Multigrid Regularized Image Reconstruction for Limited-Data Tomography
Institute of Telecommunications, Teleinformatics and Acoustics
Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland
e-mail: Rafal.Zdunek@pwr.wroc.pl
Received:
Rec. January 11, 2007
DOI: 10.12921/cmst.2007.13.01.67-77
OAI: oai:lib.psnc.pl:633
Abstract:
Limited-data tomography, to which electromagnetic geotomography belongs, is analyzed in this paper. In this technique, a discrete forward projection model may be expressed by a rank-deficient system of linear equations whose the nullspace is non-trivial. This means that some image components may fall into the nullspace, and hence the minimal-norm least-square solution, to which many image reconstructions methods converge, may be different from the true one. The Algebraic Reconstruction Technique (ART), Simultaneous Iterative Reconstruction Technique (SIRT), or Conjugate Gradients Least Squares (CGLS) are examples of such methods. In this paper, we deal with the question of how to partially recover the missing image components. First, we analyze the advantages of using the iterative Tikhonov regularization and the Maximum A Posteriori (MAP) algorithm with Gibbs prior. Then, we conclude that the missing (nullspace) image components can be partially recovered if the MAP algorithm is implemented through a multigrid technique. The results, which are presented for synthetic noise-free and noisy data, demonstrate the validity of our assumption. The problem
of estimating the regularization and scaling parameters in the MAP algorithm is also addressed.
Key words:
electromagnetic geotomography, hyperparameter estimation, limited-data tomography, multigrid image reconstruction
References:
[1] K. A. Dines and R. J. Lytle, Computerized geophysical tomography, Proc. IEEE 67, 1065-1073 (1979).
[2] D. L. Lager and R. J. Lytle, Determining a subsurface electromagnetic profile from high frequency measurements by applying reconstruction technique algorithms, Radio Science
12, 249-260 (1977).
[3] R. J. Lytle, E. F. Laine, D. L. Lager and D. J. Davis, Using cross borehole electromagnetic probing to locate high contrast anomalies, Geophysics 44, 1667-1676 (1979).
[4] N. Pendock, Radio-wave tomography for geological mapping: Underground and obscured object imaging and detection, SPIE 1942, 96-104 (1993).
[5] S. F. Somerstein et al., Radio-frequency geotomography for remotely probing the interiors of operating mini- and commercial-sized oil-shale retorts, Geophysics 49, 1288-1300
(1984).
[6] A. Pralat and R. Zdunek, Electromagnetic geotomography – selection of measuring frequency, IEEE Sensors Journal 5, 242-250 (2005).
[7] Å Björck, Numerical Methods for Least-Squares Problems, SIAM, Philadelphia (1996).
[8] Y. Censor. Finite series-expansion reconstruction methods, Proc. IEEE 71, 409-419 (1983).
[9] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.
[10] C. Paige C and M. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software 8, 43-71 (1982).
[11] P. S. Rowbotham and R. G. Pratt, Improved inversion through use of the null space, Geophysics 62, 869-883, (1997).
[12] I. Koltracht, P. Lancaster and D. Smith, The structure of some matrices arising in tomography, Linear Algebra Appl. 130, 193-218 (1990).
[13] T. A. Sanny and K. Sassa, Detection of fault structure under a near-surface low velocity layer by seismic tomography: synthetics studies, Journal of Applied Geophysics 35, 117-131 (1996).
[14] S. Geman and D. McClure, Bayesian image analysis: an application to single photon emission tomography, Proc. Amer. Statist. Assoc. Stat. Comp. Sect. 12-18 (1985).
[15] P. J. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Trans. Med. Imaging 9(1), 84-93 (1990).
[16] T. Hebert and R. Leahy, A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors, IEEE Trans. Med. Imag. 8, 194-202 (1989).
[17] K. Lange, Convergence of EM image reconstruction algorithms with Gibbs smoothing, IEEE Trans. Med. Imag. 9(4), 439-446 (1990).
[18] M. V. Ranganath, A. P. Dhawan and N. Mullani, A multigrid expectation maximization reconstruction algorithm for positron emission tomography, IEEE Trans. Med. Imag. 7, 273-278 (1988).
[19] T. S. Pan and A. E. Yagle, Numerical study of multigrid implementations of some iterative image reconstruction algorithms, IEEE Trans. Med. Imag. 10, 572-588 (1991).
[20] R. Zdunek, Optimization of methods of reconstructing images of electromagnetic wave attenuation coefficient distribution in earth, PhD thesis, Wroclaw University of Technology, Wroclaw, Poland, February 2002. in Polish.
[21] C. Popa and R. Zdunek, Kaczmarz extended algorithm for tomographic image reconstruction from limited-data, Mathematics and Computers in Simulation 65(6), 579-598 (2004).
[22] A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization, SIAM Rev. 40, 636-666 (1998).
[23] R. Zdunek and A. Pralat, Multigrid implementation of some regularized image reconstruction methods for limited-data tomography, Industrial Process Tomography 553-558, Banff, Canada, September 2003.
[24] L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imag. MI1, 113-122 (1982).
[25] K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comp. Assisted Tomo. 8(2), 306-316 (1984).
[26] A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society 39(1), 1-38 (1977).
[27] D. S. Lalush and B. M. W. Tsui, Simulation evaluation of Gibbs prior distributions for use in maximum a posteriori SPECT reconstructions, IEEE Trans. Med. Imaging 11(2), 267-275 (1992).
[28] D. M. Higdon et al., Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data, IEEE Trans. Med. Imag. 16(5), 516-525 (1997).
