Abstract

In infrared species tomography, the unknown concentration distribution of a species is inferred from the attenuation of multiple collimated light beams shone through the measurement field. The resulting set of linear equations is rank-deficient, so prior assumptions about the smoothness and nonnegativity of the distribution must be imposed to recover a solution. This paper describes how the Kalman filter can be used to incorporate additional information about the time evolution of the distribution into the reconstruction. Results show that, although performing a series of static reconstructions is more accurate at low levels of measurement noise, the Kalman filter becomes advantageous when the measurements are corrupted with high levels of noise. The Kalman filter also enables signal multiplexing, which can help achieve the high sampling rates needed to resolve turbulent flow phenomena.

© 2011 Optical Society of America

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  1. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Overseas Publishers Association, 1996), Chap. 7.
  2. R. Sadanandan, M. Stöhr, and W. Meir, “Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor,” Appl. Phys. B 90, 609–618 (2008).
    [CrossRef]
  3. A. E. Klingbeil, J. B. Jeffries, and R. K. Hanson, “Temperature- and pressure-dependent absorption cross sections of gaseous hydrocarbons at 3.39 μm,” Meas. Sci. Technol. 17, 1950–1957(2006).
    [CrossRef]
  4. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 191–200.
  5. S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
    [CrossRef] [PubMed]
  6. L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1971).
  7. R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
    [CrossRef]
  8. B. Gillet, Y. Hardalupas, C. Kavounides, and A. M. K. P. Taylor, “Infrared absorption for measurement of hydrocarbon concentration in fuel/air mixtures (MAST-B-LIQUID),” Appl. Therm. Eng. 24, 1633–1653 (2004).
    [CrossRef]
  9. E. Bryner, G. S. Diskin, C. P. Goyne, and J. C. McDaniel, “Development of an infrared laser absorption tomography system for a scramjet combustor,” in AIAA Aerodynamics Measurement Technology and Ground Testing Conference (AIAA, 2006), paper CP3445.
  10. C. A. Garcia-Stewart, N. Polydorides, K. B. Ozanyan, and H. McCann, “Image reconstruction algorithms for high-speed chemical species tomography,” in Third World Conference on Industrial Process Tomography (Virtual Centre for Industrial Process Tomography, 2003), pp. 80–86.
  11. P. C. Hansen, Rank Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
    [CrossRef]
  12. P. Wright, C. A. Garcia-Stewart, S. J. Carey, F. P. Hindle, S. H. Pegrum, S. M. Colbourne, P. J. Turner, W. J. Hurr, T. J. Litt, S. C. Murray, S. D. Crossley, K. B. Ozanyan, and H. McCann, “Toward in-cylinder absorption tomography in a production engine,” Appl. Opt. 44, 6578–6592 (2005).
    [CrossRef] [PubMed]
  13. P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
    [CrossRef]
  14. N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
    [CrossRef]
  15. K. J. Daun, “Infrared species limited data tomography through Tikhonov reconstruction,” J. Quant. Spectrosc. Radiat. Transfer 111, 105–115 (2010).
    [CrossRef]
  16. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).
  17. A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
    [CrossRef]
  18. K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
    [CrossRef]
  19. A. Seppänen, A. Voutilainen, and J. P. Kaipio, “State estimation in process tomography-reconstruction of velocity fields using EIT,” Inverse Probl. 25, 1–23 (2009).
    [CrossRef]
  20. J. P. Kaipio, S. Duncan, and A. Seppänen, “State estimation,” in Process Imaging for Automatic Control, D.M.Scott and H.McCann, eds. (CRC, 2005), pp. 207–235.
  21. R. Zdunek, “Multigrid regularized image reconstruction for limited-data tomography,” Comput. Methods Sci. Technol. 13, 67–77 (2007).
  22. P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
    [CrossRef]
  23. G. H. Golub, M. Heath, and G. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [CrossRef]
  24. V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417 (1966).
  25. D. Simon, Optimal State Estimation (Wiley, 2006).
    [CrossRef]
  26. FDS-SMV, “Fire dynamics simulator,” Software Package, Version 5.5, National Institute for Standards and Technology, 2010.
  27. K. McGrattan and G. Forney, Fire Dynamics Simulator (Version 5) Technical Reference Guide (U. S. Government Printing Office, 2009).
  28. Z. H. Yan, “A numerical study of effect of initial condition on large eddy simulation of thermal plume,” Numer. Heat Transf. B 43, 167–178 (2003).
    [CrossRef]
  29. Z. H. Yan, “Large eddy simulations of a turbulent thermal plume,” Heat Mass Transfer 43, 503–514 (2007).
    [CrossRef]
  30. X. Zhou, K. H. Luo, and J. J. R. Williams, “Large-eddy simulation of a turbulent forced plume,” Eur. J. Mech. B, Fluids 20, 233–254 (2001).
    [CrossRef]
  31. L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
    [CrossRef]

