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

The Bayesian formulation of EIT: Analysis and algorithms

Matthew M. Dunlop 1, and  Andrew M. Stuart 1,

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.
Keywords: Inverse problems, Bayesian regularisation, level set methodology., ill-posed problems, electrical impedance tomography, geometric priors.
Mathematics Subject Classification: Primary: 62G05, 65N21; Secondary: 92C5.
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), S25-S42. doi: 10.1088/0967-3334/27/5/S03.  Google Scholar

[2]

G. Alessandrini, Stable determination of an inclusion by boundary measurements, Applicable Analysis: An International Journal, 27 (1988), 153-172. doi: 10.1080/00036818808839730.  Google Scholar

[3]

M. Bédard, Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234, Stochastic Processes and their Applications, 118 (2008), 2198-2222. doi: 10.1016/j.spa.2007.12.005.  Google Scholar

[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), 727-737. doi: 10.1007/s11222-015-9556-7.  Google Scholar

[5]

A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges, Stochastics and Dynamics, 8 (2008), 319-350. doi: 10.1142/S0219493708002378.  Google Scholar

[6]

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

[7]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.  Google Scholar

[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), 203-222. doi: 10.1137/120894877.  Google Scholar

[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), 424-446. doi: 10.1214/13-STS421.  Google Scholar

[10]

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems, In Handbook of Uncertainty Quantification, Springer, 2016, 1-118. doi: 10.1007/978-3-319-11259-6_7-1.  Google Scholar

[11]

S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid monte carlo, Physics Letters B, 195 (1987), 216-222. doi: 10.1016/0370-2693(87)91197-X.  Google Scholar

[12]

J. N. Franklin, Well posed stochastic extensions of ill posed linear problems}, Journal of Mathematical Analysis and Applications, 31 (1970), 682-716, URL http://adsabs.harvard.edu/abs/1970JIMIA..31..682F. doi: 10.1016/0022-247X(70)90017-X.  Google Scholar

[13]

M. Gehre, B. Jin and X. Lu, An analysis of finite element approximation in electrical impedance tomography, Inverse Problems, 30 (2014), 045013, 24pp, URL http://stacks.iop.org/0266-5611/30/i=4/a=045013?key=crossref.284cda5f9645598cb83fccc9d226940a. doi: 10.1088/0266-5611/30/4/045013.  Google Scholar

[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), 2455-2490. doi: 10.1214/13-AAP982.  Google Scholar

[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), 250-254. doi: 10.1109/TBME.1978.326329.  Google Scholar

[16]

M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp, URL http://stacks.iop.org/0266-5611/29/i=4/a=045001?key=crossref.a70b58896eed5588086da7ad3d04954b. doi: 10.1088/0266-5611/29/4/045001.  Google Scholar

[17]

M. A. Iglesias, Y. Lu and A. M. Stuart, A bayesian level set method for geometric inverse problems, Interfaces Free Bound., 18 (2016), 181-217, URL http://arxiv.org/abs/1504.00313. doi: 10.4171/IFB/362.  Google Scholar

[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), 1487-1522. doi: 10.1088/0266-5611/16/5/321.  Google Scholar

[19]

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729. doi: 10.1088/0266-5611/15/3/306.  Google Scholar

[20]

J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005.  Google Scholar

[21]

R. E. Kass, Markov Chain Monte Carlo in practice: A roundtable discussion, The American Statistician, 52 (1998), 93-100. doi: 10.2307/2685466.  Google Scholar

[22]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data on manifolds and applications, Analysis and PDE, 6 (2013), 2003-2048. doi: 10.2140/apde.2013.6.2003.  Google Scholar

[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), 893-913. doi: 10.1137/060656930.  Google Scholar

[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), 012029. doi: 10.1088/1742-6596/124/1/012029.  Google Scholar

[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), 599-624. doi: 10.3934/ipi.2009.3.599.  Google Scholar

[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), 217-242. doi: 10.3934/ipi.2013.7.217.  Google Scholar

[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), 81-101. doi: 10.1016/j.jcp.2015.12.032.  Google Scholar

[28]

