2016, 10(4): 1007-1036. doi: 10.3934/ipi.2016030

The Bayesian formulation of EIT: Analysis and algorithms

1. 

Computing & Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, United States, United States

Received  August 2015 Revised  July 2016 Published  October 2016

We provide a rigorous Bayesian formulation of the EIT problem in an infinite dimensional setting, leading to well-posedness in the Hellinger metric with respect to the data. We focus particularly on the reconstruction of binary fields where the interface between different media is the primary unknown. We consider three different prior models -- log-Gaussian, star-shaped and level set. Numerical simulations based on the implementation of MCMC are performed, illustrating the advantages and disadvantages of each type of prior in the reconstruction, in the case where the true conductivity is a binary field, and exhibiting the properties of the resulting posterior distribution.
Citation: Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems & Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030
References:
[1]

A. Adler and W. R. B. Lionheart, Uses and abuses of EIDORS: An extensible software base for EIT,, Physiological Measurement, 27 (2006). doi: 10.1088/0967-3334/27/5/S03.

[2]

G. Alessandrini, Stable determination of an inclusion by boundary measurements,, Applicable Analysis: An International Journal, 27 (1988), 153. doi: 10.1080/00036818808839730.

[3]

M. Bédard, Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234,, Stochastic Processes and their Applications, 118 (2008), 2198. doi: 10.1016/j.spa.2007.12.005.

[4]

A. Beskos, A. Jasra, E. A. Muzaffer and A. M. Stuart, Sequential Monte Carlo methods for Bayesian elliptic inverse problems,, Statistics and Computing, 25 (2015), 727. doi: 10.1007/s11222-015-9556-7.

[5]

A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges,, Stochastics and Dynamics, 8 (2008), 319. doi: 10.1142/S0219493708002378.

[6]

V. I. Bogachev, Measure Theory Volume I,, Springer, (2007). doi: 10.1007/978-3-540-34514-5.

[7]

L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). doi: 10.1088/0266-5611/18/6/201.

[8]

T. Bui-Thanh and O. Ghattas, An analysis of infinite dimensional bayesian inverse shape acoustic scattering and its numerical approximation,, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 203. doi: 10.1137/120894877.

[9]

S. L. Cotter, G. O. Roberts, A. M. Stuart and D. White, MCMC methods for functions modifying old algorithms to make them faster,, Statistical Science, 28 (2013), 424. doi: 10.1214/13-STS421.

[10]

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems,, In Handbook of Uncertainty Quantification, (2016), 1. doi: 10.1007/978-3-319-11259-6_7-1.

[11]

S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid monte carlo,, Physics Letters B, 195 (1987), 216. doi: 10.1016/0370-2693(87)91197-X.

[12]

J. N. Franklin, Well posed stochastic extensions of ill posed linear problems},, Journal of Mathematical Analysis and Applications, 31 (1970), 682. doi: 10.1016/0022-247X(70)90017-X.

[13]

M. Gehre, B. Jin and X. Lu, An analysis of finite element approximation in electrical impedance tomography,, Inverse Problems, 30 (2014), 0266. doi: 10.1088/0266-5611/30/4/045013.

[14]

M. Hairer, A. M. Stuart and S. J. Vollmer, Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions,, The Annals of Applied Probability, 24 (2014), 2455. doi: 10.1214/13-AAP982.

[15]

R. P. Henderson and J. G. Webster, An impedance camera for spatially specific measurements of the thorax,, IEEE Transactions on Bio-Medical Engineering, 25 (1978), 250. doi: 10.1109/TBME.1978.326329.

[16]

M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems,, Inverse Problems, 29 (2013), 0266. doi: 10.1088/0266-5611/29/4/045001.

[17]

M. A. Iglesias, Y. Lu and A. M. Stuart, A bayesian level set method for geometric inverse problems,, Interfaces Free Bound., 18 (2016), 181. doi: 10.4171/IFB/362.

[18]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography,, Inverse Problems, 16 (2000), 1487. doi: 10.1088/0266-5611/16/5/321.

[19]

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information,, Inverse Problems, 15 (1999), 713. doi: 10.1088/0266-5611/15/3/306.

[20]

J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, Springer, (2005).

[21]

R. E. Kass, Markov Chain Monte Carlo in practice: A roundtable discussion,, The American Statistician, 52 (1998), 93. doi: 10.2307/2685466.

[22]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data on manifolds and applications,, Analysis and PDE, 6 (2013), 2003. doi: 10.2140/apde.2013.6.2003.

[23]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities,, SIAM Journal on Applied Mathematics, 67 (2007), 893. doi: 10.1137/060656930.

[24]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography,, Journal of Physics: Conference Series, 124 (2008). doi: 10.1088/1742-6596/124/1/012029.

[25]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599.

[26]

V. Kolehmainen, M. Lassas, P. Ola and S. Siltanen, Recovering boundary shape and conductivity in electrical impedance tomography,, Inverse Problems and Imaging, 7 (2013), 217. doi: 10.3934/ipi.2013.7.217.