2010 (2)

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

K. J. Daun, “Infrared species limited data tomography through Tikhonov reconstruction,” J. Quant. Spectrosc. Radiat. Transfer 111, 105–115 (2010).
[CrossRef]

2009 (2)

A. Seppänen, A. Voutilainen, and J. P. Kaipio, “State estimation in process tomography-reconstruction of velocity fields using EIT,” Inverse Probl. 25, 1–23 (2009).
[CrossRef]

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

2008 (2)

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

R. Sadanandan, M. Stöhr, and W. Meir, “Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor,” Appl. Phys. B 90, 609–618 (2008).
[CrossRef]

2007 (2)

R. Zdunek, “Multigrid regularized image reconstruction for limited-data tomography,” Comput. Methods Sci. Technol. 13, 67–77 (2007).

Z. H. Yan, “Large eddy simulations of a turbulent thermal plume,” Heat Mass Transfer 43, 503–514 (2007).
[CrossRef]

2006 (1)

A. E. Klingbeil, J. B. Jeffries, and R. K. Hanson, “Temperature- and pressure-dependent absorption cross sections of gaseous hydrocarbons at 3.39 μm,” Meas. Sci. Technol. 17, 1950–1957(2006).
[CrossRef]

2005 (1)

2004 (1)

B. Gillet, Y. Hardalupas, C. Kavounides, and A. M. K. P. Taylor, “Infrared absorption for measurement of hydrocarbon concentration in fuel/air mixtures (MAST-B-LIQUID),” Appl. Therm. Eng. 24, 1633–1653 (2004).
[CrossRef]

2003 (2)

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

Z. H. Yan, “A numerical study of effect of initial condition on large eddy simulation of thermal plume,” Numer. Heat Transf. B 43, 167–178 (2003).
[CrossRef]

2001 (3)

A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
[CrossRef]

K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
[CrossRef]

X. Zhou, K. H. Luo, and J. J. R. Williams, “Large-eddy simulation of a turbulent forced plume,” Eur. J. Mech. B, Fluids 20, 233–254 (2001).
[CrossRef]

1993 (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

1981 (1)

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
[CrossRef]

1979 (1)

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

1971 (1)

L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1971).

1966 (1)

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417 (1966).

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Barbe, A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Benner, D. C.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Bernath, P. F.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 191–200.

Birk, M.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 191–200.

Boudon, V.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Brown, L. R.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Bryner, E.

E. Bryner, G. S. Diskin, C. P. Goyne, and J. C. McDaniel, “Development of an infrared laser absorption tomography system for a scramjet combustor,” in AIAA Aerodynamics Measurement Technology and Ground Testing Conference (AIAA, 2006), paper CP3445.

Campargue, A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Carey, S. J.

Champion, J.-P.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Chance, K.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Colbourne, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

Colbourne, S. M.

