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2014, 8(4): 1053-1072. doi: 10.3934/ipi.2014.8.1053

A data-driven edge-preserving D-bar method for electrical impedance tomography

1. 

Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, WI 53233, United States

2. 

Department of Mathematics and Statistics, University of Helsinki, Helsinki, 00014, Finland

3. 

University of Helsinki, Department of Mathematics and Statistics, FI-00014 Helsinki

Received  December 2013 Revised  October 2014 Published  November 2014

In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known a priori that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called CGO sinogram. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.
Citation: Sarah Jane Hamilton, Andreas Hauptmann, Samuli Siltanen. A data-driven edge-preserving D-bar method for electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (4) : 1053-1072. doi: 10.3934/ipi.2014.8.1053
References:
[1]

R. Alicandro, A. Braides and J. Shah, Approximation, of non-convex functionals in GBV, (1998).

[2]

L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence,, Communications on Pure and Applied Mathematics, 43 (1990), 999. doi: 10.1002/cpa.3160430805.

[3]

K. Astala, J. Mueller, L. Päivärinta, A. Perämäki and S. Siltanen, Direct electrical impedance tomography for nonsmooth conductivities,, Inverse Problems and Imaging, 5 (2011), 531. doi: 10.3934/ipi.2011.5.531.

[4]

K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem,, in Proc. 7th Internat. Conference on Harmonic Analysis, (2006), 127.

[5]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Annals of Mathematics, 163 (2006), 265. doi: 10.4007/annals.2006.163.265.

[6]

J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015002.

[7]

R. M. Brown, Global uniqueness in the impedance imaging problem for less regular conductivities,, SIAM Journal on Mathematical Analysis, 27 (1996), 1049. doi: 10.1137/S0036141094271132.

[8]

A.-P. Calderón, On an inverse boundary value problem,, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, (1980), 65.

[9]

A. Chambolle, Image segmentation by variational methods: Mumford and shah functional and the discrete approximations,, SIAM Journal on Applied Mathematics, 55 (1995), 827. doi: 10.1137/S0036139993257132.

[10]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613.

[11]

H. Cornean, K. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT,, Journal of Inverse and Ill-Posed Problems, 14 (2006), 111. doi: 10.1515/156939406777571102.

[12]

I. N. R. Council, Dielectric properties of body tissues, 2013,, , ().

[13]

E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set,, Archive for Rational Mechanics and Analysis, 108 (1989), 195. doi: 10.1007/BF01052971.

[14]

F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms,, Applicable Analysis, 91 (2012), 737. doi: 10.1080/00036811.2011.598863.

[15]

E. Erdem and S. Tari, Mumford-shah regularizer with contextual feedback,, Journal of Mathematical Imaging and Vision, 33 (2009), 67. doi: 10.1007/s10851-008-0109-y.

[16]

L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Soviet Physics Doklady, 10 (1966), 1033.

[17]

D. E. Finkel, DIRECT Optimization Algorithm User Guide,, Technical report, (2003).

[18]

E. Francini, Recovering a complex coefficient in a planar domain from Dirichlet-to-Neumann map,, Inverse Problems, 16 (2000), 107. doi: 10.1088/0266-5611/16/1/309.

[19]

S. J. Hamilton and J. L. Mueller, Direct EIT reconstructions of complex admittivities on a chest-shaped domain in 2-D,, IEEE Transactions on Medical Imaging, 32 (2013), 757. doi: 10.1109/TMI.2012.2237389.

[20]

S. Hamilton, C. Herrera, J. L. Mueller and A. Von Herrmann, A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095005.

[21]

L. Harhanen, N. Hyvönen, H. Majander and S. Staboulis, Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography,, ArXiv e-prints, ().

[22]

T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the mumford-shah functional,, Inverse problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015008.

[23]

D. Isaacson, J. Mueller, J. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography,, Physiological Measurement, 27 (2006). doi: 10.1088/0967-3334/27/5/S04.

