# American Institute of Mathematical Sciences

May  2013, 7(2): 397-416. doi: 10.3934/ipi.2013.7.397

## Gaussian Markov random field priors for inverse problems

 1 Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812, United States

Received  January 2012 Revised  March 2013 Published  May 2013

In this paper, our focus is on the connections between the methods of (quadratic) regularization for inverse problems and Gaussian Markov random field (GMRF) priors for problems in spatial statistics. We begin with the most standard GMRFs defined on a uniform computational grid, which correspond to the oft-used discrete negative-Laplacian regularization matrix. Next, we present a class of GMRFs that allow for the formation of edges in reconstructed images, and then draw concrete connections between these GMRFs and numerical discretizations of more general diffusion operators. The benefit of the GMRF interpretation of quadratic regularization is that a GMRF is built-up from concrete statistical assumptions about the values of the unknown at each pixel given the values of its neighbors. Thus the regularization term corresponds to a concrete spatial statistical model for the unknown, encapsulated in the prior. Throughout our discussion, strong ties between specific GMRFs, numerical discretizations of diffusion operators, and corresponding regularization matrices, are established. We then show how such GMRF priors can be used for edge-preserving reconstruction of images, in both image deblurring and medical imaging test cases. Moreover, we demonstrate the effectiveness of GMRF priors for data arising from both Gaussian and Poisson noise models.
Citation: Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems & Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397
##### References:
 [1] J. M. Bardsley, D. Calvetti and E. Somersalo, Hierarchical regularization for edge-preserving reconstruction of PET images,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/3/035010. Google Scholar [2] J. M. Bardsley and J. Goldes, An iterative method for edge-preserving MAP estimation when data-noise is poisson,, SIAM J. Sci. Comput., 32 (2010), 171. doi: 10.1137/080726884. Google Scholar [3] J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed poisson maximum likelihood estimation,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/9/095005. Google Scholar [4] J. M. Bardsley and C. R. Vogel, A nonnnegatively constrained convex programming method for image reconstruction,, SIAM J. Sci. Comput., 25 (2004), 1326. doi: 10.1137/S1064827502410451. Google Scholar [5] J. Besag, Spatial interaction and the statistical analysis of lattice systems,, J. R. Stat. Soc. Ser. B, 36 (1974), 192. Google Scholar [6] D. Calvetti and E. Somersalo, "Introduction to Bayesian Scientific Computing,", Springer, (2007). Google Scholar [7] D. Calvetti and E. Somersalo, A Gaussian hypermodel to recover blocky objects,, Inverse Problems, 23 (2007), 733. doi: 10.1088/0266-5611/23/2/016. Google Scholar [8] D. Calvetti and E. Somersalo, Hypermodels in the bayesian imaging framework,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/034013. Google Scholar [9] D. Calvetti, J. Kaipio and E. Somersalo, Aristotelian prior boundary conditions,, Intl. J. Math. Comp. Sci., 1 (2005), 63. Google Scholar [10] H. K. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Kluwer, (1996). Google Scholar [11] P. C. Hansen, "Discrete Inverse Problems: Insight and Algorithms,", SIAM, (2010). doi: 10.1137/1.9780898718836. Google Scholar [12] J. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information,, Inverse Problems, (1999), 713. doi: 10.1088/0266-5611/15/3/306. Google Scholar [13] J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005). Google Scholar [14] V. Kolehmainen, T. Tarvainen, S. R. Arridge and J. P. Kaipio, Marginalization of uninteresting distributed parameters in inverse problems-application to diffuse optical tomography,, Intl. J. Uncertainty Quantification, 1 (2011), 1. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.10. Google Scholar [15] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM, (2001). doi: 10.1137/1.9780898718324. Google Scholar [16] J. M. Ollinger and J. A. Fessler, Positron-emission tomography,, IEEE Signal Processing Magazine, (1997), 43. doi: 10.1109/79.560323. Google Scholar [17] H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications,", Chapman and Hall/CRC, (2005). doi: 10.1201/9780203492024. Google Scholar [18] C. R. Vogel, "Computational Methods for Inverse Problems,", SIAM, (2002). doi: 10.1137/1.9780898717570. Google Scholar [19] C. R. Vogel and M. E. Oman, A fast, robust algorithm for total variation based reconstruction of noisy, blurred images,, IEEE Trans. Image Process., 7 (1998), 813. doi: 10.1109/83.679423. Google Scholar [20] D. Watkins, "Fundamentals of Matrix Computations,", Wiley, (2010). Google Scholar

