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May  2012, 6(2): 289-313. doi: 10.3934/ipi.2012.6.289

Inverse diffusion problems with redundant internal information

François Monard 1, and  Guillaume Bal 1,

1. 

Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027

Received  June 2011 Published  May 2012

This paper concerns the reconstruction of a scalar diffusion coefficient $\sigma(x)$ from redundant functionals of the form $H_i(x)=\sigma^{2\alpha}(x)|\nabla u_i|^2(x)$ where $\alpha\in\mathbb{R}$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot \sigma \nabla u_i=0$ for $1\leq i\leq I$. The case $\alpha=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT), ultrasound modulated optical tomography (UMOT) as well as impedance acoustic computerized tomography (ImpACT). The case $\alpha=1$ finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT).
    We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in [10] and [5], respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.
Keywords: hybrid methods, power density measurements, differential geometry., Inverse conductivity, strongly coupled elliptic systems, Calderón's problem.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J45, 53B2.
Citation: François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems & Imaging, 2012, 6 (2) : 289-313. doi: 10.3934/ipi.2012.6.289
References:
[1]

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.  Google Scholar

[2]

G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171. doi: 10.1007/PL00004242.  Google Scholar

[3]

G. Bal, Hybrid inverse problems and internal functionals (review paper), in "Inside Out'' (ed. Gunther Uhlmann), Cambridge University Press, 2012. Google Scholar

[4]

_____, Cauchy problem and Ultrasound modulated EIT,, submitted., ().   Google Scholar

[5]

G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, in press, 2012. Google Scholar

[6]

G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055007.  Google Scholar

[7]

G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010. doi: 10.1088/0266-5611/26/8/085010.  Google Scholar

[8]

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.  Google Scholar

[9]

C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'' Third edition, AMS, Chelsea, 1999. Google Scholar

[10]

Y. Capdeboscq, J. Fehrenbach, F. de Gournay and O. Kavian, Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements, SIAM Journal on Imaging Sciences, 2 (2009), 1003-1030.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, Vol. 19, AMS, 1998.  Google Scholar

[12]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2009), 565-576. doi: 10.1137/080715123.  Google Scholar

[13]

S. Kim, O. Kwon, J. K. Seo and J.-R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM J. Math. Anal., 34 (2002), 511-526. doi: 10.1137/S0036141001391354.  Google Scholar

[14]

P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013.  Google Scholar

[15]

J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'' Graduate Texts in Mathematics, Vol. 176, Springer, 1997.  Google Scholar

[16]

F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'' Ph.D. thesis, Columbia University, New York, 2012. Google Scholar

[17]

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.  Google Scholar

[18]

_____, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014.  Google Scholar

[19]

_____, Current density impedance imaging, Contemporary Mathematics, American Mathematical Society, in press, 2012. Google Scholar

[20]

O. Scherzer, "Handbook of Mathematical Methods in Imaging,'' Springer Verlag, New York, 2011. Google Scholar

[21]

M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'' Second edition, Publish or perish, 1990. Google Scholar

[22]

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.  Google Scholar

[23]

M. Taylor, "Partial Differential Equations I, Basic Theory,'' Springer, New York, 1996.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

G. Alessandrini and V. Nesi, Univalent $e^\sigma$-harmonic mappings, Arch. Rat. Mech. Anal., 158 (2001), 155-171. doi: 10.1007/PL00004242.  Google Scholar

[3]

G. Bal, Hybrid inverse problems and internal functionals (review paper), in "Inside Out'' (ed. Gunther Uhlmann), Cambridge University Press, 2012. Google Scholar

[4]

_____, Cauchy problem and Ultrasound modulated EIT,, submitted., ().   Google Scholar

[5]

G. Bal, E. Bonnetier, F. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, in press, 2012. Google Scholar

[6]

G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055007.  Google Scholar

[7]

G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics, Inverse Problems, 26 (2010), 085010. doi: 10.1088/0266-5611/26/8/085010.  Google Scholar

[8]

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.  Google Scholar

[9]

C. Carathéodory, "Calculus of Variations and Partial Differential Equations of the First Order,'' Third edition, AMS, Chelsea, 1999. Google Scholar

[10]

Y. Capdeboscq, J. Fehrenbach, F. de Gournay and O. Kavian, Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements, SIAM Journal on Imaging Sciences, 2 (2009), 1003-1030.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, Vol. 19, AMS, 1998.  Google Scholar

[12]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Applied Math., 69 (2009), 565-576. doi: 10.1137/080715123.  Google Scholar

[13]

S. Kim, O. Kwon, J. K. Seo and J.-R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM J. Math. Anal., 34 (2002), 511-526. doi: 10.1137/S0036141001391354.  Google Scholar

[14]

P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013.  Google Scholar

[15]

J. M. Lee, "Riemannian Manifolds. An Introduction to Curvature,'' Graduate Texts in Mathematics, Vol. 176, Springer, 1997.  Google Scholar

[16]

F. Monard, "Taming Unstable Inverse Problems. Mathematical Routes Toward High-Resolution Medical Imaging Modalities,'' Ph.D. thesis, Columbia University, New York, 2012. Google Scholar

[17]

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.  Google Scholar

[18]

_____, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014.  Google Scholar

[19]

_____, Current density impedance imaging, Contemporary Mathematics, American Mathematical Society, in press, 2012. Google Scholar

[20]

O. Scherzer, "Handbook of Mathematical Methods in Imaging,'' Springer Verlag, New York, 2011. Google Scholar

[21]

M. Spivak, "A Comprehensive Introduction to Differential Geometry, Vol. 2,'' Second edition, Publish or perish, 1990. Google Scholar

[22]

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.  Google Scholar

[23]

M. Taylor, "Partial Differential Equations I, Basic Theory,'' Springer, New York, 1996.  Google Scholar

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