American Institute of Mathematical Sciences

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

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

Henrik Garde 1, and  Kim Knudsen 2,

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.
Keywords: sparsity, numerical reconstruction., partial data, prior information, Impedance tomography.
Mathematics Subject Classification: Primary: 65N20, 65N2.
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, Pure and Applied Mathematics, Amsterdam, 2003.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.

[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), 2041-2058.

[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), 173-193.

[5]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668.

[6]

A.-P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., (1980), 65-73.

[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), 1413-1457.

[8]

H. Garde and K. Knudsen, Sparsity prior for electrical impedance tomography with partial data, Inverse Probl. Sci. Eng., (2015), DOI: 10.1080/17415977.2015.1047365.

[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), 2126-2136.

[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), 1505-1518.

[11]

B. von Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403.

[12]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787-1796.

[13]

V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.

[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), 337-353.

[15]

B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 1027-1048.

[16]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2), 165 (2007), 567-591.

[17]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.

[18]

K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57-71.

[19]

A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer, Heidelberg, 2012.

[20]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-257.

[21]

S. J. Wright, R. D. Nowak and M. A. T. Figueiredo, Sparse reconstruction by separable approximation, IEEE Trans. Signal Process., 57 (2009), 2479-2493.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Pure and Applied Mathematics, Amsterdam, 2003.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.

[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), 2041-2058.

[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), 173-193.

[5]

A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668.

[6]

A.-P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat., (1980), 65-73.

[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), 1413-1457.

[8]

H. Garde and K. Knudsen, Sparsity prior for electrical impedance tomography with partial data, Inverse Probl. Sci. Eng., (2015), DOI: 10.1080/17415977.2015.1047365.

[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), 2126-2136.

[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), 1505-1518.

[11]

B. von Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403.

[12]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787-1796.

[13]

V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.

[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), 337-353.

[15]

B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 1027-1048.

[16]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2), 165 (2007), 567-591.

[17]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.

[18]

K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57-71.

[19]

A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer, Heidelberg, 2012.

[20]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-257.

[21]

S. J. Wright, R. D. Nowak and M. A. T. Figueiredo, Sparse reconstruction by separable approximation, IEEE Trans. Signal Process., 57 (2009), 2479-2493.

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