Abstract
A new nonlinear optimization approach is proposed for the sparse reconstruction of log-conductivities in current density impedance imaging. This framework comprises of minimizing an objective functional involving a least squares fit of the interior electric field data corresponding to two boundary voltage measurements, where the conductivity and the electric potential are related through an elliptic PDE arising in electrical impedance tomography. Further, the objective functional consists of a \(L^1\) regularization term that promotes sparsity patterns in the conductivity and a Perona–Malik anisotropic diffusion term that enhances the edges to facilitate high contrast and resolution. This framework is motivated by a similar recent approach to solve an inverse problem in acousto-electric tomography. Several numerical experiments and comparison with an existing method demonstrate the effectiveness of the proposed method for superior image reconstructions of a wide variety of log-conductivity patterns.
Similar content being viewed by others
References
Adesokan, B., Knudsen, K., Krishnan, V.P., Roy, S.: A fully non-linear optimization approach to acousto-electric tomography. Inverse Probl. 34, 104004 (2018)
Alessandrini, G., Nesi, V.: Univalent \(\sigma \)-harmonic mappings. Arch. Ration. Mech. Anal. 158, 155–171 (2001)
Ambartsoumian, G., Gouia-Zarrad, R., Krishnan, V.P., Roy, S.: Image reconstruction from radially incomplete spherical Radon data. Eur. J. Appl. Math. 29(3), 470–493 (2018)
Ammari, H., Garnier, J., Jing, W., Nguyen, L.H.: Quantitative thermo-acoustic imaging: an exact reconstruction formula. J. Differ. Equ. 254(3), 1375–1395 (2013)
Ammari, H., Grasland-Mongrain, P., Millien, P., Seppecher, L., Seo, J.-K.: A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography. Journal de Mathématiques Pures et Appliquées 103(6), 1390–1409 (2015)
Ammari, H., Boulmier, S., Millien, P.: A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction. J. Differ. Equ. 259(10), 5379–5405 (2015)
Ammari, H., Qiu, L., Santosa, F., Zhang, W.: Determining anisotropic conductivity using diffusion tensor imaging data in magneto-acoustic tomography with magnetic induction. Inverse Probl. 33, 125006 (2017)
Bal, G., Guo, C., Monard, F.: Imaging of anisotropic conductivities from current densities in two dimensions. SIAM J. Imaging Sci. 7(4), 2538–2557 (2014)
Bal, G., Guo, C., Monard, F.: Inverse anisotropic conductivity from internal current densities. Inverse Probl. 30, 025001 (2014)
Candes, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59, 1207–1223 (2006)
Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992)
Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)
Cheney, M., Issacson, D., Newell, J.C.: Electrical impedance tomography. SIAM Rev. 41, 85–101 (1999)
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)
Gao, H., Osher, S., Zhao, H.: Quantitative Photoacoustic Tomography, Mathematical Modeling in Biomedical Imaging II, Lecture Notes in Mathematics, vol. 2035, pp. 131–158. Springer, Heidelberg (2012)
Garmatter, D., Harrach, B.: Magnetic resonance electrical impedance tomography (MREIT): convergence and reduced basis approach. SIAM J. Imaging Sci. 11(1), 863–887 (2018)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Hasanov, K.F., Ma, A.W., Nachman, A., Joy, M.L.G.: Current density impedance imaging. IEEE Trans. Med. Imaging 27(9), 1301–1309 (2008)
Hoffman, K., Knudsen, K.: Iterative reconstruction methods for hybrid inverse problems in impedance tomography. Sens. Imaging 15, 96 (2014)
Khang, H.S., Lee, B.I., Oh, S.H., Woo, E.J., Lee, S.Y., Cho, M.H., Kwon, O., Yoon, J.R., Seo, J.K.: \(J\)-substitution algorithm in magnetic resonance electrical impedance tomography (MREIT): phaontom experiments for static resistivity images. IEEE Trans. Med. Imaging 21, 695–702 (2002)
Knudsen, K., Lassas, M., Mueller, J.L., Siltanen, S.: Regularized D-bar method for the inverse conductivity problem. Inverse Probl. Imaging 3, 599–624 (2009)
Kuchment, P., Steinhauer, D.: Stabilizing inverse problems by internal data. Inverse Probl. 28(8), 084007 (2012)
Kwon, O., Woo, E.J., Yoon, J.R., Seo, J.K.: Magnetic resonance electrical impedance tomography (MREIT): simulation study of \(J\)-substitution algorithm. IEEE Trans. Biomed. Eng. 49, 160–167 (2002)
Kim, Y.J., Kwon, O., Seo, J.K., Woo, E.J.: Uniqueness and convergence of conductivity image reconstruction in magnetic resonance electrical impedance tomography. Inverse Probl. 19, 1213–1225 (2003)
Kim, S., Kwon, O., Seo, J.K., Yoon, J.-R.: On a non-linear partial differential equation arising in magnetic resonance electrical impedance tomography. SIAM J. Math. Anal. 34(3), 511–526 (2002)
