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 and 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), 035010. doi: 10.1088/0266-5611/26/3/035010.

[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), 171-185. doi: 10.1137/080726884.

[3]

J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed poisson maximum likelihood estimation, Inverse Problems, 25 (2009), 095005. doi: 10.1088/0266-5611/25/9/095005.

[4]

J. M. Bardsley and C. R. Vogel, A nonnnegatively constrained convex programming method for image reconstruction, SIAM J. Sci. Comput., 25 (2004), 1326-1343. doi: 10.1137/S1064827502410451.

[5]

J. Besag, Spatial interaction and the statistical analysis of lattice systems, J. R. Stat. Soc. Ser. B, 36 (1974), 192-236.

[6]

D. Calvetti and E. Somersalo, "Introduction to Bayesian Scientific Computing," Springer, New York, 2007.

[7]

D. Calvetti and E. Somersalo, A Gaussian hypermodel to recover blocky objects, Inverse Problems, 23 (2007), 733-754. doi: 10.1088/0266-5611/23/2/016.

[8]

D. Calvetti and E. Somersalo, Hypermodels in the bayesian imaging framework, Inverse Problems, 24 (2008), 034013. doi: 10.1088/0266-5611/24/3/034013.

[9]

D. Calvetti, J. Kaipio and E. Somersalo, Aristotelian prior boundary conditions, Intl. J. Math. Comp. Sci., 1 (2005), 63-81.

[10]

H. K. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Kluwer, The Netherlands, 1996.

[11]

P. C. Hansen, "Discrete Inverse Problems: Insight and Algorithms," SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898718836.

[12]

J. 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.

[13]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, New York, 2005.

[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), 1-17. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.10.

[15]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898718324.

[16]

J. M. Ollinger and J. A. Fessler, Positron-emission tomography, IEEE Signal Processing Magazine, (1997), 43-55. doi: 10.1109/79.560323.

[17]

H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications," Chapman and Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9780203492024.

[18]

C. R. Vogel, "Computational Methods for Inverse Problems," SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898717570.

[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), 813-824. doi: 10.1109/83.679423.

[20]

D. Watkins, "Fundamentals of Matrix Computations," Wiley, Hoboken, NJ, 2010.

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), 035010. doi: 10.1088/0266-5611/26/3/035010.

[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), 171-185. doi: 10.1137/080726884.

[3]

J. M. Bardsley and J. Goldes, Regularization parameter selection methods for ill-posed poisson maximum likelihood estimation, Inverse Problems, 25 (2009), 095005. doi: 10.1088/0266-5611/25/9/095005.

[4]

J. M. Bardsley and C. R. Vogel, A nonnnegatively constrained convex programming method for image reconstruction, SIAM J. Sci. Comput., 25 (2004), 1326-1343. doi: 10.1137/S1064827502410451.

[5]

J. Besag, Spatial interaction and the statistical analysis of lattice systems, J. R. Stat. Soc. Ser. B, 36 (1974), 192-236.

[6]

D. Calvetti and E. Somersalo, "Introduction to Bayesian Scientific Computing," Springer, New York, 2007.

[7]

D. Calvetti and E. Somersalo, A Gaussian hypermodel to recover blocky objects, Inverse Problems, 23 (2007), 733-754. doi: 10.1088/0266-5611/23/2/016.

[8]

D. Calvetti and E. Somersalo, Hypermodels in the bayesian imaging framework, Inverse Problems, 24 (2008), 034013. doi: 10.1088/0266-5611/24/3/034013.

[9]

D. Calvetti, J. Kaipio and E. Somersalo, Aristotelian prior boundary conditions, Intl. J. Math. Comp. Sci., 1 (2005), 63-81.

[10]

H. K. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Kluwer, The Netherlands, 1996.

[11]

P. C. Hansen, "Discrete Inverse Problems: Insight and Algorithms," SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898718836.

[12]

J. 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.

[13]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, New York, 2005.

[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), 1-17. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.10.

[15]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898718324.

[16]

J. M. Ollinger and J. A. Fessler, Positron-emission tomography, IEEE Signal Processing Magazine, (1997), 43-55. doi: 10.1109/79.560323.

[17]

H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications," Chapman and Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9780203492024.

[18]

C. R. Vogel, "Computational Methods for Inverse Problems," SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898717570.

[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), 813-824. doi: 10.1109/83.679423.

[20]

D. Watkins, "Fundamentals of Matrix Computations," Wiley, Hoboken, NJ, 2010.

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