# American Institute of Mathematical Sciences

2015, 2015(special): 495-504. doi: 10.3934/proc.2015.0495

## 3D reconstruction for partial data electrical impedance tomography using a sparsity prior

 1 Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark 2 Danmarks Tekniske Universitet, Department of Applied Mathematics and Computer Science, Matematiktorvet, Building 303 B, DK - 2800 Kgs. Lyngby

Received  September 2014 Revised  August 2015 Published  November 2015

In electrical impedance tomography the electrical conductivity inside a physical body is computed from electro-static boundary measurements. The focus of this paper is to extend recent results for the 2D problem to 3D: prior information about the sparsity and spatial distribution of the conductivity is used to improve reconstructions for the partial data problem with Cauchy data measured only on a subset of the boundary. A sparsity prior is enforced using the $\ell_1$ norm in the penalty term of a Tikhonov functional, and spatial prior information is incorporated by applying a spatially distributed regularization parameter. The optimization problem is solved numerically using a generalized conditional gradient method with soft thresholding. Numerical examples show the effectiveness of the suggested method even for the partial data problem with measurements affected by noise.
Citation: Henrik Garde, Kim Knudsen. 3D reconstruction for partial data electrical impedance tomography using a sparsity prior. Conference Publications, 2015, 2015 (special) : 495-504. doi: 10.3934/proc.2015.0495
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003). [2] G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153. [3] T. Bonesky, K. Bredies, D. A. Lorenz and P. Maass, A generalized conditional gradient method for nonlinear operator equations with sparsity constraints,, Inverse Problems, 23 (2007), 2041. [4] K. Bredies, D. A. Lorenz and P. Maass, A generalized conditional gradient method and its connection to an iterative shrinkage method,, Comput. Optim. Appl., 42 (2009), 173. [5] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data,, Comm. Partial Differential Equations, 27 (2002), 653. [6] A.-P. Calderón, On an inverse boundary value problem,, in Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65. [7] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,, Comm. Pure Appl. Math., 57 (2004), 1413. [8] H. Garde and K. Knudsen, Sparsity prior for electrical impedance tomography with partial data,, Inverse Probl. Sci. Eng., (2015). [9] M. Gehre, T. Kluth, A. Lipponen, B. Jin, A. Seppänen, J. P. Kaipio and P. Maass, Sparsity reconstruction in electrical impedance tomography: an experimental evaluation,, J. Comput. Appl. Math., 236 (2012), 2126. [10] B. von Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography,, SIAM J. Math. Anal., 42 (2010), 1505. [11] B. von Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography,, SIAM J. Math. Anal., 45 (2013), 3382. [12] H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data,, Inverse Problems, 22 (2006), 1787. [13] V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95. [14] B. Jin, T. Khan and P. Maass, A reconstruction algorithm for electrical impedance tomography based on sparsity regularization,, Internat. J. Numer. Methods Engrg., 89 (2012), 337. [15] B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization,, ESAIM: Control, 18 (2012), 1027. [16] C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data,, Ann. of Math. (2), 165 (2007), 567. [17] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford University Press, (2008). [18] K. Knudsen, The Calderón problem with partial data for less smooth conductivities,, Comm. Partial Differential Equations, 31 (2006), 57. [19] A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method,, Springer, (2012). [20] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,, Ann. Inst. Fourier (Grenoble), 15 (1965), 189. [21] S. J. Wright, R. D. Nowak and M. A. T. Figueiredo, Sparse reconstruction by separable approximation,, IEEE Trans. Signal Process., 57 (2009), 2479.

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##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003). [2] G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153. [3] T. Bonesky, K. Bredies, D. A. Lorenz and P. Maass, A generalized conditional gradient method for nonlinear operator equations with sparsity constraints,, Inverse Problems, 23 (2007), 2041. [4] K. Bredies, D. A. Lorenz and P. Maass, A generalized conditional gradient method and its connection to an iterative shrinkage method,, Comput. Optim. Appl., 42 (2009), 173. [5] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data,, Comm. Partial Differential Equations, 27 (2002), 653. [6] A.-P. Calderón, On an inverse boundary value problem,, in Seminar on Numerical Analysis and its Applications to Continuum Physics, (1980), 65. [7] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,, Comm. Pure Appl. Math., 57 (2004), 1413. [8] H. Garde and K. Knudsen, Sparsity prior for electrical impedance tomography with partial data,, Inverse Probl. Sci. Eng., (2015). [9] M. Gehre, T. Kluth, A. Lipponen, B. Jin, A. Seppänen, J. P. Kaipio and P. Maass, Sparsity reconstruction in electrical impedance tomography: an experimental evaluation,, J. Comput. Appl. Math., 236 (2012), 2126. [10] B. von Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography,, SIAM J. Math. Anal., 42 (2010), 1505. [11] B. von Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography,, SIAM J. Math. Anal., 45 (2013), 3382. [12] H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data,, Inverse Problems, 22 (2006), 1787. [13] V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95. [14] B. Jin, T. Khan and P. Maass, A reconstruction algorithm for electrical impedance tomography based on sparsity regularization,, Internat. J. Numer. Methods Engrg., 89 (2012), 337. [15] B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization,, ESAIM: Control, 18 (2012), 1027. [16] C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data,, Ann. of Math. (2), 165 (2007), 567. [17] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, Oxford University Press, (2008). [18] K. Knudsen, The Calderón problem with partial data for less smooth conductivities,, Comm. Partial Differential Equations, 31 (2006), 57. [19] A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method,, Springer, (2012). [20] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,, Ann. Inst. Fourier (Grenoble), 15 (1965), 189. [21] S. J. Wright, R. D. Nowak and M. A. T. Figueiredo, Sparse reconstruction by separable approximation,, IEEE Trans. Signal Process., 57 (2009), 2479.
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