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Stable determination of surface impedance on a rough obstacle by far field data
Inverse diffusion from knowledge of power densities
1. | Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027 |
2. | Laboratoire Jean Kuntzmann, Université de Joseph Fourier, CNRS, 38041 Grenoble Cedex 9, France |
We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.
References:
[1] |
G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[2] |
H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573.
doi: 10.1137/070686408. |
[3] |
S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2010), 123010.
doi: 10.1088/0266-5611/25/12/123010. |
[4] |
M. Atlan, B. C. Forget, F. Ramaz, A. C. Boccara and M. Gross, Pulsed acousto-optic imaging in dynamic scattering media with heterodyne parallel speckle detection, Optics Letters, 30 (2005), 1360-1362.
doi: 10.1364/OL.30.001360. |
[5] |
G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001.
doi: 10.1088/0266-5611/25/5/053001. |
[6] |
_____, Cauchy problem for Ultrasound modulated EIT, to appear in Analysis & PDE, (2012). |
[7] |
G. Bal, K. Ren, G. Uhlmann and T. Zhou, Quantitative thermo-acoustics and related problems, submitted, 27 (2011), 055007.
doi: 10.1088/0266-5611/27/5/055007. |
[8] |
G. Bal and J. C. Schotland, Inverse scattering and Acousto-Optics imaging, Phys. Rev. Letters, 104 (2010), 043902.
doi: 10.1103/PhysRevLett.104.043902. |
[9] |
G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010.
doi: 10.1088/0266-5611/26/8/085010. |
[10] |
M. Briane, G. W. Milton and V. Nesi, Change of sign of the corrector's determinant for homogenization in three-dimensional conductivity, Arch. Ration. Mech. Anal., 173 (2004), 133-150.
doi: 10.1007/s00205-004-0315-8. |
[11] |
A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, (1980), 65-73. |
[12] |
Y. Capdeboscq, J. Fehrenbach, F. de Gournay and O. Kavian, Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements, SIAM J. Imaging Sciences, 2 (2009), 1003-1030.
doi: 10.1137/080723521. |
[13] |
B. T. Cox, S. R. Arridge and P. C. Beard, Estimating chromophore distributions from multiwavelength photoacoustic images, J. Opt. Soc. Am. A, 26 (2009), 443-455.
doi: 10.1364/JOSAA.26.000443. |
[14] |
B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2008), 565-576.
doi: 10.1137/080715123. |
[15] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1977. |
[16] |
M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, New York, 1974. |
[17] |
M. Kempe, M. Larionov, D. Zaslavsky and A. Z. Genack, Acousto-optic tomography with multiply scattered light, J. Opt. Soc. Am. A, 14 (1997), 1151-1158.
doi: 10.1364/JOSAA.14.001151. |
[18] |
P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 21 pp, 055013.
doi: 10.1088/0266-5611/27/5/055013. |
[19] |
O. Kwon, E. J. Woo, J.-R. Yoon and J. K. Seo, Magnetic resonance electrical impedance tomography (mreit): Simulation study of j-substitution algorithm, IEEE Transactions on Biomedical Engineering, 49 (2002), 160-167. |
[20] |
F. Monard and G. Bal, Inverse diffusion problems with redundant internal information, Inverse Problems and Imaging, 6 (2012), 289-313.
doi: 10.3934/ipi.2012.6.289. |
[21] |
A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563.
doi: 10.1088/0266-5611/23/6/017. |
[22] |
A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014.
doi: 10.1088/0266-5611/25/3/035014. |
[23] |
S. Patch and O. Scherzer, Photo- and thermo- acoustic imaging, Inverse Problems, 23 (2007), S1-S10.
doi: 10.1088/0266-5611/23/6/S01. |
[24] |
J. Stuelpnagel, On the parametrization of the three-dimensional rotation group, SIAM Review, 6 (1964), 422–-430.
doi: 10.1137/1006093. |
[25] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[26] |
F. Triki, Uniqueness and stability for the inverse medium problem with internal data, Inverse Problems, 26 (2010), 095014.
doi: 10.1088/0266-5611/26/9/095014. |
[27] |
G. Uhlmann, Calderón's problem and electrical impedance tomography, Inverse Problems, 25 (2009), 123011.
doi: 10.1088/0266-5611/25/12/123011. |
[28] |
L. V. Wang, Ultrasound-mediated biophotonic imaging: A review of acousto-optical tomography and photo-acoustic tomography, Journal of Disease Markers, 19 (2004), 123-138. |
[29] |
H. Zhang and L. Wang, Acousto-electric tomography, Proc. SPIE, 5320 (2004), 14514.
doi: 10.1117/12.532610. |
show all references
References:
[1] |
G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[2] |
H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573.
