April  2017, 11(2): 339-353. doi: 10.3934/ipi.2017016

Stability in conductivity imaging from partial measurements of one interior current

1. 

Deparment of Mathematics, University of Washingotn, Seattle, WA 98195-4350, USA

2. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

* Corresponding author: Carlos Montalto

Received  April 2016 Revised  July 2016 Published  March 2017

Fund Project: A. Tamasan was supported by the NSF Grant DMS 1312883.

We prove a stability result in the hybrid inverse problem of recovering the electrical conductivity from partial knowledge of one current density field generated inside a body by an imposed boundary voltage. The region of stable reconstruction is well defined by a combination of the exact and perturbed data. This work explains the high resolution and accuracy reconstructions in some existing numerical experiments that use partial interior data.

Citation: Carlos Montalto, Alexandru Tamasan. Stability in conductivity imaging from partial measurements of one interior current. Inverse Problems and Imaging, 2017, 11 (2) : 339-353. doi: 10.3934/ipi.2017016
References:
[1]

G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 14 (1987), 229-256. 

[2]

G. Alessandrini, Global stability for a coupled physics inverse problem, Inverse Problems, 30 (2014), 075008, 10pp. doi: 10.1088/0266-5611/30/7/075008.

[3]

G. Bal, Hybrid inverse problems and internal functionals, Inverse problems and applications: Inside out. II, 325-368, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013

[4]

G. Bal, Hybrid inverse problems and redundant systems of partial differential equations, Inverse Problems and Applications, 15-47, Contemp. Math., 615, Amer. Math. Soc., Providence, RI, 2014. doi: 10.1090/conm/615/12289.

[5]

G. BalE. BonnetierF. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Problems and Imaging, 7 (2013), 353-375.  doi: 10.3934/ipi.2013.7.353.

[6]

G. Bal, C. Guo and F. Monard, Inverse anisotropic conductivity from internal current densities Inverse Problems, 30 (2014), 025001, 21pp. doi: 10.1088/0266-5611/30/2/025001.

[7]

Y. CapdeboscqJ. FehrenbachF. 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.  doi: 10.1137/080723521.

[8]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Appl. Math., 69 (2008), 565-576.  doi: 10.1137/080715123.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.

[10]

K. Hasanov, A. Ma, R. Yoon, A. Nachman and M. Joy, A new approach to current density impedance imaging, Engineering in Medicine and Biology Society, IEMBS '04. 26th Annual International Conference of the IEEE, 1 (2004), 1321-1324. doi: 10.1109/IEMBS.2004.1403415.

[11]

N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for single interior measurement Inverse Problems, 30 (2014), 055001, 19pp. doi: 10.1088/0266-5611/30/5/055001.

[12]

S. KimO. KwonJ. K. Seo and J.-R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM journal on Mathematical Analysis, 34 (2002), 511-526.  doi: 10.1137/S0036141001391354.

[13]

Y. -J. Kim and M. -G. Lee, Well-posedness of the conductivity reconstruction from an interior current density in terms of Schauder theory, Quart. Appl. Math., 73 (2015), 419-433. doi: 10.1090/qam/1368.

[14]

Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data Inverse Problems, 28 (2012), 084007, 20pp. doi: 10.1088/0266-5611/28/8/084007.

[15]

Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data. Ⅱ: Non-local internal data and generic linearized uniqueness, Anal. Math. Phys., 5 (2015), 391-425.  doi: 10.1007/s13324-015-0104-6.

[16]

O. KwonJ.-Y. Lee and J.-R. Yoon, Equipotential line method for magnetic resonance electrical impedance tomography, Inverse Problems, 18 (2002), 1089-1100.  doi: 10.1088/0266-5611/18/4/310.

[17]

J.-Y. Lee, A reconstruction formula and uniqueness of conductivity in MREIT using two internal current distributions, Inverse Problems, 20 (2004), 847-858.  doi: 10.1088/0266-5611/20/3/012.

[18]

F. Monard and G. Bal, Inverse anisotropic diffusion from power density measurements in two dimensions Inverse Problems, 28 (2012), 084001, 20pp. doi: 10.1088/0266-5611/28/8/084001.

[19]

F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n≥q3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207.  doi: 10.1080/03605302.2013.787089.

[20]

C. Montalto and Stefanov, Stability of coupled-physics inverse problems with one internal measurement Inverse Problems, 29 (2013), 125004, 13pp. doi: 10.1088/0266-5611/29/12/125004.

