes

• Measured voltage between xm

+ and xm − for the emitting pair (xe +, xe −) :

Uem

• (xe

+, xe −) =

⇒ Induced potential ue = ⇒ Density ψe

• ∂D supposed to have a regular parametrization z(t) , t ∈ [0, 2π] =

⇒ density ψe expressed as a function ϕe of t ∈ [0, 2π] (ϕe(t) = ψe(z(t)))

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 14 / 26

Equations of the inverse scheme

Conductivity σ1 of D supposed to be known.

• Explanations when considering one emitting pair (xe

+, xe −) and one

measuring pair (xm

+, xm −) of electrodes

• Measured voltage between xm

+ and xm − for the emitting pair (xe +, xe −) :

Uem

• (xe

+, xe −) =

⇒ Induced potential ue = ⇒ Density ψe

• ∂D supposed to have a regular parametrization z(t) , t ∈ [0, 2π] =

⇒ density ψe expressed as a function ϕe of t ∈ [0, 2π] (ϕe(t) = ψe(z(t)))

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 14 / 26

Equations of the inverse scheme

Recall of the scheme and the notations

D σ1 Ω σ0 ∂D (z)

b b b b b b b b b b b b b b b b

Γ = ∂Ω

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 15 / 26

Equations of the inverse scheme

Recall of the scheme and the notations

D σ1 Ω σ0 ∂D (z)

b b b b b b b b b b b b b b b b

Γ = ∂Ω xe

+

xe

−

I

ϕe

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 15 / 26

Equations of the inverse scheme

Recall of the scheme and the notations

D σ1 Ω σ0 ∂D (z)

b b b b b b b b b b b b b b b b

Γ = ∂Ω xe

+

xe

−

I

ϕe

V xm

+

xm

−

Uem

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 15 / 26

Equations of the inverse scheme

Parameterized versions of S, Sc, K′, Kc′

˜ S(z, ϕe, x) = 2π Φ(x, z(t))ϕe(t)|z′(t)|dt ˜ Sc(z, ϕe, x) = 2π Φc(x, z(t))ϕe(t)|z′(t)|dt ˜ K′ (z, ϕe, t) = 2π

• ∇1Φ(z(t), z(τ))
• z′T (t)
• ϕe(τ)|z′(τ)|

|z′(t)| dτ ˜ Kc′ (z, ϕe, t) = 2π

• ∇1Φc(z(t), z(τ))
• z′T (t)
• ϕe(τ)|z′(τ)|

|z′(t)| dτ

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 16 / 26

Equations of the inverse scheme

Parameterized versions of S, Sc, K′, Kc′

˜ S(z, ϕe, x) = 2π Φ(x, z(t))ϕe(t)|z′(t)|dt ˜ Sc(z, ϕe, x) = 2π Φc(x, z(t))ϕe(t)|z′(t)|dt ˜ K′ (z, ϕe, t) = 2π

• ∇1Φ(z(t), z(τ))
• z′T (t)
• ϕe(τ)|z′(τ)|

|z′(t)| dτ ˜ Kc′ (z, ϕe, t) = 2π

• ∇1Φc(z(t), z(τ))
• z′T (t)
• ϕe(τ)|z′(τ)|

|z′(t)| dτ

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 16 / 26

Equations of the inverse scheme

Parameterized versions of S, Sc, K′, Kc′

˜ S(z, ϕe, x) = 2π Φ(x, z(t))ϕe(t)|z′(t)|dt ˜ Sc(z, ϕe, x) = 2π Φc(x, z(t))ϕe(t)|z′(t)|dt ˜ K′ (z, ϕe, t) = 2π

• ∇1Φ(z(t), z(τ))
• z′T (t)
• ϕe(τ)|z′(τ)|

|z′(t)| dτ ˜ Kc′ (z, ϕe, t) = 2π

• ∇1Φc(z(t), z(τ))
• z′T (t)
• ϕe(τ)|z′(τ)|

|z′(t)| dτ

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 16 / 26

Equations of the inverse scheme

• Find z and ϕe such that

ϕe(t) − 2µ ˜ K′(z, ϕe, t) − 2µ ˜ Kc′(z, ϕe, t) = 2µ

• ∇1Λ(z(t), xe

+, xe −)

• z′T (t)

|z′(t)|

• , t ∈ [0, 2π]

(2)

• And (since Φ(x, y) = Φc(x, y) for x ∈ Γ and y ∈ Ω)
• Λ(xm

+, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• −
• Λ(xm

−, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• = Uem

(3)