[27]

S. Lan, T. Bui-Thanh, M. Christie and M. Girolami, Emulation of Higher-Order Tensors in Manifold Monte Carlo Methods for Bayesian Inverse Problems,, Journal of Computational Physics, 308 (2016), 81. doi: 10.1016/j.jcp.2015.12.032.

[28]

R. E. Langer, An inverse problem in differential equations,, Bulletin of the American Mathematical Society, 39 (1933), 814. doi: 10.1090/S0002-9904-1933-05752-X.

[29]

S. Lasanen, Non-gaussian statistical inverse problems. part I: Posterior distributions,, Inverse Problems & Imaging, 6 (2012), 215. doi: 10.3934/ipi.2012.6.215.

[30]

S. Lasanen, Non-gaussian statistical inverse problems. part II: Posterior convergence for approximated unknowns,, Inverse Problems & Imaging, 6 (2012), 267. doi: 10.3934/ipi.2012.6.267.

[31]

S. Lasanen, J. M. J. Huttunen and L. Roininen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography,, Inverse Problems and Imaging, 8 (2014), 561. doi: 10.3934/ipi.2014.8.561.

[32]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors,, Inverse Problems and Imaging, 3 (2009), 87. doi: 10.3934/ipi.2009.3.87.

[33]

M. S. Lehtinen, L. Paivarinta and E. Somersalo, Linear inverse problems for generalised random variables,, Inverse Problems, 5 (1989), 599. doi: 10.1088/0266-5611/5/4/011.

[34]

A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space,, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65 (1984), 385. doi: 10.1007/BF00533743.

[35]

A. I. Nachman, Reconstructions from boundary measurements,, Annals of Mathematics, 128 (1988), 531. doi: 10.2307/1971435.

[36]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Annals of Mathematics, 143 (1996), 71. doi: 10.2307/2118653.

[37]

G. C. Papanicolaou and L. Borcea, Network approximation for transport properties of high contrast materials,, SIAM Journal on Applied Mathematics, 58 (1998), 501. doi: 10.1137/S0036139996301891.

[38]

M. Salo, Calderón problem,, Lecture Notes., ().

[39]

D. Schymura, An upper bound on the volume of the symmetric difference of a body and a congruent copy,, Advances in Geometry, 14 (2014), 287. doi: 10.1515/advgeom-2013-0029.

[40]

E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM Journal on Applied Mathematics, 52 (1992), 1023. doi: 10.1137/0152060.

[41]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numerica, 19 (2010), 451. doi: 10.1017/S0962492910000061.

[42]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value,, Annals of Mathematics, 125 (1987), 153. doi: 10.2307/1971291.

[43]

L. Tierney, Markov chains for exploring posterior distributions,, Annals of Statistics, 22 (1994), 1701. doi: 10.1214/aos/1176325750.

show all references

References:
[1]

A. Adler and W. R. B. Lionheart, Uses and abuses of EIDORS: An extensible software base for EIT,, Physiological Measurement, 27 (2006). doi: 10.1088/0967-3334/27/5/S03.

[2]

G. Alessandrini, Stable determination of an inclusion by boundary measurements,, Applicable Analysis: An International Journal, 27 (1988), 153. doi: 10.1080/00036818808839730.

[3]

M. Bédard, Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234,, Stochastic Processes and their Applications, 118 (2008), 2198. doi: 10.1016/j.spa.2007.12.005.

[4]

A. Beskos, A. Jasra, E. A. Muzaffer and A. M. Stuart, Sequential Monte Carlo methods for Bayesian elliptic inverse problems,, Statistics and Computing, 25 (2015), 727. doi: 10.1007/s11222-015-9556-7.

[5]

A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges,, Stochastics and Dynamics, 8 (2008), 319. doi: 10.1142/S0219493708002378.

[6]

V. I. Bogachev, Measure Theory Volume I,, Springer, (2007). doi: 10.1007/978-3-540-34514-5.

[7]

L. Borcea, Electrical impedance tomography,, Inverse Problems, 18 (2002). doi: 10.1088/0266-5611/18/6/201.

[8]

T. Bui-Thanh and O. Ghattas, An analysis of infinite dimensional bayesian inverse shape acoustic scattering and its numerical approximation,, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 203. doi: 10.1137/120894877.

[9]

S. L. Cotter, G. O. Roberts, A. M. Stuart and D. White, MCMC methods for functions modifying old algorithms to make them faster,, Statistical Science, 28 (2013), 424. doi: 10.1214/13-STS421.

[10]

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems,, In Handbook of Uncertainty Quantification, (2016), 1. doi: 10.1007/978-3-319-11259-6_7-1.

[11]

S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid monte carlo,, Physics Letters B, 195 (1987), 216. doi: 10.1016/0370-2693(87)91197-X.

[12]

J. N. Franklin, Well posed stochastic extensions of ill posed linear problems},, Journal of Mathematical Analysis and Applications, 31 (1970), 682. doi: 10.1016/0022-247X(70)90017-X.

