Inverse diffusion from knowledge of power densities

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May 2013, 7(2): 353-375. doi: 10.3934/ipi.2013.7.353

Inverse diffusion from knowledge of power densities

Guillaume Bal ^1, , Eric Bonnetier ^2, , François Monard ^1, and Faouzi Triki ^2,

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

Received March 2011 Revised February 2012 Published May 2013

Abstract
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This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography.
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.

Keywords: Calderón's problem, inverse diffusion, hybrid methods, Inverse conductivity, power density measurements, optical tomography., complex geometrical optics solutions.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35J1.

Citation: Guillaume Bal, Eric Bonnetier, François Monard, Faouzi Triki. Inverse diffusion from knowledge of power densities. Inverse Problems and Imaging, 2013, 7 (2) : 353-375. doi: 10.3934/ipi.2013.7.353

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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