Coudert, L. H.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Crossley, S. D.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

P. Wright, C. A. Garcia-Stewart, S. J. Carey, F. P. Hindle, S. H. Pegrum, S. M. Colbourne, P. J. Turner, W. J. Hurr, T. J. Litt, S. C. Murray, S. D. Crossley, K. B. Ozanyan, and H. McCann, “Toward in-cylinder absorption tomography in a production engine,” Appl. Opt. 44, 6578–6592 (2005).
[CrossRef] [PubMed]

Dana, V.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Daun, K. J.

K. J. Daun, “Infrared species limited data tomography through Tikhonov reconstruction,” J. Quant. Spectrosc. Radiat. Transfer 111, 105–115 (2010).
[CrossRef]

Davidson, J. L.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

Devi, V. M.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Diskin, G. S.

E. Bryner, G. S. Diskin, C. P. Goyne, and J. C. McDaniel, “Development of an infrared laser absorption tomography system for a scramjet combustor,” in AIAA Aerodynamics Measurement Technology and Ground Testing Conference (AIAA, 2006), paper CP3445.

Duncan, S.

J. P. Kaipio, S. Duncan, and A. Seppänen, “State estimation,” in Process Imaging for Automatic Control, D.M.Scott and H.McCann, eds. (CRC, 2005), pp. 207–235.

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Overseas Publishers Association, 1996), Chap. 7.

Emmerman, P. J.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
[CrossRef]

Fally, S.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Flaud, J.-M.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Forney, G.

K. McGrattan and G. Forney, Fire Dynamics Simulator (Version 5) Technical Reference Guide (U. S. Government Printing Office, 2009).

Gamache, R. R.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Garcia-Castillo, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

Garcia-Stewart, C. A.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

P. Wright, C. A. Garcia-Stewart, S. J. Carey, F. P. Hindle, S. H. Pegrum, S. M. Colbourne, P. J. Turner, W. J. Hurr, T. J. Litt, S. C. Murray, S. D. Crossley, K. B. Ozanyan, and H. McCann, “Toward in-cylinder absorption tomography in a production engine,” Appl. Opt. 44, 6578–6592 (2005).
[CrossRef] [PubMed]

C. A. Garcia-Stewart, N. Polydorides, K. B. Ozanyan, and H. McCann, “Image reconstruction algorithms for high-speed chemical species tomography,” in Third World Conference on Industrial Process Tomography (Virtual Centre for Industrial Process Tomography, 2003), pp. 80–86.

Gillet, B.

B. Gillet, Y. Hardalupas, C. Kavounides, and A. M. K. P. Taylor, “Infrared absorption for measurement of hydrocarbon concentration in fuel/air mixtures (MAST-B-LIQUID),” Appl. Therm. Eng. 24, 1633–1653 (2004).
[CrossRef]

Goldman, A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Golub, G. H.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Gordon, I. E.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Goulard, R.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
[CrossRef]

Goyne, C. P.

E. Bryner, G. S. Diskin, C. P. Goyne, and J. C. McDaniel, “Development of an infrared laser absorption tomography system for a scramjet combustor,” in AIAA Aerodynamics Measurement Technology and Ground Testing Conference (AIAA, 2006), paper CP3445.

Hansen, P. C.

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

P. C. Hansen, Rank Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
[CrossRef]

Hanson, R. K.

A. E. Klingbeil, J. B. Jeffries, and R. K. Hanson, “Temperature- and pressure-dependent absorption cross sections of gaseous hydrocarbons at 3.39 μm,” Meas. Sci. Technol. 17, 1950–1957(2006).
[CrossRef]

Hardalupas, Y.

B. Gillet, Y. Hardalupas, C. Kavounides, and A. M. K. P. Taylor, “Infrared absorption for measurement of hydrocarbon concentration in fuel/air mixtures (MAST-B-LIQUID),” Appl. Therm. Eng. 24, 1633–1653 (2004).
[CrossRef]

Heath, M.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Hindle, F. P.

Hurr, W. J.

Jacquemart, D.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Järvenpää, S.

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

Jeffries, J. B.

