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Sparse Reconstruction of Log-Conductivity in Current Density Impedance Tomography

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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.

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References

  1. Adesokan, B., Knudsen, K., Krishnan, V.P., Roy, S.: A fully non-linear optimization approach to acousto-electric tomography. Inverse Probl. 34, 104004 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Alessandrini, G., Nesi, V.: Univalent \(\sigma \)-harmonic mappings. Arch. Ration. Mech. Anal. 158, 155–171 (2001)

    MathSciNet  MATH  Google Scholar 

  3. 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)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    MathSciNet  MATH  Google Scholar 

  5. 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)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    MathSciNet  MATH  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. 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)

    MathSciNet  MATH  Google Scholar 

  9. Bal, G., Guo, C., Monard, F.: Inverse anisotropic conductivity from internal current densities. Inverse Probl. 30, 025001 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Candes, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59, 1207–1223 (2006)

    MathSciNet  MATH  Google Scholar 

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. 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)

    MathSciNet  MATH  Google Scholar 

  13. Cheney, M., Issacson, D., Newell, J.C.: Electrical impedance tomography. SIAM Rev. 41, 85–101 (1999)

    MathSciNet  MATH  Google Scholar 

  14. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  15. 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)

    Google Scholar 

  16. Garmatter, D., Harrach, B.: Magnetic resonance electrical impedance tomography (MREIT): convergence and reduced basis approach. SIAM J. Imaging Sci. 11(1), 863–887 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

    MATH  Google Scholar 

  18. 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)

    Google Scholar 

  19. Hoffman, K., Knudsen, K.: Iterative reconstruction methods for hybrid inverse problems in impedance tomography. Sens. Imaging 15, 96 (2014)

    Google Scholar 

  20. 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)

    Google Scholar 

  21. 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)

    MathSciNet  MATH  Google Scholar 

  22. Kuchment, P., Steinhauer, D.: Stabilizing inverse problems by internal data. Inverse Probl. 28(8), 084007 (2012)

    MathSciNet  MATH  Google Scholar 

  23. 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)

    Google Scholar 

  24. 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)

    MathSciNet  MATH  Google Scholar 

  25. 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)

    MathSciNet  MATH  Google Scholar 

  26. Kruger, M.V.P.: Tomography as a metrology technique for semiconductor manufacturing. Ph.D. Thesis, University of California, Berkeley (2003)

  27. Lee, J.Y.: A reconstruction formula and uniqueness of conductivity in MREIT using two internal current distributions. Inverse Probl. 20, 847–858 (2004)

    MathSciNet  MATH  Google Scholar 

  28. 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)

    Google Scholar 

  29. 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)

    MathSciNet  MATH  Google Scholar 

  30. 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)

    MathSciNet  MATH  Google Scholar 

  31. Monard, F., Bal, G.: Inverse diffusion problems with redundant information. Inverse Probl. Imaging 6(2), 289–313 (2012)

    MathSciNet  MATH  Google Scholar 

  32. 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)

    MathSciNet  MATH  Google Scholar 

  33. Nachman, A., Tamasan, A., Timonov, A.: Conductivity imaging with a single measurement of boundary and interior data. Inverse Probl. 23, 2551–2563 (2007)

    MathSciNet  MATH  Google Scholar 

  34. Nachman, A., Tamasan, A., Timonov, A.: Recovering the conductivity from a single measurement of interior data. Inverse Probl. 25(3), 035014 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Nachman, A., Tamasan, A., Timonov, A.: Current density impedance imaging. Tomogr. Inverse Transp. Theory 559, 135–149 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Google Scholar 

  37. Qiu, L., Santosa, F.: Analysis of the magnetoacoustic tomography with magnetic induction. SIAM J. Imaging Sci. 8(3), 2070–2086 (2015)

    MathSciNet  MATH  Google Scholar 

  38. 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)

    MathSciNet  Google Scholar 

  39. Roy, S., Annunziato, M., Borzì, A., Klingenberg, C.: A Fokker–Planck approach to control collective motion. Comput. Optim. Appl. 69(2), 423–459 (2018)

    MathSciNet  MATH  Google Scholar 

  40. 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)

    MathSciNet  MATH  Google Scholar 

  41. 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)

    MATH  Google Scholar 

  42. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    MathSciNet  MATH  Google Scholar 

  43. Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. J. Math. Imaging Vis. 12, 43–63 (2000)

    MathSciNet  MATH  Google Scholar 

  44. Schindele, A., Borzí, A.: Proximal methods for elliptic optimal control problems with sparsity cost functional. Appl. Math. 7, 967–992 (2016)

    Google Scholar 

  45. 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

    Article  MathSciNet  MATH  Google Scholar 

  46. 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)

    Google Scholar 

  47. 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)

    Google Scholar 

  48. Seo, J.K., Woo, E.J.: Magnetic resonance electrical impedance tomogrpahy. SIAM Rev. 53, 40–68 (2011)

    MathSciNet  MATH  Google Scholar 

  49. 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)

    MathSciNet  MATH  Google Scholar 

  50. Tamasan, A., Veras, J.: Conductivity imaging by the method of characteristics in the 1-Laplacian. Inverse Probl. 28, 084006 (2012)

    MathSciNet  MATH  Google Scholar 

  51. Waterfall, R.C., He, R., Beck, C.M.: Visualizing combustion using electrical impedance tomography. Chem. Eng. Sci. 52(13), 2129–2138 (1997)

    Google Scholar 

  52. 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)

    Google Scholar 

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Acknowledgements

S. Roy was partly supported by the National Cancer Institute, National Institutes of Health, Grant Number: 1R21CA242933-01.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Arlington, 655 W. Mitchell Street, 222H SEIR Building, Arlington, TX, 76010, USA

    Madhu Gupta

  2. Department of Mathematics, University of Texas at Arlington, 655 W. Mitchell Street, 222B SEIR Building, Arlington, TX, 76010, USA

    Rohit Kumar Mishra

  3. Department of Mathematics, University of Texas at Arlington, 655 W. Mitchell Street, 219 SEIR Building, Arlington, TX, 76010, USA

    Souvik Roy

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  1. Madhu Gupta
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Correspondence to Souvik Roy.

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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

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