R. E. Langer, An inverse problem in differential equations, Bulletin of the American Mathematical Society, 39 (1933), 814-820. doi: 10.1090/S0002-9904-1933-05752-X.  Google Scholar

[29]

S. Lasanen, Non-gaussian statistical inverse problems. part I: Posterior distributions, Inverse Problems & Imaging, 6 (2012), 215-266. doi: 10.3934/ipi.2012.6.215.  Google Scholar

[30]

S. Lasanen, Non-gaussian statistical inverse problems. part II: Posterior convergence for approximated unknowns, Inverse Problems & Imaging, 6 (2012), 267-287. doi: 10.3934/ipi.2012.6.267.  Google Scholar

[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), 561-586, URL http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=9912. doi: 10.3934/ipi.2014.8.561.  Google Scholar

[32]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122. doi: 10.3934/ipi.2009.3.87.  Google Scholar

[33]

M. S. Lehtinen, L. Paivarinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612. doi: 10.1088/0266-5611/5/4/011.  Google Scholar

[34]

A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65 (1984), 385-397. doi: 10.1007/BF00533743.  Google Scholar

[35]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576, URL http://cat.inist.fr/?aModele=afficheN&cpsidt=7131706. doi: 10.2307/1971435.  Google Scholar

[36]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[37]

G. C. Papanicolaou and L. Borcea, Network approximation for transport properties of high contrast materials, SIAM Journal on Applied Mathematics, 58 (1998), 501-539. doi: 10.1137/S0036139996301891.  Google Scholar

[38]

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

[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), 287-298. doi: 10.1515/advgeom-2013-0029.  Google Scholar

[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), 1023-1040. doi: 10.1137/0152060.  Google Scholar

[41]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559, URL http://webcat.warwick.ac.uk:80/record=b2341209. doi: 10.1017/S0962492910000061.  Google Scholar

[42]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, Annals of Mathematics, 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[43]

L. Tierney, Markov chains for exploring posterior distributions, Annals of Statistics, 22 (1994), 1701-1728. doi: 10.1214/aos/1176325750.  Google Scholar

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), S25-S42. doi: 10.1088/0967-3334/27/5/S03.  Google Scholar

[2]

G. Alessandrini, Stable determination of an inclusion by boundary measurements, Applicable Analysis: An International Journal, 27 (1988), 153-172. doi: 10.1080/00036818808839730.  Google Scholar

[3]

M. Bédard, Optimal acceptance rates for Metropolis algorithms: Moving beyond 0.234, Stochastic Processes and their Applications, 118 (2008), 2198-2222. doi: 10.1016/j.spa.2007.12.005.  Google Scholar

[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), 727-737. doi: 10.1007/s11222-015-9556-7.  Google Scholar

[5]

A. Beskos, G. O. Roberts, A. M. Stuart and J. Voss, MCMC methods for diffusion bridges, Stochastics and Dynamics, 8 (2008), 319-350. doi: 10.1142/S0219493708002378.  Google Scholar

[6]

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

[7]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.  Google Scholar

[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), 203-222. doi: 10.1137/120894877.  Google Scholar

[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), 424-446. doi: 10.1214/13-STS421.  Google Scholar

[10]

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems, In Handbook of Uncertainty Quantification, Springer, 2016, 1-118. doi: 10.1007/978-3-319-11259-6_7-1.  Google Scholar

[11]

S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid monte carlo, Physics Letters B, 195 (1987), 216-222. doi: 10.1016/0370-2693(87)91197-X.  Google Scholar

[12]

J. N. Franklin, Well posed stochastic extensions of ill posed linear problems}, Journal of Mathematical Analysis and Applications, 31 (1970), 682-716, URL http://adsabs.harvard.edu/abs/1970JIMIA..31..682F. doi: 10.1016/0022-247X(70)90017-X.  Google Scholar

[13]

M. Gehre, B. Jin and X. Lu, An analysis of finite element approximation in electrical impedance tomography, Inverse Problems, 30 (2014), 045013, 24pp, URL http://stacks.iop.org/0266-5611/30/i=4/a=045013?key=crossref.284cda5f9645598cb83fccc9d226940a. doi: 10.1088/0266-5611/30/4/045013.  Google Scholar

