Home Imaging of complex-valued tensors for two-dimensional Maxwell’s equations
Article
Licensed
Unlicensed Requires Authentication

Imaging of complex-valued tensors for two-dimensional Maxwell’s equations

  • Chenxi Guo EMAIL logo and Guillaume Bal
Published/Copyright: May 12, 2016

Abstract

This paper concerns the imaging of a complex-valued anisotropic tensor γ=σ+𝜾ωε from knowledge of several inter magnetic fields H, where H satisfies the anisotropic Maxwell system on a bounded domain X2 with prescribed boundary conditions on X. We show that γ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition H. A minimum number of five well-chosen functionals guaranties a local reconstruction of γ in dimension two. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice of boundary conditions on the stability of the reconstruction. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.

MSC 2010: 35J99; 35R30; 92C55

References

[1] Ammari H., Bonnetier E., Capdeboscq Y., Tanter M. and Fink M., Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math. 68 (2008), 1557–1573. 10.1137/070686408Search in Google Scholar

[2] Bal G., Guo C. and Monard F., Imaging of anisotropic conductivities from current densities in two dimensions, SIAM J. Imaging Sci. 7 (2014), no. 4, 2538–2557. 10.1137/140961754Search in Google Scholar

[3] Bal G., Guo C. and Monard F., Inverse anisotropic conductivity from internal current densities, Inverse Problems 30 (2014), no. 2, Article ID 025001. 10.1088/0266-5611/30/2/025001Search in Google Scholar

[4] Bal G., Guo C. and Monard F., Linearized internal functionals for anisotropic conductivities, Inverse Probl. Imaging 8 (2014), no. 1, 1–22. 10.3934/ipi.2014.8.1Search in Google Scholar

[5] Bal G. and Uhlmann G., Inverse diffusion theory of photoacoustics, Inverse Problems 26 (2010), no. 8, Article ID 085010. 10.1088/0266-5611/26/8/085010Search in Google Scholar

[6] Bal G. and Uhlmann G., Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Comm. Pure Appl. Math. 66 (2013), no. 10, 1629–1652.10.1002/cpa.21453Search in Google Scholar

[7] Caro P., Ola P. and Salo M., Inverse boundary value problem for Maxwell equations with local data, Comm. Partial Differential Equations 34 (2009), 1452–1464. 10.1080/03605300903296272Search in Google Scholar

[8] Daytray R. and Lions J. L., Mathematical Analysis and Numerical Methods for Science and Technology. Volume 3: Spectral Theory and Applications, Springer, Berlin, 2000. Search in Google Scholar

[9] Goldstein T. and Osher S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci. 2 (2009), 323–343. 10.1137/080725891Search in Google Scholar

[10] Guo C. and Bal G., Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields, Inverse Probl. Imaging 8 (2014), no. 4, 1033–1051. 10.3934/ipi.2014.8.1033Search in Google Scholar

[11] Ider Y. and Muftuler L., Measurement of AC magnetic field distribution using magnetic resonance imaging, IEEE Trans. Medical Imag. 16 (1997), 617–622. 10.1109/42.640752Search in Google Scholar PubMed

[12] Kabanikhin S. I., Inverse and Ill-Posed Problems. Theory and Applications, De Gruyter, Berlin, 2011. 10.1515/9783110224016Search in Google Scholar

[13] Kenig C. E., Salo M. and Uhlmann G., Inverse problems for the anisotropic Maxwell equations, Duke Math. J. 157 (2011), 369–419. 10.1215/00127094-1272903Search in Google Scholar

[14] Kohn R. and Vogelius M., Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), 289–298. 10.1002/cpa.3160370302Search in Google Scholar

[15] Kuchment P. and Kunyansky L., 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems 27 (2011), no. 5, Article ID 055013. 10.1088/0266-5611/27/5/055013Search in Google Scholar

[16] Kuchment P. and Steinhauer D., Stabilizing inverse problems by internal data, Inverse Problems 28 (2012), no. 8, Article ID 084007. 10.1088/0266-5611/28/8/084007Search in Google Scholar

[17] Monard F. and Bal G., Inverse anisotropic conductivity from power densities in dimension n3, Comm. Partial Differential Equations 38 (2013), no. 7, 1183–1207. 10.1080/03605302.2013.787089Search in Google Scholar

[18] Nachman A., Tamasan A. and Timonov A., Recovering the conductivity from a single measurement of interior data, Inverse Problems 25 (2009), Article ID 035014. 10.1088/0266-5611/25/3/035014Search in Google Scholar

[19] Novikov R., The ¯-approach to approximate inverse scattering at fixed energy in three dimensions, Int. Math. Res. Pap. IMRP 2005 (2005), no. 6, 287–349. 10.1155/IMRP.2005.287Search in Google Scholar

[20] Ola P., Päivärinta L. and Somersalo E., An inverse boundary value problem in electromagnetics, Duke Math. J. 70 (1993), 617–653. 10.1215/S0012-7094-93-07014-7Search in Google Scholar

[21] Ola P. and Somersalo E., Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math. 56 (1996), 1129–1145. 10.1137/S0036139995283948Search in Google Scholar

[22] Seo J. K., Kim D.-H., Lee J., Kwon O. I., Sajib S. Z. K. and Woo E. J., Electrical tissue property imaging using MRI at DC and Larmor frequency, Inverse Problems 28 (2012), Article ID 084002. 10.1088/0266-5611/28/8/084002Search in Google Scholar

[23] Somersalo E., Isaacson D. and Cheney M., A linearized inverse boundary value problem for Maxwell’s equations, J. Comput. Appl. Math 42 (2012), 123–136. 10.1016/0377-0427(92)90167-VSearch in Google Scholar

[24] Sylvester J. and Uhlmann G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. 125 (1987), no. 1, 153–169. 10.2307/1971291Search in Google Scholar

[25] Takhtadzhan L. A. and Faddeev L. D., The quantum method of the inverse problem and the Heisenberg XYZ model, Russian Math. Surveys 34 (1979), 11–68. 10.1142/9789812815453_0008Search in Google Scholar

[26] Uhlmann G., Calderón’s problem and electrical impedance tomography, Inverse Problems 25 (2009), Article ID 123011. 10.1088/0266-5611/25/12/123011Search in Google Scholar

Received: 2015-1-24
Revised: 2015-12-28
Accepted: 2016-3-30
Published Online: 2016-5-12
Published in Print: 2017-4-1

© 2017 by De Gruyter

Downloaded on 28.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jiip-2015-0013/html
Scroll to top button