[21]

A. Moradifam, A. Nachman and A. Tamasan, Uniqueness of minimizers of weighted least gradient problems arising in conductivity imaging, preprint, 2014.

[22]

A. NachmanA. 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.

[23]

A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data Inverse Problems 25 (2009), 035014, 16pp. doi: 10.1088/0266-5611/25/3/035014.

[24]

A. NachmanA. Tamasan and A. Timonov, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362.  doi: 10.1137/10079241X.

[25]

A. Nachman, A. Tamasan and A. Timonov, Current density impedance imaging, Tomography and inverse transport theory, vol. 559 of Contem Math., Amer. Math. Soc., Providence, RI, 2011,135-149. doi: 10.1090/conm/559/11076.

[26]

M. Z. Nashed and A. Tamasan, Structural stability in a minimizationproblem and applications to conductivity imaging, Inverse Probl. Imaging, 5 (2011), 219-236.  doi: 10.3934/ipi.2011.5.219.

[27]

G. C. ScottM. L. G. JoyR. L. Armstrong and R. M. Henkelman, Measurement of nonuniform current density by magnetic resonance, IEEE Transactions on Medical Imaging, 10 (1991), 362-374.  doi: 10.1109/42.97586.

[28]

J. K. Seo and E. J. Woo, Magnetic resonance electrical impedance tomography (MREIT), SIAM Rev., 53 (2011), 40-68.  doi: 10.1137/080742932.

[29]

A. Tamasan and A. Timonov, Coupled physics electrical conductivity imaging, Eurasian J. Math. Com Appl., 2 (2014), 5-29. 

[30]

A. TamasanA. Timonov and J. Veras, Stable reconstruction of regular 1-harmonic maps with a given trace at the boundary, Appl. Anal, 94 (2015), 1098-1115.  doi: 10.1080/00036811.2014.918260.

[31]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth, Heidelberg, 2nd ed., 1995.

[32]

T. Widlak and O. Scherzer, Hybrid tomography for conductivity imaging Inverse Problems, 28 (2012), 084008, 28pp. doi: 10.1088/0266-5611/28/8/084008.

[33]

N. Zhang, Electrical impedance tomography based on current density imaging, MSc. thesis, University of Toronto, 1992.

show all references

References:
[1]

G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 14 (1987), 229-256. 

[2]

G. Alessandrini, Global stability for a coupled physics inverse problem, Inverse Problems, 30 (2014), 075008, 10pp. doi: 10.1088/0266-5611/30/7/075008.

[3]

G. Bal, Hybrid inverse problems and internal functionals, Inverse problems and applications: Inside out. II, 325-368, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013

[4]

G. Bal, Hybrid inverse problems and redundant systems of partial differential equations, Inverse Problems and Applications, 15-47, Contemp. Math., 615, Amer. Math. Soc., Providence, RI, 2014. doi: 10.1090/conm/615/12289.

[5]

G. BalE. BonnetierF. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Problems and Imaging, 7 (2013), 353-375.  doi: 10.3934/ipi.2013.7.353.

[6]

G. Bal, C. Guo and F. Monard, Inverse anisotropic conductivity from internal current densities Inverse Problems, 30 (2014), 025001, 21pp. doi: 10.1088/0266-5611/30/2/025001.

[7]

Y. CapdeboscqJ. FehrenbachF. 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.  doi: 10.1137/080723521.

[8]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM J. Appl. Math., 69 (2008), 565-576.  doi: 10.1137/080715123.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.

[10]

K. Hasanov, A. Ma, R. Yoon, A. Nachman and M. Joy, A new approach to current density impedance imaging, Engineering in Medicine and Biology Society, IEMBS '04. 26th Annual International Conference of the IEEE, 1 (2004), 1321-1324. doi: 10.1109/IEMBS.2004.1403415.

[11]

N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for single interior measurement Inverse Problems, 30 (2014), 055001, 19pp. doi: 10.1088/0266-5611/30/5/055001.

[12]

S. KimO. KwonJ. K. Seo and J.-R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM journal on Mathematical Analysis, 34 (2002), 511-526.  doi: 10.1137/S0036141001391354.

[13]

Y. -J. Kim and M. -G. Lee, Well-posedness of the conductivity reconstruction from an interior current density in terms of Schauder theory, Quart. Appl. Math., 73 (2015), 419-433. doi: 10.1090/qam/1368.

[14]

Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data Inverse Problems, 28 (2012), 084007, 20pp. doi: 10.1088/0266-5611/28/8/084007.

