Electrical impedance tomography with punctual electrodes Fabrice - PowerPoint PPT Presentation

Electrical impedance tomography with punctual electrodes Fabrice Delbary 1 , Rainer Kreß 1 1 Universit¨ at G¨ ottingen AIP 2009 Vienna F. Delbary (Universit¨ at G¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) EIT AIP 2009 Vienna 5 / 26

b b b b b b b b b b b b b b b b Problem and method electrodes Γ = ∂ Ω ∂D D σ 1 Ω σ 0 Problem : Find the location, the shape and the conductivity σ 1 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¨ ottingen) EIT AIP 2009 Vienna 6 / 26

b b b b b b b b b b b b b b b b Problem and method electrodes Γ = ∂ Ω ∂D D σ 1 Ω σ 0 Problem : Find the location, the shape and the conductivity σ 1 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¨ ottingen) EIT AIP 2009 Vienna 6 / 26

b b b b b b b b b b b b b b b b Problem and method Γ = ∂ Ω I ∂D D σ 1 U V Ω σ 0 Problem : Find the location, the shape and the conductivity σ 1 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¨ ottingen) 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¨ ottingen) 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¨ ottingen) EIT AIP 2009 Vienna 8 / 26

Equations of the potential Notations R : radius of Ω Piecewise constant conductivity in Ω � σ 0 in Ω \ D � σ = � in σ 1 D � Fundamental solution of the Laplace equation ∀ x, y ∈ R 2 , x � = y , Φ( x, y ) = − 1 2 π 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¨ ottingen) EIT AIP 2009 Vienna 9 / 26

Equations of the potential Notations R : radius of Ω Piecewise constant conductivity in Ω � σ 0 in Ω \ D � σ = � in σ 1 D � Fundamental solution of the Laplace equation ∀ x, y ∈ R 2 , x � = y , Φ( x, y ) = − 1 2 π 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¨ ottingen) EIT AIP 2009 Vienna 9 / 26