show all references

##### References:
 [1] J. M. Bardsley, D. Calvetti and E. Somersalo, Hierarchical regularization for edge-preserving reconstruction of PET images,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/3/035010. Google Scholar [2] J. M. Bardsley and J. Goldes, An iterative method for edge-preserving MAP estimation when data-noise is poisson,, SIAM J. Sci. Comput., 32 (2010), 171. doi: 10.1137/080726884. Google Scholar [3] J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed poisson maximum likelihood estimation,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/9/095005. Google Scholar [4] J. M. Bardsley and C. R. Vogel, A nonnnegatively constrained convex programming method for image reconstruction,, SIAM J. Sci. Comput., 25 (2004), 1326. doi: 10.1137/S1064827502410451. Google Scholar [5] J. Besag, Spatial interaction and the statistical analysis of lattice systems,, J. R. Stat. Soc. Ser. B, 36 (1974), 192. Google Scholar [6] D. Calvetti and E. Somersalo, "Introduction to Bayesian Scientific Computing,", Springer, (2007). Google Scholar [7] D. Calvetti and E. Somersalo, A Gaussian hypermodel to recover blocky objects,, Inverse Problems, 23 (2007), 733. doi: 10.1088/0266-5611/23/2/016. Google Scholar [8] D. Calvetti and E. Somersalo, Hypermodels in the bayesian imaging framework,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/3/034013. Google Scholar [9] D. Calvetti, J. Kaipio and E. Somersalo, Aristotelian prior boundary conditions,, Intl. J. Math. Comp. Sci., 1 (2005), 63. Google Scholar [10] H. K. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Kluwer, (1996). Google Scholar [11] P. C. Hansen, "Discrete Inverse Problems: Insight and Algorithms,", SIAM, (2010). doi: 10.1137/1.9780898718836. Google Scholar [12] J. Kaipio, V. Kolehmainen, M. Vauhkonen and E. Somersalo, Inverse problems with structural prior information,, Inverse Problems, (1999), 713. doi: 10.1088/0266-5611/15/3/306. Google Scholar [13] J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005). Google Scholar [14] V. Kolehmainen, T. Tarvainen, S. R. Arridge and J. P. Kaipio, Marginalization of uninteresting distributed parameters in inverse problems-application to diffuse optical tomography,, Intl. J. Uncertainty Quantification, 1 (2011), 1. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.10. Google Scholar [15] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM, (2001). doi: 10.1137/1.9780898718324. Google Scholar [16] J. M. Ollinger and J. A. Fessler, Positron-emission tomography,, IEEE Signal Processing Magazine, (1997), 43. doi: 10.1109/79.560323. Google Scholar [17] H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications,", Chapman and Hall/CRC, (2005). doi: 10.1201/9780203492024. Google Scholar [18] C. R. Vogel, "Computational Methods for Inverse Problems,", SIAM, (2002). doi: 10.1137/1.9780898717570. Google Scholar [19] C. R. Vogel and M. E. Oman, A fast, robust algorithm for total variation based reconstruction of noisy, blurred images,, IEEE Trans. Image Process., 7 (1998), 813. doi: 10.1109/83.679423. Google Scholar [20] D. Watkins, "Fundamentals of Matrix Computations,", Wiley, (2010). Google Scholar
 [1] Lacramioara Grecu, Constantin Popa. Constrained SART algorithm for inverse problems in image reconstruction. Inverse Problems & Imaging, 2013, 7 (1) : 199-216. doi: 10.3934/ipi.2013.7.199 [2] Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013 [3] Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183 [4] Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81 [5] Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271 [6] Vladimir Kazakov. Sampling - reconstruction procedure with jitter of markov continuous processes formed by stochastic differential equations of the first order. Conference Publications, 2009, 2009 (Special) : 433-441. doi: 10.3934/proc.2009.2009.433 [7] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [8] Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229 [9] Yunhai Xiao, Junfeng Yang, Xiaoming Yuan. Alternating algorithms for total variation image reconstruction from random projections. Inverse Problems & Imaging, 2012, 6 (3) : 547-563. doi: 10.3934/ipi.2012.6.547 [10] Evangelos Evangelou. Approximate Bayesian inference for geostatistical generalised linear models. Foundations of Data Science, 2019, 1 (1) : 39-60. doi: 10.3934/fods.2019002 [11] Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems & Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449 [12] Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems & Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27 [13] Kui Lin, Shuai Lu, Peter Mathé. Oracle-type posterior contraction rates in Bayesian inverse problems. Inverse Problems & Imaging, 2015, 9 (3) : 895-915. doi: 10.3934/ipi.2015.9.895 [14] Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 [15] Tom Goldstein, Xavier Bresson, Stan Osher. Global minimization of Markov random fields with applications to optical flow. Inverse Problems & Imaging, 2012, 6 (4) : 623-644. doi: 10.3934/ipi.2012.6.623 [16] Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428 [17] Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399 [18] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [19] Igor G. Vladimirov. The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 575-600. doi: 10.3934/dcdsb.2013.18.575 [20] Mikko Kaasalainen. Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction. Inverse Problems & Imaging, 2011, 5 (1) : 37-57. doi: 10.3934/ipi.2011.5.37

2018 Impact Factor: 1.469

## Metrics

• PDF downloads (8)
• HTML views (0)
• Cited by (7)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]