[15]

Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data. Ⅱ: Non-local internal data and generic linearized uniqueness, Anal. Math. Phys., 5 (2015), 391-425.  doi: 10.1007/s13324-015-0104-6.

[16]

O. KwonJ.-Y. Lee and J.-R. Yoon, Equipotential line method for magnetic resonance electrical impedance tomography, Inverse Problems, 18 (2002), 1089-1100.  doi: 10.1088/0266-5611/18/4/310.

[17]

J.-Y. Lee, A reconstruction formula and uniqueness of conductivity in MREIT using two internal current distributions, Inverse Problems, 20 (2004), 847-858.  doi: 10.1088/0266-5611/20/3/012.

[18]

F. Monard and G. Bal, Inverse anisotropic diffusion from power density measurements in two dimensions Inverse Problems, 28 (2012), 084001, 20pp. doi: 10.1088/0266-5611/28/8/084001.

[19]

F. Monard and G. Bal, Inverse anisotropic conductivity from power densities in dimension $n≥q3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207.  doi: 10.1080/03605302.2013.787089.

[20]

C. Montalto and Stefanov, Stability of coupled-physics inverse problems with one internal measurement Inverse Problems, 29 (2013), 125004, 13pp. doi: 10.1088/0266-5611/29/12/125004.

[21]

A. Moradifam, A. Nachman and A. Tamasan, Uniqueness of minimizers of weighted least gradient problems arising in conductivity imaging, preprint, 2014.

[22]

A. NachmanA. 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.

[23]

A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data Inverse Problems 25 (2009), 035014, 16pp. doi: 10.1088/0266-5611/25/3/035014.

[24]

A. NachmanA. Tamasan and A. Timonov, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM J. Appl. Math., 70 (2010), 3342-3362.  doi: 10.1137/10079241X.

[25]

A. Nachman, A. Tamasan and A. Timonov, Current density impedance imaging, Tomography and inverse transport theory, vol. 559 of Contem Math., Amer. Math. Soc., Providence, RI, 2011,135-149. doi: 10.1090/conm/559/11076.

[26]

M. Z. Nashed and A. Tamasan, Structural stability in a minimizationproblem and applications to conductivity imaging, Inverse Probl. Imaging, 5 (2011), 219-236.  doi: 10.3934/ipi.2011.5.219.

[27]

G. C. ScottM. L. G. JoyR. L. Armstrong and R. M. Henkelman, Measurement of nonuniform current density by magnetic resonance, IEEE Transactions on Medical Imaging, 10 (1991), 362-374.  doi: 10.1109/42.97586.

[28]

J. K. Seo and E. J. Woo, Magnetic resonance electrical impedance tomography (MREIT), SIAM Rev., 53 (2011), 40-68.  doi: 10.1137/080742932.

[29]

A. Tamasan and A. Timonov, Coupled physics electrical conductivity imaging, Eurasian J. Math. Com Appl., 2 (2014), 5-29. 

[30]

A. TamasanA. Timonov and J. Veras, Stable reconstruction of regular 1-harmonic maps with a given trace at the boundary, Appl. Anal, 94 (2015), 1098-1115.  doi: 10.1080/00036811.2014.918260.

[31]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth, Heidelberg, 2nd ed., 1995.

[32]

T. Widlak and O. Scherzer, Hybrid tomography for conductivity imaging Inverse Problems, 28 (2012), 084008, 28pp. doi: 10.1088/0266-5611/28/8/084008.

[33]

N. Zhang, Electrical impedance tomography based on current density imaging, MSc. thesis, University of Toronto, 1992.

Figure 2.  The injectivity region $\mathcal{I}(\Gamma,u)$ is the light grey region that contains the stability region $\mathcal{S}(\Gamma,u)$ in dark grey.
Figure 1.  Illustration of the segment $l_p$ and the surface $\Sigma_p$.
Figure 3.  When $\Gamma$ is connected the visible region and the trajectory region can be the same.
Figure 4.  We illustrate how we can controlled the visible region and the projection of the current density. The underlying idea is to chose voltages potential $f$ at the boundary to induced specific level curves $u_0 =$const. By doing so, we get a better understanding of the required direction of the current density to obtain an stable reconstruction. Here, $\delta \bf{J}_0$ denote $\bf{J}(\sigma) - \bf{J}(\sigma_0)$ and ${{\bf{w}}_{0}} = \Pi_{\nabla (u_0)}(\delta \bf{J}_0) $.
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