Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


IMPACT FACTOR 2017: 0.941
5-year IMPACT FACTOR: 0.953

CiteScore 2017: 0.91

SCImago Journal Rank (SJR) 2017: 0.461
Source Normalized Impact per Paper (SNIP) 2017: 1.022

Mathematical Citation Quotient (MCQ) 2017: 0.49

Online
ISSN
1569-3945
See all formats and pricing
More options …
Volume 25, Issue 2

Issues

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

Chenxi Guo
  • Corresponding author
  • Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Guillaume Bal
Published Online: 2016-05-12 | DOI: https://doi.org/10.1515/jiip-2015-0013

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.

Keywords: Anisotropic inverse problems; coupled-physics medical imaging; Maxwell’s equations

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. 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. 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. 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. Google Scholar

  • [5]

    Bal G. and Uhlmann G., Inverse diffusion theory of photoacoustics, Inverse Problems 26 (2010), no. 8, Article ID 085010. 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.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. 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. Google Scholar

  • [9]

    Goldstein T. and Osher S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci. 2 (2009), 323–343. 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. 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. Google Scholar

  • [12]

    Kabanikhin S. I., Inverse and Ill-Posed Problems. Theory and Applications, De Gruyter, Berlin, 2011. 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. Google Scholar

  • [14]

    Kohn R. and Vogelius M., Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), 289–298. 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. Google Scholar

  • [16]

    Kuchment P. and Steinhauer D., Stabilizing inverse problems by internal data, Inverse Problems 28 (2012), no. 8, Article ID 084007. 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. 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. 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. Google Scholar

  • [20]

    Ola P., Päivärinta L. and Somersalo E., An inverse boundary value problem in electromagnetics, Duke Math. J. 70 (1993), 617–653. Google Scholar

  • [21]

    Ola P. and Somersalo E., Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math. 56 (1996), 1129–1145. 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. 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. 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. 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. Google Scholar

  • [26]

    Uhlmann G., Calderón’s problem and electrical impedance tomography, Inverse Problems 25 (2009), Article ID 123011. Google Scholar

About the article

Received: 2015-01-24

Revised: 2015-12-28

Accepted: 2016-03-30

Published Online: 2016-05-12

Published in Print: 2017-04-01


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 2, Pages 195–205, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2015-0013.

Export Citation

© 2017 by De Gruyter.Get Permission

Comments (0)

Please log in or register to comment.
Log in