doi: 10.1137/070686408. |
[3] |
S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2010), 123010.
doi: 10.1088/0266-5611/25/12/123010. |
[4] |
M. Atlan, B. C. Forget, F. Ramaz, A. C. Boccara and M. Gross, Pulsed acousto-optic imaging in dynamic scattering media with heterodyne parallel speckle detection, Optics Letters, 30 (2005), 1360-1362.
doi: 10.1364/OL.30.001360. |
[5] |
G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001.
doi: 10.1088/0266-5611/25/5/053001. |
[6] |
_____, Cauchy problem for Ultrasound modulated EIT, to appear in Analysis & PDE, (2012). |
[7] |
G. Bal, K. Ren, G. Uhlmann and T. Zhou, Quantitative thermo-acoustics and related problems, submitted, 27 (2011), 055007.
doi: 10.1088/0266-5611/27/5/055007. |
[8] |
G. Bal and J. C. Schotland, Inverse scattering and Acousto-Optics imaging, Phys. Rev. Letters, 104 (2010), 043902.
doi: 10.1103/PhysRevLett.104.043902. |
[9] |
G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010.
doi: 10.1088/0266-5611/26/8/085010. |
[10] |
M. Briane, G. W. Milton and V. Nesi, Change of sign of the corrector's determinant for homogenization in three-dimensional conductivity, Arch. Ration. Mech. Anal., 173 (2004), 133-150.
doi: 10.1007/s00205-004-0315-8. |
[11] |
A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, (1980), 65-73. |
[12] |
Y. Capdeboscq, J. Fehrenbach, F. de Gournay and O. Kavian, Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements, SIAM J. Imaging Sciences, 2 (2009), 1003-1030.
doi: 10.1137/080723521. |
[13] |
B. T. Cox, S. R. Arridge and P. C. Beard, Estimating chromophore distributions from multiwavelength photoacoustic images, J. Opt. Soc. Am. A, 26 (2009), 443-455.
doi: 10.1364/JOSAA.26.000443. |
[14] |
B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2008), 565-576.
doi: 10.1137/080715123. |
[15] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1977. |
[16] |
M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, New York, 1974. |
[17] |
M. Kempe, M. Larionov, D. Zaslavsky and A. Z. Genack, Acousto-optic tomography with multiply scattered light, J. Opt. Soc. Am. A, 14 (1997), 1151-1158.
doi: 10.1364/JOSAA.14.001151. |
[18] |
P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 21 pp, 055013.
doi: 10.1088/0266-5611/27/5/055013. |
[19] |
O. Kwon, E. J. Woo, J.-R. Yoon and J. K. Seo, Magnetic resonance electrical impedance tomography (mreit): Simulation study of j-substitution algorithm, IEEE Transactions on Biomedical Engineering, 49 (2002), 160-167. |
[20] |
F. Monard and G. Bal, Inverse diffusion problems with redundant internal information, Inverse Problems and Imaging, 6 (2012), 289-313.
doi: 10.3934/ipi.2012.6.289. |
[21] |
A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563.
doi: 10.1088/0266-5611/23/6/017. |
[22] |
A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014.
doi: 10.1088/0266-5611/25/3/035014. |
[23] |
S. Patch and O. Scherzer, Photo- and thermo- acoustic imaging, Inverse Problems, 23 (2007), S1-S10.
doi: 10.1088/0266-5611/23/6/S01. |
[24] |
J. Stuelpnagel, On the parametrization of the three-dimensional rotation group, SIAM Review, 6 (1964), 422–-430.
doi: 10.1137/1006093. |
[25] |
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[26] |
F. Triki, Uniqueness and stability for the inverse medium problem with internal data, Inverse Problems, 26 (2010), 095014.
doi: 10.1088/0266-5611/26/9/095014. |
[27] |
G. Uhlmann, Calderón's problem and electrical impedance tomography, Inverse Problems, 25 (2009), 123011.
doi: 10.1088/0266-5611/25/12/123011. |
[28] |
L. V. Wang, Ultrasound-mediated biophotonic imaging: A review of acousto-optical tomography and photo-acoustic tomography, Journal of Disease Markers, 19 (2004), 123-138. |
[29] |
H. Zhang and L. Wang, Acousto-electric tomography, Proc. SPIE, 5320 (2004), 14514.
doi: 10.1117/12.532610. |
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