[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), 2455-2490. doi: 10.1214/13-AAP982.  Google Scholar

[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), 250-254. doi: 10.1109/TBME.1978.326329.  Google Scholar

[16]

M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp, URL http://stacks.iop.org/0266-5611/29/i=4/a=045001?key=crossref.a70b58896eed5588086da7ad3d04954b. doi: 10.1088/0266-5611/29/4/045001.  Google Scholar

[17]

M. A. Iglesias, Y. Lu and A. M. Stuart, A bayesian level set method for geometric inverse problems, Interfaces Free Bound., 18 (2016), 181-217, URL http://arxiv.org/abs/1504.00313. doi: 10.4171/IFB/362.  Google Scholar

[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), 1487-1522. doi: 10.1088/0266-5611/16/5/321.  Google Scholar

[19]

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information, Inverse Problems, 15 (1999), 713-729. doi: 10.1088/0266-5611/15/3/306.  Google Scholar

[20]

J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005.  Google Scholar

[21]

R. E. Kass, Markov Chain Monte Carlo in practice: A roundtable discussion, The American Statistician, 52 (1998), 93-100. doi: 10.2307/2685466.  Google Scholar

[22]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data on manifolds and applications, Analysis and PDE, 6 (2013), 2003-2048. doi: 10.2140/apde.2013.6.2003.  Google Scholar

[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), 893-913. doi: 10.1137/060656930.  Google Scholar

[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), 012029. doi: 10.1088/1742-6596/124/1/012029.  Google Scholar

[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), 599-624. doi: 10.3934/ipi.2009.3.599.  Google Scholar

[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), 217-242. doi: 10.3934/ipi.2013.7.217.  Google Scholar

[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), 81-101. doi: 10.1016/j.jcp.2015.12.032.  Google Scholar

[28]

R. E. Langer, An inverse problem in differential equations, Bulletin of the American Mathematical Society, 39 (1933), 814-820. doi: 10.1090/S0002-9904-1933-05752-X.  Google Scholar

[29]

S. Lasanen, Non-gaussian statistical inverse problems. part I: Posterior distributions, Inverse Problems & Imaging, 6 (2012), 215-266. doi: 10.3934/ipi.2012.6.215.  Google Scholar

[30]

S. Lasanen, Non-gaussian statistical inverse problems. part II: Posterior convergence for approximated unknowns, Inverse Problems & Imaging, 6 (2012), 267-287. doi: 10.3934/ipi.2012.6.267.  Google Scholar

[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), 561-586, URL http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=9912. doi: 10.3934/ipi.2014.8.561.  Google Scholar

[32]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122. doi: 10.3934/ipi.2009.3.87.  Google Scholar

[33]

M. S. Lehtinen, L. Paivarinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612. doi: 10.1088/0266-5611/5/4/011.  Google Scholar

[34]

A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65 (1984), 385-397. doi: 10.1007/BF00533743.  Google Scholar

[35]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576, URL http://cat.inist.fr/?aModele=afficheN&cpsidt=7131706. doi: 10.2307/1971435.  Google Scholar

[36]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96. doi: 10.2307/2118653.  Google Scholar

[37]

G. C. Papanicolaou and L. Borcea, Network approximation for transport properties of high contrast materials, SIAM Journal on Applied Mathematics, 58 (1998), 501-539. doi: 10.1137/S0036139996301891.  Google Scholar

[38]

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

[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), 287-298. doi: 10.1515/advgeom-2013-0029.  Google Scholar

[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), 1023-1040. doi: 10.1137/0152060.  Google Scholar

[41]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559, URL http://webcat.warwick.ac.uk:80/record=b2341209. doi: 10.1017/S0962492910000061.  Google Scholar

[42]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, Annals of Mathematics, 125 (1987), 153-169. doi: 10.2307/1971291.  Google Scholar

[43]

L. Tierney, Markov chains for exploring posterior distributions, Annals of Statistics, 22 (1994), 1701-1728. doi: 10.1214/aos/1176325750.  Google Scholar

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