[24]

D. R. Jones, C. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the lipschitz constant,, Journal of Optimization Theory and Applications, 79 (1993), 157. doi: 10.1007/BF00941892.

[25]

M. Jung, X. Bresson, T. F. Chan and L. A. Vese, Nonlocal Mumford-Shah regularizers for color image restoration,, IEEE Trans. Image Process., 20 (2011), 1583. doi: 10.1109/TIP.2010.2092433.

[26]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane,, Physiological Measurement, 24 (2003), 391. doi: 10.1088/0967-3334/24/2/351.

[27]

K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599.

[28]

K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane,, Communications in Partial Differential Equations, 29 (2004), 361. doi: 10.1081/PDE-120030401.

[29]

J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, vol. 10 of Computational Science and Engineering,, SIAM, (2012). doi: 10.1137/1.9781611972344.

[30]

J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements,, SIAM Journal on Scientific Computing, 24 (2003), 1232. doi: 10.1137/S1064827501394568.

[31]

D. Mumford and J. Shah, Boundary detection by minimizing functionals,, in IEEE Conference on Computer Vision and Pattern Recognition, (1985).

[32]

M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/4/045004.

[33]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Annals of Mathematics, 143 (1996), 71. doi: 10.2307/2118653.

[34]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629. doi: 10.1109/34.56205.

[35]

R. Ramlau and W. Ring, A mumford-shah level-set approach for the inversion and segmentation of x-ray tomography data,, Journal of Computational Physics, 221 (2007), 539. doi: 10.1016/j.jcp.2006.06.041.

[36]

L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional,, ESAIM Control Optim. Calc. Var., 6 (2001), 517. doi: 10.1051/cocv:2001121.

[37]

J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in IEEE Conference on Computer Vision and Pattern Recognition, (1996), 136. doi: 10.1109/CVPR.1996.517065.

[38]

S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of a. nachman for the 2-d inverse conductivity problem,, Inverse Problems, 16 (2000), 681. doi: 10.1088/0266-5611/16/3/310.

[39]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Annals of Mathematics, 125 (1987), 153. doi: 10.2307/1971291.

[40]

J. Weickert, Anisotropic Diffusion in Image Processing,, Teubner Stuttgart, (1998).

show all references

References:
[1]

R. Alicandro, A. Braides and J. Shah, Approximation, of non-convex functionals in GBV, (1998).

[2]

L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence,, Communications on Pure and Applied Mathematics, 43 (1990), 999. doi: 10.1002/cpa.3160430805.

[3]

K. Astala, J. Mueller, L. Päivärinta, A. Perämäki and S. Siltanen, Direct electrical impedance tomography for nonsmooth conductivities,, Inverse Problems and Imaging, 5 (2011), 531. doi: 10.3934/ipi.2011.5.531.

[4]

K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem,, in Proc. 7th Internat. Conference on Harmonic Analysis, (2006), 127.

[5]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Annals of Mathematics, 163 (2006), 265. doi: 10.4007/annals.2006.163.265.

[6]

J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015002.

[7]

R. M. Brown, Global uniqueness in the impedance imaging problem for less regular conductivities,, SIAM Journal on Mathematical Analysis, 27 (1996), 1049. doi: 10.1137/S0036141094271132.

[8]

A.-P. Calderón, On an inverse boundary value problem,, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, (1980), 65.

[9]

A. Chambolle, Image segmentation by variational methods: Mumford and shah functional and the discrete approximations,, SIAM Journal on Applied Mathematics, 55 (1995), 827. doi: 10.1137/S0036139993257132.

[10]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613.

[11]

H. Cornean, K. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT,, Journal of Inverse and Ill-Posed Problems, 14 (2006), 111. doi: 10.1515/156939406777571102.

