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November  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. Google Scholar [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), 999-1036. doi: 10.1002/cpa.3160430805.  Google Scholar [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), 531-549. doi: 10.3934/ipi.2011.5.531.  Google Scholar [4] K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem, in Proc. 7th Internat. Conference on Harmonic Analysis, Collectanea Mathematica, (2006), 127-139.  Google Scholar [5] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.  Google Scholar [6] J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 015002, 19 pp. doi: 10.1088/0266-5611/27/1/015002.  Google Scholar [7] R. M. Brown, Global uniqueness in the impedance imaging problem for less regular conductivities, SIAM Journal on Mathematical Analysis, 27 (1996), 1049-1056. doi: 10.1137/S0036141094271132.  Google Scholar [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), Soc. Brasil. Mat., Rio de Janeiro, (1980), 65-73.  Google Scholar [9] A. Chambolle, Image segmentation by variational methods: Mumford and shah functional and the discrete approximations, SIAM Journal on Applied Mathematics, 55 (1995), 827-863. doi: 10.1137/S0036139993257132.  Google Scholar [10] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101. doi: 10.1137/S0036144598333613.  Google Scholar [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), 111-134. doi: 10.1515/156939406777571102.  Google Scholar [12] I. N. R. Council, Dielectric properties of body tissues, 2013,, , ().   Google Scholar [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), 195-218. doi: 10.1007/BF01052971.  Google Scholar [14] F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms, Applicable Analysis, 91 (2012), 737-755. doi: 10.1080/00036811.2011.598863.  Google Scholar [15] E. Erdem and S. Tari, Mumford-shah regularizer with contextual feedback, Journal of Mathematical Imaging and Vision, 33 (2009), 67-84. doi: 10.1007/s10851-008-0109-y.  Google Scholar [16] L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035. Google Scholar [17] D. E. Finkel, DIRECT Optimization Algorithm User Guide, Technical report, Center for Research in Scientific Computation, North Carolina State University, 2003. Google Scholar [18] E. Francini, Recovering a complex coefficient in a planar domain from Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107-119. doi: 10.1088/0266-5611/16/1/309.  Google Scholar [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), 757-769. doi: 10.1109/TMI.2012.2237389.  Google Scholar [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), 095005, 24 pp. doi: 10.1088/0266-5611/28/9/095005.  Google Scholar [21] L. Harhanen, N. Hyvönen, H. Majander and S. Staboulis, Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography,, ArXiv e-prints, ().   Google Scholar [22] T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the mumford-shah functional, Inverse problems, 27 (2011), 015008, 32 pp. doi: 10.1088/0266-5611/27/1/015008.  Google Scholar [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), S43-S50. doi: 10.1088/0967-3334/27/5/S04.  Google Scholar [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), 157-181. doi: 10.1007/BF00941892.  Google Scholar [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), 1583-1598. doi: 10.1109/TIP.2010.2092433.  Google Scholar [26] K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiological Measurement, 24 (2003), 391-403. doi: 10.1088/0967-3334/24/2/351.  Google Scholar [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), 599-624. doi: 10.3934/ipi.2009.3.599.  Google Scholar [28] K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Communications in Partial Differential Equations, 29 (2004), 361-381. doi: 10.1081/PDE-120030401.  Google Scholar [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.  Google Scholar [30] J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), 1232-1266. doi: 10.1137/S1064827501394568.  Google Scholar [31] D. Mumford and J. Shah, Boundary detection by minimizing functionals, in IEEE Conference on Computer Vision and Pattern Recognition, 1985. Google Scholar [32] M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials, Inverse Problems, 29 (2013), 045004, 25 pp. doi: 10.1088/0266-5611/29/4/045004.  Google Scholar [33] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar [34] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205.  Google Scholar [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), 539-557. doi: 10.1016/j.jcp.2006.06.041.  Google Scholar [36] L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM Control Optim. Calc. Var., 6 (2001), 517-538. doi: 10.1051/cocv:2001121.  Google Scholar [37] J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (1996), 136-142. doi: 10.1109/CVPR.1996.517065.  Google Scholar [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), 681-699. doi: 10.1088/0266-5611/16/3/310.  Google Scholar [39] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar [40] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner Stuttgart, 1998.  Google Scholar

show all references