• Several pairs of emitting/measuring electrodes =

⇒ Set of systems (2)-(3)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 17 / 26

Equations of the inverse scheme

• Find z and ϕe such that

ϕe(t) − 2µ ˜ K′(z, ϕe, t) − 2µ ˜ Kc′(z, ϕe, t) = 2µ

• ∇1Λ(z(t), xe

+, xe −)

• z′T (t)

|z′(t)|

• , t ∈ [0, 2π]

(2)

• And (since Φ(x, y) = Φc(x, y) for x ∈ Γ and y ∈ Ω)
• Λ(xm

+, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• −
• Λ(xm

−, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• = Uem

(3)

• Several pairs of emitting/measuring electrodes =

⇒ Set of systems (2)-(3)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 17 / 26

Equations of the inverse scheme

• Find z and ϕe such that

ϕe(t) − 2µ ˜ K′(z, ϕe, t) − 2µ ˜ Kc′(z, ϕe, t) = 2µ

• ∇1Λ(z(t), xe

+, xe −)

• z′T (t)

|z′(t)|

• , t ∈ [0, 2π]

(2)

• And (since Φ(x, y) = Φc(x, y) for x ∈ Γ and y ∈ Ω)
• Λ(xm

+, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• −
• Λ(xm

−, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• = Uem

(3)

• Several pairs of emitting/measuring electrodes =

⇒ Set of systems (2)-(3)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 17 / 26

Equations of the inverse scheme

• Find z and ϕe such that

ϕe(t) − 2µ ˜ K′(z, ϕe, t) − 2µ ˜ Kc′(z, ϕe, t) = 2µ

• ∇1Λ(z(t), xe

+, xe −)

• z′T (t)

|z′(t)|

• , t ∈ [0, 2π]

(2)

• And (since Φ(x, y) = Φc(x, y) for x ∈ Γ and y ∈ Ω)
• Λ(xm

+, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• −
• Λ(xm

−, xe +, xe −) + 2 ˜

S(z, ϕe, xm

+)

• = Uem

(3)

• Several pairs of emitting/measuring electrodes =

⇒ Set of systems (2)-(3)

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 17 / 26

Equations of the inverse scheme

Unknowns and equations

= ⇒ Unknowns : Boundary z of ∂D Densities ϕe : One density for each couple of emitting electrodes = ⇒ Equations : Number of (2)-equations = Number of emitting couples of electrodes Number of (3)-equations = One equation for each emission-measurement couple

Remark

The equations (2) and (3) are non linear with respect to z.

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 18 / 26

Equations of the inverse scheme

Unknowns and equations

= ⇒ Unknowns : Boundary z of ∂D Densities ϕe : One density for each couple of emitting electrodes = ⇒ Equations : Number of (2)-equations = Number of emitting couples of electrodes Number of (3)-equations = One equation for each emission-measurement couple

Remark

The equations (2) and (3) are non linear with respect to z.

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 18 / 26

Equations of the inverse scheme

Newton-like scheme

We consider an initial guess z0. Solve the set of boundary integral equations (2) to find the densities ϕe corresponding to the curve z0 for each emitting couple of electrodes. ˜ S and ˜ Sc have a Fr´ echet derivative with respect to z (R. Potthast) = ⇒ Linearize the set of equations (3) in the neighborhood of z0 = ⇒ z = z0 + ξ = ⇒ Solve the linear system with unknown ξ. Update the shape : z1 = z0 + ξ. Iterate the scheme

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 19 / 26

Equations of the inverse scheme

Newton-like scheme

We consider an initial guess z0. Solve the set of boundary integral equations (2) to find the densities ϕe corresponding to the curve z0 for each emitting couple of electrodes. ˜ S and ˜ Sc have a Fr´ echet derivative with respect to z (R. Potthast) = ⇒ Linearize the set of equations (3) in the neighborhood of z0 = ⇒ z = z0 + ξ = ⇒ Solve the linear system with unknown ξ. Update the shape : z1 = z0 + ξ. Iterate the scheme

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 19 / 26

Equations of the inverse scheme

Newton-like scheme

We consider an initial guess z0. Solve the set of boundary integral equations (2) to find the densities ϕe corresponding to the curve z0 for each emitting couple of electrodes. ˜ S and ˜ Sc have a Fr´ echet derivative with respect to z (R. Potthast) = ⇒ Linearize the set of equations (3) in the neighborhood of z0 = ⇒ z = z0 + ξ = ⇒ Solve the linear system with unknown ξ. Update the shape : z1 = z0 + ξ. Iterate the scheme