[12]

I. N. R. Council, Dielectric properties of body tissues, 2013,, , ().

[13]

E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set,, Archive for Rational Mechanics and Analysis, 108 (1989), 195. doi: 10.1007/BF01052971.

[14]

F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms,, Applicable Analysis, 91 (2012), 737. doi: 10.1080/00036811.2011.598863.

[15]

E. Erdem and S. Tari, Mumford-shah regularizer with contextual feedback,, Journal of Mathematical Imaging and Vision, 33 (2009), 67. doi: 10.1007/s10851-008-0109-y.

[16]

L. D. Faddeev, Increasing solutions of the Schrödinger equation,, Soviet Physics Doklady, 10 (1966), 1033.

[17]

D. E. Finkel, DIRECT Optimization Algorithm User Guide,, Technical report, (2003).

[18]

E. Francini, Recovering a complex coefficient in a planar domain from Dirichlet-to-Neumann map,, Inverse Problems, 16 (2000), 107. doi: 10.1088/0266-5611/16/1/309.

[19]

S. J. Hamilton and J. L. Mueller, Direct EIT reconstructions of complex admittivities on a chest-shaped domain in 2-D,, IEEE Transactions on Medical Imaging, 32 (2013), 757. doi: 10.1109/TMI.2012.2237389.

[20]

S. Hamilton, C. Herrera, J. L. Mueller and A. Von Herrmann, A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095005.

[21]

L. Harhanen, N. Hyvönen, H. Majander and S. Staboulis, Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography,, ArXiv e-prints, ().

[22]

T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the mumford-shah functional,, Inverse problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015008.

[23]

D. Isaacson, J. Mueller, J. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography,, Physiological Measurement, 27 (2006). doi: 10.1088/0967-3334/27/5/S04.

[24]

D. R. Jones, C. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the lipschitz constant,, Journal of Optimization Theory and Applications, 79 (1993), 157. doi: 10.1007/BF00941892.

[25]

M. Jung, X. Bresson, T. F. Chan and L. A. Vese, Nonlocal Mumford-Shah regularizers for color image restoration,, IEEE Trans. Image Process., 20 (2011), 1583. doi: 10.1109/TIP.2010.2092433.

[26]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane,, Physiological Measurement, 24 (2003), 391. doi: 10.1088/0967-3334/24/2/351.

[27]

K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599.

[28]

K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane,, Communications in Partial Differential Equations, 29 (2004), 361. doi: 10.1081/PDE-120030401.

[29]

J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, vol. 10 of Computational Science and Engineering,, SIAM, (2012). doi: 10.1137/1.9781611972344.

[30]

J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements,, SIAM Journal on Scientific Computing, 24 (2003), 1232. doi: 10.1137/S1064827501394568.

[31]

D. Mumford and J. Shah, Boundary detection by minimizing functionals,, in IEEE Conference on Computer Vision and Pattern Recognition, (1985).

[32]

M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/4/045004.

[33]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Annals of Mathematics, 143 (1996), 71. doi: 10.2307/2118653.

[34]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629. doi: 10.1109/34.56205.

[35]

R. Ramlau and W. Ring, A mumford-shah level-set approach for the inversion and segmentation of x-ray tomography data,, Journal of Computational Physics, 221 (2007), 539. doi: 10.1016/j.jcp.2006.06.041.

[36]

L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional,, ESAIM Control Optim. Calc. Var., 6 (2001), 517. doi: 10.1051/cocv:2001121.

[37]

J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in IEEE Conference on Computer Vision and Pattern Recognition, (1996), 136. doi: 10.1109/CVPR.1996.517065.

[38]

S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of a. nachman for the 2-d inverse conductivity problem,, Inverse Problems, 16 (2000), 681. doi: 10.1088/0266-5611/16/3/310.

[39]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Annals of Mathematics, 125 (1987), 153. doi: 10.2307/1971291.

[40]

J. Weickert, Anisotropic Diffusion in Image Processing,, Teubner Stuttgart, (1998).

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