##### References:
 [1] R. Alicandro, A. Braides and J. Shah, Approximation of non-convex functionals in GBV, 1998. Google Scholar [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), 999-1036. doi: 10.1002/cpa.3160430805.  Google Scholar [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), 531-549. doi: 10.3934/ipi.2011.5.531.  Google Scholar [4] K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem, in Proc. 7th Internat. Conference on Harmonic Analysis, Collectanea Mathematica, (2006), 127-139.  Google Scholar [5] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.  Google Scholar [6] J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 015002, 19 pp. doi: 10.1088/0266-5611/27/1/015002.  Google Scholar [7] R. M. Brown, Global uniqueness in the impedance imaging problem for less regular conductivities, SIAM Journal on Mathematical Analysis, 27 (1996), 1049-1056. doi: 10.1137/S0036141094271132.  Google Scholar [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), Soc. Brasil. Mat., Rio de Janeiro, (1980), 65-73.  Google Scholar [9] A. Chambolle, Image segmentation by variational methods: Mumford and shah functional and the discrete approximations, SIAM Journal on Applied Mathematics, 55 (1995), 827-863. doi: 10.1137/S0036139993257132.  Google Scholar [10] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101. doi: 10.1137/S0036144598333613.  Google Scholar [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), 111-134. doi: 10.1515/156939406777571102.  Google Scholar [12] I. N. R. Council, Dielectric properties of body tissues, 2013,, , ().   Google Scholar [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), 195-218. doi: 10.1007/BF01052971.  Google Scholar [14] F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms, Applicable Analysis, 91 (2012), 737-755. doi: 10.1080/00036811.2011.598863.  Google Scholar [15] E. Erdem and S. Tari, Mumford-shah regularizer with contextual feedback, Journal of Mathematical Imaging and Vision, 33 (2009), 67-84. doi: 10.1007/s10851-008-0109-y.  Google Scholar [16] L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035. Google Scholar [17] D. E. Finkel, DIRECT Optimization Algorithm User Guide, Technical report, Center for Research in Scientific Computation, North Carolina State University, 2003. Google Scholar [18] E. Francini, Recovering a complex coefficient in a planar domain from Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107-119. doi: 10.1088/0266-5611/16/1/309.  Google Scholar [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), 757-769. doi: 10.1109/TMI.2012.2237389.  Google Scholar [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), 095005, 24 pp. doi: 10.1088/0266-5611/28/9/095005.  Google Scholar [21] L. Harhanen, N. Hyvönen, H. Majander and S. Staboulis, Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography,, ArXiv e-prints, ().   Google Scholar [22] T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the mumford-shah functional, Inverse problems, 27 (2011), 015008, 32 pp. doi: 10.1088/0266-5611/27/1/015008.  Google Scholar [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), S43-S50. doi: 10.1088/0967-3334/27/5/S04.  Google Scholar [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), 157-181. doi: 10.1007/BF00941892.  Google Scholar [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), 1583-1598. doi: 10.1109/TIP.2010.2092433.  Google Scholar [26] K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiological Measurement, 24 (2003), 391-403. doi: 10.1088/0967-3334/24/2/351.  Google Scholar [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), 599-624. doi: 10.3934/ipi.2009.3.599.  Google Scholar [28] K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Communications in Partial Differential Equations, 29 (2004), 361-381. doi: 10.1081/PDE-120030401.  Google Scholar [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.  Google Scholar [30] J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), 1232-1266. doi: 10.1137/S1064827501394568.  Google Scholar [31] D. Mumford and J. Shah, Boundary detection by minimizing functionals, in IEEE Conference on Computer Vision and Pattern Recognition, 1985. Google Scholar [32] M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials, Inverse Problems, 29 (2013), 045004, 25 pp. doi: 10.1088/0266-5611/29/4/045004.  Google Scholar [33] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar [34] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205.  Google Scholar [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), 539-557. doi: 10.1016/j.jcp.2006.06.041.  Google Scholar [36] L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM Control Optim. Calc. Var., 6 (2001), 517-538. doi: 10.1051/cocv:2001121.  Google Scholar [37] J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (1996), 136-142. doi: 10.1109/CVPR.1996.517065.  Google Scholar [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), 681-699. doi: 10.1088/0266-5611/16/3/310.  Google Scholar [39] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar [40] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner Stuttgart, 1998.  Google Scholar
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