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 19 / 26

Equations of the inverse scheme

Newton-like scheme

We consider an initial guess z0. Solve the set of boundary integral equations (2) to find the densities ϕe corresponding to the curve z0 for each emitting couple of electrodes. ˜ S and ˜ Sc have a Fr´ echet derivative with respect to z (R. Potthast) = ⇒ Linearize the set of equations (3) in the neighborhood of z0 = ⇒ z = z0 + ξ = ⇒ Solve the linear system with unknown ξ. Update the shape : z1 = z0 + ξ. Iterate the scheme

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 19 / 26

Equations of the inverse scheme

Newton-like scheme

We consider an initial guess z0. Solve the set of boundary integral equations (2) to find the densities ϕe corresponding to the curve z0 for each emitting couple of electrodes. ˜ S and ˜ Sc have a Fr´ echet derivative with respect to z (R. Potthast) = ⇒ Linearize the set of equations (3) in the neighborhood of z0 = ⇒ z = z0 + ξ = ⇒ Solve the linear system with unknown ξ. Update the shape : z1 = z0 + ξ. Iterate the scheme

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 19 / 26

Equations of the inverse scheme

Fr´ echet derivative of ˜ S + ˜ Sc

• One has only to derive the kernel (R. Potthast)

= ⇒ For ξ ∈ C2([0, 2π]) (remember that Φ(x, y) = Φc(x, y) for x ∈ Γ and y ∈ Ω)

• d2( ˜

S + ˜ Sc)(ϕe, z, x)

• (ξ)

= − 1 π 2π (z(t) − x|ξ(t)) |z(t) − x|2 |z′(t)|ϕe(t)dt − 1 π 2π (z′(t)|ξ′(t)) |z′(t)| ln(|z(t) − x|)ϕe(t)dt ,

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 20 / 26

Equations of the inverse scheme

Fr´ echet derivative of ˜ S + ˜ Sc

• One has only to derive the kernel (R. Potthast)

= ⇒ For ξ ∈ C2([0, 2π]) (remember that Φ(x, y) = Φc(x, y) for x ∈ Γ and y ∈ Ω)

• d2( ˜

S + ˜ Sc)(ϕe, z, x)

• (ξ)

= − 1 π 2π (z(t) − x|ξ(t)) |z(t) − x|2 |z′(t)|ϕe(t)dt − 1 π 2π (z′(t)|ξ′(t)) |z′(t)| ln(|z(t) − x|)ϕe(t)dt ,

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 20 / 26

Numerical results

Radius of the circle : R = 4 Number of electrodes : 16 Conductivity of the background medium Ω : σ0 = 1 Conductivity of the inclusion D : σ1 = 2 Different shapes for the inclusion D : kite-shaped, bean-shaped, square-shaped, ”heart-lungs”-shaped 2% Gaussian noise on the synthetic data

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 21 / 26

Numerical results

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 22 / 26

Red : electrodes Green : inclusion Blue : reconstruction Iteration n°00

Numerical results

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 23 / 26

Red : electrodes Green : inclusion Blue : reconstruction Iteration n°00

Numerical results

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 24 / 26

Red : electrodes Green : inclusion Blue : reconstruction Iteration n°00

Numerical results

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 25 / 26

Red : electrodes Green : inclusion Blue : reconstruction Iteration n°00

Conclusion and future work

• Quite good reconstructions, relatively fast. Of course, for smaller

inclusions, the reconstructions are not so good (for noisy data). One has to take more electrodes.

• Following of the work...

Implement the construction of an appropriate initial guess, for example using the Factorization Method Treat the case where the conductivity σ1 of the inclusion D is unknown Treat the case of multiple inclusions Test the scheme with real data measurements

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 26 / 26

Conclusion and future work

• Quite good reconstructions, relatively fast. Of course, for smaller

inclusions, the reconstructions are not so good (for noisy data). One has to take more electrodes.

• Following of the work...

Implement the construction of an appropriate initial guess, for example using the Factorization Method Treat the case where the conductivity σ1 of the inclusion D is unknown Treat the case of multiple inclusions Test the scheme with real data measurements

• F. Delbary (Universit¨

at G¨

• ttingen)

EIT AIP 2009 Vienna 26 / 26