[29] Z. Zhou, R. M. Leahy and J. Qi, Approximate maximum likelihood hyperparameter estimation for Gibbs priors, IEEE Trans. Med. Imag. 6, 844-861 (1997).
Limited-data tomography, to which electromagnetic geotomography belongs, is analyzed in this paper. In this technique, a discrete forward projection model may be expressed by a rank-deficient system of linear equations whose the nullspace is non-trivial. This means that some image components may fall into the nullspace, and hence the minimal-norm least-square solution, to which many image reconstructions methods converge, may be different from the true one. The Algebraic Reconstruction Technique (ART), Simultaneous Iterative Reconstruction Technique (SIRT), or Conjugate Gradients Least Squares (CGLS) are examples of such methods. In this paper, we deal with the question of how to partially recover the missing image components. First, we analyze the advantages of using the iterative Tikhonov regularization and the Maximum A Posteriori (MAP) algorithm with Gibbs prior. Then, we conclude that the missing (nullspace) image components can be partially recovered if the MAP algorithm is implemented through a multigrid technique. The results, which are presented for synthetic noise-free and noisy data, demonstrate the validity of our assumption. The problem
of estimating the regularization and scaling parameters in the MAP algorithm is also addressed.
Key words:
electromagnetic geotomography, hyperparameter estimation, limited-data tomography, multigrid image reconstruction
References:
[1] K. A. Dines and R. J. Lytle, Computerized geophysical tomography, Proc. IEEE 67, 1065-1073 (1979).
[2] D. L. Lager and R. J. Lytle, Determining a subsurface electromagnetic profile from high frequency measurements by applying reconstruction technique algorithms, Radio Science
12, 249-260 (1977).
[3] R. J. Lytle, E. F. Laine, D. L. Lager and D. J. Davis, Using cross borehole electromagnetic probing to locate high contrast anomalies, Geophysics 44, 1667-1676 (1979).
[4] N. Pendock, Radio-wave tomography for geological mapping: Underground and obscured object imaging and detection, SPIE 1942, 96-104 (1993).
[5] S. F. Somerstein et al., Radio-frequency geotomography for remotely probing the interiors of operating mini- and commercial-sized oil-shale retorts, Geophysics 49, 1288-1300
(1984).
[6] A. Pralat and R. Zdunek, Electromagnetic geotomography – selection of measuring frequency, IEEE Sensors Journal 5, 242-250 (2005).
[7] Å Björck, Numerical Methods for Least-Squares Problems, SIAM, Philadelphia (1996).
[8] Y. Censor. Finite series-expansion reconstruction methods, Proc. IEEE 71, 409-419 (1983).
[9] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.
[10] C. Paige C and M. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software 8, 43-71 (1982).
[11] P. S. Rowbotham and R. G. Pratt, Improved inversion through use of the null space, Geophysics 62, 869-883, (1997).
[12] I. Koltracht, P. Lancaster and D. Smith, The structure of some matrices arising in tomography, Linear Algebra Appl. 130, 193-218 (1990).
[13] T. A. Sanny and K. Sassa, Detection of fault structure under a near-surface low velocity layer by seismic tomography: synthetics studies, Journal of Applied Geophysics 35, 117-131 (1996).
[14] S. Geman and D. McClure, Bayesian image analysis: an application to single photon emission tomography, Proc. Amer. Statist. Assoc. Stat. Comp. Sect. 12-18 (1985).
[15] P. J. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Trans. Med. Imaging 9(1), 84-93 (1990).
[16] T. Hebert and R. Leahy, A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors, IEEE Trans. Med. Imag. 8, 194-202 (1989).
[17] K. Lange, Convergence of EM image reconstruction algorithms with Gibbs smoothing, IEEE Trans. Med. Imag. 9(4), 439-446 (1990).
[18] M. V. Ranganath, A. P. Dhawan and N. Mullani, A multigrid expectation maximization reconstruction algorithm for positron emission tomography, IEEE Trans. Med. Imag. 7, 273-278 (1988).
[19] T. S. Pan and A. E. Yagle, Numerical study of multigrid implementations of some iterative image reconstruction algorithms, IEEE Trans. Med. Imag. 10, 572-588 (1991).
[20] R. Zdunek, Optimization of methods of reconstructing images of electromagnetic wave attenuation coefficient distribution in earth, PhD thesis, Wroclaw University of Technology, Wroclaw, Poland, February 2002. in Polish.
[21] C. Popa and R. Zdunek, Kaczmarz extended algorithm for tomographic image reconstruction from limited-data, Mathematics and Computers in Simulation 65(6), 579-598 (2004).
[22] A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization, SIAM Rev. 40, 636-666 (1998).
[23] R. Zdunek and A. Pralat, Multigrid implementation of some regularized image reconstruction methods for limited-data tomography, Industrial Process Tomography 553-558, Banff, Canada, September 2003.
[24] L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imag. MI1, 113-122 (1982).
[25] K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comp. Assisted Tomo. 8(2), 306-316 (1984).
[26] A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society 39(1), 1-38 (1977).
[27] D. S. Lalush and B. M. W. Tsui, Simulation evaluation of Gibbs prior distributions for use in maximum a posteriori SPECT reconstructions, IEEE Trans. Med. Imaging 11(2), 267-275 (1992).
[28] D. M. Higdon et al., Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data, IEEE Trans. Med. Imag. 16(5), 516-525 (1997).
[29] Z. Zhou, R. M. Leahy and J. Qi, Approximate maximum likelihood hyperparameter estimation for Gibbs priors, IEEE Trans. Med. Imag. 6, 844-861 (1997).