Kruger, M.V.P.: Tomography as a metrology technique for semiconductor manufacturing. Ph.D. Thesis, University of California, Berkeley (2003)
Lee, J.Y.: A reconstruction formula and uniqueness of conductivity in MREIT using two internal current distributions. Inverse Probl. 20, 847–858 (2004)
Li, M., Yang, H., Kudo, H.: An accurate iterative reconstruction algorithm for sparse objects: application to 3D blood vessel reconstruction from a limited number of projections. Phys. Med. Biol. 47, 2599–2609 (2002)
Liu, J.J., Seo, J.K., Sini, M., Woo, E.J.: On the convergence of the harmonic \(B_z\) algorithm in magentic resonance electrical impedance tomography. SIAM J. Appl. Math. 67, 1259–1282 (2007)
Liu, J.J., Seo, J.K., Woo, E.J.: A posteriori error estimate and convergence analysis for conductivity image reconstruction in MREIT. SIAM J. Appl. Math. 70, 2883–2903 (2010)
Monard, F., Bal, G.: Inverse diffusion problems with redundant information. Inverse Probl. Imaging 6(2), 289–313 (2012)
Moradifam, A., Nachman, A., Timonov, A.: A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data. Inverse Probl. 28(8), 084003 (2012)
Nachman, A., Tamasan, A., Timonov, A.: Conductivity imaging with a single measurement of boundary and interior data. Inverse Probl. 23, 2551–2563 (2007)
Nachman, A., Tamasan, A., Timonov, A.: Recovering the conductivity from a single measurement of interior data. Inverse Probl. 25(3), 035014 (2009)
Nachman, A., Tamasan, A., Timonov, A.: Current density impedance imaging. Tomogr. Inverse Transp. Theory 559, 135–149 (2011)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)
Qiu, L., Santosa, F.: Analysis of the magnetoacoustic tomography with magnetic induction. SIAM J. Imaging Sci. 8(3), 2070–2086 (2015)
Roy, S., Annunziato, M., Borzì, A.: A Fokker–Planck feedback control-constrained approach for modelling crowd motion. J. Comput. Theor. Transp. 45, 442–458 (2016)
Roy, S., Annunziato, M., Borzì, A., Klingenberg, C.: A Fokker–Planck approach to control collective motion. Comput. Optim. Appl. 69(2), 423–459 (2018)
Roy, S., Borzì, A.: A new optimisation approach to sparse reconstruction of log-conductivity in acousto-electric tomography. SIAM J. Imaging Sci. 11(2), 1759–1784 (2018)
Roy, S., Krishnan, V.P., Chandrasekhar, P., Vasudeva Murthy, A.S.: An efficient numerical algorithm for Radon transform inversion with applications in ultrasound imaging. J. Math. Imaging Vis. 53, 78–91 (2015)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. J. Math. Imaging Vis. 12, 43–63 (2000)
Schindele, A., Borzí, A.: Proximal methods for elliptic optimal control problems with sparsity cost functional. Appl. Math. 7, 967–992 (2016)
Schindele, A., Borzí, A.: Proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals. Int. J. Control 90, 2349–2367 (2016). https://doi.org/10.1080/00207179.2016.1245870
Scott, G.C., Joy, M.L.G., Armstrong, R.L., Henkelman, R.M.: Measurement of nonuniform current density by magnetic resonance. IEEE Trans. Med. Imaging 10, 262–374 (1991)
Seagar, A.D., Barber, D.C., Brown, B.H.: Theoretical limits to sensitivity and resolution in impedance imaging. Clin. Phys. Physiol. Meas. 8(Suppl. A), 13–31 (1987)
Seo, J.K., Woo, E.J.: Magnetic resonance electrical impedance tomogrpahy. SIAM Rev. 53, 40–68 (2011)
Stadler, G.: Elliptic optimal control problems with \(L^1\)-control costs and applications for the placement of control devices. Comput. Optim. Appl. 44, 159–181 (2009)
Tamasan, A., Veras, J.: Conductivity imaging by the method of characteristics in the 1-Laplacian. Inverse Probl. 28, 084006 (2012)
Waterfall, R.C., He, R., Beck, C.M.: Visualizing combustion using electrical impedance tomography. Chem. Eng. Sci. 52(13), 2129–2138 (1997)
Zain, N.M., Chelliah, K.K.: Breast imaging using electrical impedance tomography: correlation of quantitative assessment with visual interpretation. Asian Pac. J. Cancer Prev. 15(3), 1327–1331 (2014)
Acknowledgements
S. Roy was partly supported by the National Cancer Institute, National Institutes of Health, Grant Number: 1R21CA242933-01.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gupta, M., Mishra, R.K. & Roy, S. Sparse Reconstruction of Log-Conductivity in Current Density Impedance Tomography. J Math Imaging Vis 62, 189–205 (2020). https://doi.org/10.1007/s10851-019-00929-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-019-00929-5
Keywords
- Inverse problems
- PDE-constrained optimization
- Proximal methods
- Edge enhancement
- Sparsity patterns
- Current density impedance imaging