2014, 8(2): 561-586. doi: 10.3934/ipi.2014.8.561

Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography

1. 

University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä

2. 

University of Eastern Finland, Department of Applied Physics, Yliopistonranta 1 F, FI-70211 Kuopio, Finland

3. 

Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu

Received  April 2013 Revised  February 2014 Published  May 2014

We study flexible and proper smoothness priors for Bayesian statistical inverse problems by using Whittle-Matérn Gaussian random fields. We review earlier results on finite-difference approximations of certain Whittle-Matérn random field in $\mathbb{R}^2$. Then we derive finite-element method approximations and show that the discrete approximations can be expressed as solutions of sparse stochastic matrix equations. Such equations are known to be computationally efficient and useful in inverse problems with a large number of unknowns.
    The presented construction of Whittle-Matérn correlation functions allows both isotropic or anisotropic priors with adjustable parameters in correlation length and variance. These parameters can be used, for example, to model spatially varying structural information of unknowns.
    As numerical examples, we apply the developed priors to two-dimensional electrical impedance tomography problems.
Citation: Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561
References:
[1]

D. C. Barber and B. H. Brown, Applied potential tomography,, J. Phys. E: Sci. Instrum., 17 (1984), 723. doi: 10.1088/0022-3735/17/9/002.

[2]

P. Bettess, Infinite elements,, International Journal for Numerical Methods in Engineering, 11 (1977), 53. doi: 10.1002/nme.1620110107.

[3]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613.

[4]

K-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Trans. Biomed. Eng., 36 (1989), 918.

[5]

I. M. Gelfand and G. E. Shilov, Generalized Functions. Vol. 1. Properties and Operations,, Academic Press, (1964).

[6]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,, Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. With one CD-ROM (Windows, (2007).

[7]

I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes I,, Springer-Verlag, (2004).

[8]

T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems,, Inverse Problems and Imaging, 3 (2009), 567. doi: 10.3934/ipi.2009.3.567.

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (2013).

[10]

T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,, With the collaboration of Robert M. Ferencz and Arthur M. Raefsky. Prentice Hall, (1987).

[11]

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.

[12]

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

[13]

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

[14]

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

[15]

S. Lasanen, Discretizations of generalized random variables with applications to inverse problems,, Ph.D thesis, 130 (2002).

[16]

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.

[17]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?,, Inverse Problems, 20 (2004), 1537. doi: 10.1088/0266-5611/20/5/013.

[18]

F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian Markov random fields: The stochastic partial differential equation approach,, Journal of the Royal Statistical Society: Series B, 73 (2011), 423. doi: 10.1111/j.1467-9868.2011.00777.x.

[19]

B. Matérn, Spatial Variation. Vol. 36 of Lecture Notes in Statistics,, $2^{nd}$ edition, (1986).

[20]

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning,, Adaptive Computation and Machine Learning, (2006).

[21]

L. Roininen, M. Lehtinen, S. Lasanen, M. Orispää and M. Markkanen, Correlation priors,, Inverse Problems and Imaging, 5 (2011), 167. doi: 10.3934/ipi.2011.5.167.

[22]

L. Roininen, P. Piiroinen and M. Lehtinen, Constructing Continuous Stationary Covariances as Limits of the Second-Order Stochastic Difference Equations,, Inverse Problems and Imaging, 7 (2013), 611. doi: 10.3934/ipi.2013.7.611.

[23]

Yu. A. Rozanov, Markov Random Fields,, Springer-Verlag, (1982).

[24]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,, SIAM Journal on Scientific and Statistical Computing, 7 (1986), 856. doi: 10.1137/0907058.

[25]

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.

[26]

P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Three-dimensional electrical impedance tomography based on the complete electrode model,, IEEE Transactions on Biomedical Engineering, 46 (1999), 1150. doi: 10.1109/10.784147.

[27]

P. Whittle, Stochastic processes in several dimensions,, Bull. Inst. Int. Statist., 40 (1963), 974.

show all references

References:
[1]

D. C. Barber and B. H. Brown, Applied potential tomography,, J. Phys. E: Sci. Instrum., 17 (1984), 723. doi: 10.1088/0022-3735/17/9/002.

[2]

P. Bettess, Infinite elements,, International Journal for Numerical Methods in Engineering, 11 (1977), 53. doi: 10.1002/nme.1620110107.

[3]

M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613.

[4]

K-S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Trans. Biomed. Eng., 36 (1989), 918.

[5]

I. M. Gelfand and G. E. Shilov, Generalized Functions. Vol. 1. Properties and Operations,, Academic Press, (1964).

[6]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,, Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. With one CD-ROM (Windows, (2007).

[7]

I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes I,, Springer-Verlag, (2004).

[8]

T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems,, Inverse Problems and Imaging, 3 (2009), 567. doi: 10.3934/ipi.2009.3.567.

[9]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (2013).

[10]

T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,, With the collaboration of Robert M. Ferencz and Arthur M. Raefsky. Prentice Hall, (1987).

[11]

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.

[12]

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

[13]

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

[14]

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

[15]

S. Lasanen, Discretizations of generalized random variables with applications to inverse problems,, Ph.D thesis, 130 (2002).

[16]

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.

[17]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?,, Inverse Problems, 20 (2004), 1537. doi: 10.1088/0266-5611/20/5/013.

[18]

F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian Markov random fields: The stochastic partial differential equation approach,, Journal of the Royal Statistical Society: Series B, 73 (2011), 423. doi: 10.1111/j.1467-9868.2011.00777.x.

[19]

B. Matérn, Spatial Variation. Vol. 36 of Lecture Notes in Statistics,, $2^{nd}$ edition, (1986).

[20]

C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning,, Adaptive Computation and Machine Learning, (2006).

[21]

L. Roininen, M. Lehtinen, S. Lasanen, M. Orispää and M. Markkanen, Correlation priors,, Inverse Problems and Imaging, 5 (2011), 167. doi: 10.3934/ipi.2011.5.167.

[22]

L. Roininen, P. Piiroinen and M. Lehtinen, Constructing Continuous Stationary Covariances as Limits of the Second-Order Stochastic Difference Equations,, Inverse Problems and Imaging, 7 (2013), 611. doi: 10.3934/ipi.2013.7.611.

[23]

Yu. A. Rozanov, Markov Random Fields,, Springer-Verlag, (1982).

[24]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,, SIAM Journal on Scientific and Statistical Computing, 7 (1986), 856. doi: 10.1137/0907058.

[25]

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.

[26]

P. J. Vauhkonen, M. Vauhkonen, T. Savolainen and J. P. Kaipio, Three-dimensional electrical impedance tomography based on the complete electrode model,, IEEE Transactions on Biomedical Engineering, 46 (1999), 1150. doi: 10.1109/10.784147.

[27]

P. Whittle, Stochastic processes in several dimensions,, Bull. Inst. Int. Statist., 40 (1963), 974.

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