Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging

Ammari, Habib; Garnier, Josselin; Sølna, Knut

doi:10.1090/S0002-9939-2013-11590-X

Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging
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by Habib Ammari, Josselin Garnier and Knut Sølna PDF

Proc. Amer. Math. Soc. 141 (2013), 3431-3446 Request permission

Abstract:

In this paper we consider resolution estimates in both the linearized conductivity problem and the wave imaging problem. Our purpose is to provide explicit formulas for the resolving power of the measurements in the presence of measurement noise. We show that the low-frequency regime in wave imaging and the inverse conductivity problem are very sensitive to measurement noise, while high frequencies increase stability in wave imaging.

References

M. Abramowitz and I. Stegun (editors), Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.
Habib Ammari, Josselin Garnier, Hyeonbae Kang, Mikyoung Lim, and Knut Sølna, Multistatic imaging of extended targets, SIAM J. Imaging Sci. 5 (2012), no. 2, 564–600. MR 2971173, DOI 10.1137/10080631X
H. Ammari, J. Garnier, and K. Sølna, Limited view resolving power of conductivity imaging from boundary measurements, submitted.
Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, Mathematical Surveys and Monographs, vol. 153, American Mathematical Society, Providence, RI, 2009. MR 2488135, DOI 10.1090/surv/153
Habib Ammari, Hyeonbae Kang, Mikyoung Lim, and Habib Zribi, Conductivity interface problems. I. Small perturbations of an interface, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2435–2449. MR 2584606, DOI 10.1090/S0002-9947-09-04842-9
Gang Bao, Yu Chen, and Fuming Ma, Regularity and stability for the scattering map of a linearized inverse medium problem, J. Math. Anal. Appl. 247 (2000), no. 1, 255–271. MR 1766937, DOI 10.1006/jmaa.2000.6856
Elena Beretta and Elisa Francini, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneities, Inverse problems: theory and applications (Cortona/Pisa, 2002) Contemp. Math., vol. 333, Amer. Math. Soc., Providence, RI, 2003, pp. 49–62. MR 2032006, DOI 10.1090/conm/333/05953
M. Bertero, P. Boccacci, and M. Piana, Resolution and super-resolution in inverse diffraction, Inverse problems of wave propagation and diffraction (Aix-les-Bains, 1996) Lecture Notes in Phys., vol. 486, Springer, Berlin, 1997, pp. 1–17. MR 1601171, DOI 10.1007/BFb0105756
Liliana Borcea, Electrical impedance tomography, Inverse Problems 18 (2002), no. 6, R99–R136. MR 1955896, DOI 10.1088/0266-5611/18/6/201
Liliana Borcea, George Papanicolaou, and Fernando Guevara Vasquez, Edge illumination and imaging of extended reflectors, SIAM J. Imaging Sci. 1 (2008), no. 1, 75–114. MR 2475826, DOI 10.1137/07069290X
M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999.
David C. Dobson, Estimates on resolution and stabilization for the linearized inverse conductivity problem, Inverse Problems 8 (1992), no. 1, 71–81. MR 1153128, DOI 10.1088/0266-5611/8/1/005
David C. Dobson and Fadil Santosa, Resolution and stability analysis of an inverse problem in electrical impedance tomography: dependence on the input current patterns, SIAM J. Appl. Math. 54 (1994), no. 6, 1542–1560. MR 1301272, DOI 10.1137/S0036139992237596
Grégoire Derveaux, George Papanicolaou, and Chrysoula Tsogka, Resolution and denoising in near-field imaging, Inverse Problems 22 (2006), no. 4, 1437–1456. MR 2249472, DOI 10.1088/0266-5611/22/4/017
Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1408680, DOI 10.1007/978-94-009-1740-8
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
Bastian Harrach and Jin Keun Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography, SIAM J. Math. Anal. 42 (2010), no. 4, 1505–1518. MR 2679585, DOI 10.1137/090773970
Songming Hou, Knut Solna, and Hongkai Zhao, A direct imaging algorithm for extended targets, Inverse Problems 22 (2006), no. 4, 1151–1178. MR 2249458, DOI 10.1088/0266-5611/22/4/003
Songming Hou, Knut Solna, and Hongkai Zhao, Imaging of location and geometry for extended targets using the response matrix, J. Comput. Phys. 199 (2004), no. 1, 317–338. MR 2081006, DOI 10.1016/j.jcp.2004.02.010
David Isaacson and Eli L. Isaacson, Comment on A.-P. Calderón’s paper: “On an inverse boundary value problem” [in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980; MR0590275 (81k:35160)], Math. Comp. 52 (1989), no. 186, 553–559. MR 962208, DOI 10.1090/S0025-5718-1989-0962208-X
Victor Isakov, Increased stability in the Cauchy problem for some elliptic equations, Instability in models connected with fluid flows. I, Int. Math. Ser. (N. Y.), vol. 6, Springer, New York, 2008, pp. 339–362. MR 2459261, DOI 10.1007/978-0-387-75217-4_{8}
P. Kearey, M. Brooks, and I. Hill, An Introduction to Geophysical Exploration, 3rd edition, Blackwell Science, Oxford, 2002.
Niculae Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems 17 (2001), no. 5, 1435–1444. MR 1862200, DOI 10.1088/0266-5611/17/5/313
Sei Nagayasu, Gunther Uhlmann, and Jenn-Nan Wang, A depth-dependent stability estimate in electrical impedance tomography, Inverse Problems 25 (2009), no. 7, 075001, 14. MR 2519853, DOI 10.1088/0266-5611/25/7/075001
David Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983), no. 3, 379–393. MR 710468, DOI 10.1137/1025078
R. Wong, Asymptotic expansion of $\int ^{\pi /2}_0J^2_\nu (\lambda \,\textrm {cos}\,\theta )\,d\theta$, Math. Comp. 50 (1988), no. 181, 229–234. MR 917830, DOI 10.1090/S0025-5718-1988-0917830-2
Hongkai Zhao, Analysis of the response matrix for an extended target, SIAM J. Appl. Math. 64 (2004), no. 3, 725–745. MR 2068119, DOI 10.1137/S0036139902415282