-
Previous Article
Manakov solitons and effects of external potential wells
- PROC Home
- This Issue
-
Next Article
Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets
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 |
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. |
[1] |
Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems and Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020 |
[2] |
Ville Kolehmainen, Matthias J. Ehrhardt, Simon R. Arridge. Incorporating structural prior information and sparsity into EIT using parallel level sets. Inverse Problems and Imaging, 2019, 13 (2) : 285-307. doi: 10.3934/ipi.2019015 |
[3] |
Ville Kolehmainen, Matthias J. Ehrhardt, Simon R. Arridge. Corrigendum to "Incorporating structural prior information and sparsity into EIT using parallel level sets". Inverse Problems and Imaging, 2020, 14 (2) : 399-399. doi: 10.3934/ipi.2020018 |
[4] |
Jérémi Dardé, Harri Hakula, Nuutti Hyvönen, Stratos Staboulis. Fine-tuning electrode information in electrical impedance tomography. Inverse Problems and Imaging, 2012, 6 (3) : 399-421. doi: 10.3934/ipi.2012.6.399 |
[5] |
Benjamin Palacios. Photoacoustic tomography in attenuating media with partial data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022013 |
[6] |
Chengxiang Wang, Li Zeng, Yumeng Guo, Lingli Zhang. Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data. Inverse Problems and Imaging, 2017, 11 (6) : 917-948. doi: 10.3934/ipi.2017043 |
[7] |
Nuutti Hyvönen, Harri Hakula, Sampsa Pursiainen. Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Problems and Imaging, 2007, 1 (2) : 299-317. doi: 10.3934/ipi.2007.1.299 |
[8] |
Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems and Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021 |
[9] |
Melody Dodd, Jennifer L. Mueller. A real-time D-bar algorithm for 2-D electrical impedance tomography data. Inverse Problems and Imaging, 2014, 8 (4) : 1013-1031. doi: 10.3934/ipi.2014.8.1013 |
[10] |
Sarah Jane Hamilton, Andreas Hauptmann, Samuli Siltanen. A data-driven edge-preserving D-bar method for electrical impedance tomography. Inverse Problems and Imaging, 2014, 8 (4) : 1053-1072. doi: 10.3934/ipi.2014.8.1053 |
[11] |
Fabrice Delbary, Rainer Kress. Electrical impedance tomography using a point electrode inverse scheme for complete electrode data. Inverse Problems and Imaging, 2011, 5 (2) : 355-369. doi: 10.3934/ipi.2011.5.355 |
[12] |
Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems and Imaging, 2014, 8 (1) : 173-197. doi: 10.3934/ipi.2014.8.173 |
[13] |
Ke Zhang, Maokun Li, Fan Yang, Shenheng Xu, Aria Abubakar. Electrical impedance tomography with multiplicative regularization. Inverse Problems and Imaging, 2019, 13 (6) : 1139-1159. doi: 10.3934/ipi.2019051 |
[14] |
Bastian Gebauer. Localized potentials in electrical impedance tomography. Inverse Problems and Imaging, 2008, 2 (2) : 251-269. doi: 10.3934/ipi.2008.2.251 |
[15] |
Kari Astala, Jennifer L. Mueller, Lassi Päivärinta, Allan Perämäki, Samuli Siltanen. Direct electrical impedance tomography for nonsmooth conductivities. Inverse Problems and Imaging, 2011, 5 (3) : 531-549. doi: 10.3934/ipi.2011.5.531 |
[16] |
Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems and Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217 |
[17] |
Sarah J. Hamilton, David Isaacson, Ville Kolehmainen, Peter A. Muller, Jussi Toivanen, Patrick F. Bray. 3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods. Inverse Problems and Imaging, 2021, 15 (5) : 1135-1169. doi: 10.3934/ipi.2021032 |
[18] |
Subrata Dasgupta. Disentangling data, information and knowledge. Big Data & Information Analytics, 2016, 1 (4) : 377-389. doi: 10.3934/bdia.2016016 |
[19] |
Dong liu, Ville Kolehmainen, Samuli Siltanen, Anne-maria Laukkanen, Aku Seppänen. Estimation of conductivity changes in a region of interest with electrical impedance tomography. Inverse Problems and Imaging, 2015, 9 (1) : 211-229. doi: 10.3934/ipi.2015.9.211 |
[20] |
Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. Study of noise effects in electrical impedance tomography with resistor networks. Inverse Problems and Imaging, 2013, 7 (2) : 417-443. doi: 10.3934/ipi.2013.7.417 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]