Conductivity reconstruction from power density data in limited view
In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.
Adams, R. A., and Fournier, J. J. F., Sobolev spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.
Alberti, G. S., Bal, G., and Di Cristo, M., Critical points for elliptic equations with prescribed boundary conditions, Arch. Ration. Mech. Anal. 226 (2017), no. 1, 117–141. https://doi.org/10.1007/s00205-017-1130-3
Alberti, G. S., and Capdeboscq, Y., Lectures on elliptic methods for hybrid inverse problems, Cours Spécialisés 25, Société Mathématique de France, Paris, 2018.
Alberti, G. S., and Capdeboscq, Y., Combining the Runge approximation and the Whitney embedding theorem in hybrid imaging, Int. Math. Res. Not. IMRN 2022, no. 6, 4387–4406. https://doi.org/10.1093/imrn/rnaa162
Ammari, H., Bonnetier, E., Capdeboscq, Y., Tanter, M., and Fink, M., Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math. 68 (2008), no. 6, 1557–1573. https://doi.org/10.1137/070686408
Bal, G., Bonnetier, E., Monard, F., and Triki, F., Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging 7 (2013), no. 2, 353–375. https://doi.org/10.3934/ipi.2013.7.353
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. https://doi.org/10.1137/140961754
Bauman, P., Marini, A., and Nesi, V., Univalent solutions of an elliptic system of partial differential equations arising in homogenization, Indiana Univ. Math. J. 50 (2001), no. 2, 747–757. https://doi.org/10.1512/iumj.2001.50.1832
Di Cristo, M., and Rondi, L., Interior decay of solutions to elliptic equations with respect to frequencies at the boundary, Indiana Univ. Math. J. 70 (2021), no. 4, 1303–1334. https://doi.org/10.1512/iumj.2021.70.9367
Gebauer, B., Localized potentials in electrical impedance tomography, Inverse Probl. Imaging 2 (2008), no. 2, 251–269. https://doi.org/10.3934/ipi.2008.2.251
Giaquinta, M., and Martinazzi, L., An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Second edition. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 11. Edizioni della Normale, Pisa, 2012. https://doi.org/10.1007/978-88-7642-443-4
Gilbarg, D., and Trudinger, N. S., Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
Hubmer, S., Knudsen, K., Li, C., and Sherina, E., Limited-angle acousto-electrical tomography, Inverse Probl. Sci. Eng. 27 (2019), no. 9, 1298–1317. https://doi.org/10.1080/17415977.2018.1512983,
Jensen, B., Kirkeby, A., and Knudsen, K., Feasibility of acousto-electric tomography, arXiv:1908.04215.
Logg, A., Mardal, K.-A., and Wells, G., Automated solution of differential equations by the finite element method: The FEniCS book, Lecture Notes in Computational Science and Engineering vol. 84, Springer Science & Business Media, 2012. https://doi.org/10.1007/978-3-642-23099-8
Monard, F., and Bal, G., Inverse anisotropic diffusion from power density measurements in two dimensions, Inverse Problems 28 (2012), no. 8, 084001. https://doi.org/10.1088/0266-5611/28/8/084001
Monard, F., and Bal, G., Inverse diffusion problems with redundant internal information, Inverse Problems and Imaging 6 (2012), no. 2, 289–313. https://doi.org/10.3934/ipi.2012.6.289
Rüland, A., and Salo, M., Quantitative Runge approximation and inverse problems, Int. Math. Res. Not. IMRN 2019 (2019), no. 20, 6216–6234. https://doi.org/10.1093/imrn/rnx301
Salsa, S., Partial differential equations in action, Third edition, Springer, [Cham], 2016. https://doi.org/10.1007/978-3-319-31238-5
Zhang, H., and Wang, L. V., Acousto-electric tomography, Progress in biomedical optics and imaging - proceedings of SPIE 5 (2004), no. 9, 20, 145–149. https://doi.org/10.1117/12.532610