[13]

M. Gehre, B. Jin and X. Lu, An analysis of finite element approximation in electrical impedance tomography,, Inverse Problems, 30 (2014), 0266. doi: 10.1088/0266-5611/30/4/045013.

[14]

M. Hairer, A. M. Stuart and S. J. Vollmer, Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions,, The Annals of Applied Probability, 24 (2014), 2455. doi: 10.1214/13-AAP982.

[15]

R. P. Henderson and J. G. Webster, An impedance camera for spatially specific measurements of the thorax,, IEEE Transactions on Bio-Medical Engineering, 25 (1978), 250. doi: 10.1109/TBME.1978.326329.

[16]

M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems,, Inverse Problems, 29 (2013), 0266. doi: 10.1088/0266-5611/29/4/045001.

[17]

M. A. Iglesias, Y. Lu and A. M. Stuart, A bayesian level set method for geometric inverse problems,, Interfaces Free Bound., 18 (2016), 181. doi: 10.4171/IFB/362.

[18]

J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography,, Inverse Problems, 16 (2000), 1487. doi: 10.1088/0266-5611/16/5/321.

[19]

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information,, Inverse Problems, 15 (1999), 713. doi: 10.1088/0266-5611/15/3/306.

[20]

J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, Springer, (2005).

[21]

R. E. Kass, Markov Chain Monte Carlo in practice: A roundtable discussion,, The American Statistician, 52 (1998), 93. doi: 10.2307/2685466.

[22]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data on manifolds and applications,, Analysis and PDE, 6 (2013), 2003. doi: 10.2140/apde.2013.6.2003.

[23]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities,, SIAM Journal on Applied Mathematics, 67 (2007), 893. doi: 10.1137/060656930.

[24]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography,, Journal of Physics: Conference Series, 124 (2008). doi: 10.1088/1742-6596/124/1/012029.

[25]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599.

[26]

V. Kolehmainen, M. Lassas, P. Ola and S. Siltanen, Recovering boundary shape and conductivity in electrical impedance tomography,, Inverse Problems and Imaging, 7 (2013), 217. doi: 10.3934/ipi.2013.7.217.

[27]

S. Lan, T. Bui-Thanh, M. Christie and M. Girolami, Emulation of Higher-Order Tensors in Manifold Monte Carlo Methods for Bayesian Inverse Problems,, Journal of Computational Physics, 308 (2016), 81. doi: 10.1016/j.jcp.2015.12.032.

[28]

R. E. Langer, An inverse problem in differential equations,, Bulletin of the American Mathematical Society, 39 (1933), 814. doi: 10.1090/S0002-9904-1933-05752-X.

[29]

S. Lasanen, Non-gaussian statistical inverse problems. part I: Posterior distributions,, Inverse Problems & Imaging, 6 (2012), 215. doi: 10.3934/ipi.2012.6.215.

[30]

S. Lasanen, Non-gaussian statistical inverse problems. part II: Posterior convergence for approximated unknowns,, Inverse Problems & Imaging, 6 (2012), 267. doi: 10.3934/ipi.2012.6.267.

[31]

S. Lasanen, J. M. J. Huttunen and L. Roininen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography,, Inverse Problems and Imaging, 8 (2014), 561. doi: 10.3934/ipi.2014.8.561.

[32]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors,, Inverse Problems and Imaging, 3 (2009), 87. doi: 10.3934/ipi.2009.3.87.

[33]

M. S. Lehtinen, L. Paivarinta and E. Somersalo, Linear inverse problems for generalised random variables,, Inverse Problems, 5 (1989), 599. doi: 10.1088/0266-5611/5/4/011.

[34]

A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space,, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65 (1984), 385. doi: 10.1007/BF00533743.

[35]

A. I. Nachman, Reconstructions from boundary measurements,, Annals of Mathematics, 128 (1988), 531. doi: 10.2307/1971435.

[36]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Annals of Mathematics, 143 (1996), 71. doi: 10.2307/2118653.

[37]

G. C. Papanicolaou and L. Borcea, Network approximation for transport properties of high contrast materials,, SIAM Journal on Applied Mathematics, 58 (1998), 501. doi: 10.1137/S0036139996301891.

[38]

M. Salo, Calderón problem,, Lecture Notes., ().

[39]

D. Schymura, An upper bound on the volume of the symmetric difference of a body and a congruent copy,, Advances in Geometry, 14 (2014), 287. doi: 10.1515/advgeom-2013-0029.

[40]

E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM Journal on Applied Mathematics, 52 (1992), 1023. doi: 10.1137/0152060.

[41]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numerica, 19 (2010), 451. doi: 10.1017/S0962492910000061.

[42]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value,, Annals of Mathematics, 125 (1987), 153. doi: 10.2307/1971291.

[43]

L. Tierney, Markov chains for exploring posterior distributions,, Annals of Statistics, 22 (1994), 1701. doi: 10.1214/aos/1176325750.

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