A. E. Klingbeil, J. B. Jeffries, and R. K. Hanson, “Temperature- and pressure-dependent absorption cross sections of gaseous hydrocarbons at 3.39 μm,” Meas. Sci. Technol. 17, 1950–1957(2006).
[CrossRef]

Kaipio, J. P.

A. Seppänen, A. Voutilainen, and J. P. Kaipio, “State estimation in process tomography-reconstruction of velocity fields using EIT,” Inverse Probl. 25, 1–23 (2009).
[CrossRef]

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
[CrossRef]

J. P. Kaipio, S. Duncan, and A. Seppänen, “State estimation,” in Process Imaging for Automatic Control, D.M.Scott and H.McCann, eds. (CRC, 2005), pp. 207–235.

Kavounides, C.

B. Gillet, Y. Hardalupas, C. Kavounides, and A. M. K. P. Taylor, “Infrared absorption for measurement of hydrocarbon concentration in fuel/air mixtures (MAST-B-LIQUID),” Appl. Therm. Eng. 24, 1633–1653 (2004).
[CrossRef]

Kim, B. S.

K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
[CrossRef]

Kim, K. Y.

K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
[CrossRef]

Kim, M. C.

K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
[CrossRef]

Kleiner, I.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Klingbeil, A. E.

A. E. Klingbeil, J. B. Jeffries, and R. K. Hanson, “Temperature- and pressure-dependent absorption cross sections of gaseous hydrocarbons at 3.39 μm,” Meas. Sci. Technol. 17, 1950–1957(2006).
[CrossRef]

Koistinen, P.

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

Kolehmainen, V.

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

Lacome, N.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Lafferty, W. J.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Lassas, M.

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

Lee, Y. J.

K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
[CrossRef]

Litc, T.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

Litt, T. J.

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

P. Wright, C. A. Garcia-Stewart, S. J. Carey, F. P. Hindle, S. H. Pegrum, S. M. Colbourne, P. J. Turner, W. J. Hurr, T. J. Litt, S. C. Murray, S. D. Crossley, K. B. Ozanyan, and H. McCann, “Toward in-cylinder absorption tomography in a production engine,” Appl. Opt. 44, 6578–6592 (2005).
[CrossRef] [PubMed]

Logan, B. F.

L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1971).

Luo, K. H.

X. Zhou, K. H. Luo, and J. J. R. Williams, “Large-eddy simulation of a turbulent forced plume,” Eur. J. Mech. B, Fluids 20, 233–254 (2001).
[CrossRef]

Mandin, J.-Y.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Massie, S. T.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

McCann, H.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

P. Wright, C. A. Garcia-Stewart, S. J. Carey, F. P. Hindle, S. H. Pegrum, S. M. Colbourne, P. J. Turner, W. J. Hurr, T. J. Litt, S. C. Murray, S. D. Crossley, K. B. Ozanyan, and H. McCann, “Toward in-cylinder absorption tomography in a production engine,” Appl. Opt. 44, 6578–6592 (2005).
[CrossRef] [PubMed]

C. A. Garcia-Stewart, N. Polydorides, K. B. Ozanyan, and H. McCann, “Image reconstruction algorithms for high-speed chemical species tomography,” in Third World Conference on Industrial Process Tomography (Virtual Centre for Industrial Process Tomography, 2003), pp. 80–86.

McDaniel, J. C.

E. Bryner, G. S. Diskin, C. P. Goyne, and J. C. McDaniel, “Development of an infrared laser absorption tomography system for a scramjet combustor,” in AIAA Aerodynamics Measurement Technology and Ground Testing Conference (AIAA, 2006), paper CP3445.

McGrattan, K.

K. McGrattan and G. Forney, Fire Dynamics Simulator (Version 5) Technical Reference Guide (U. S. Government Printing Office, 2009).

Meir, W.

R. Sadanandan, M. Stöhr, and W. Meir, “Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor,” Appl. Phys. B 90, 609–618 (2008).
[CrossRef]

Mikhailenko, S. N.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Miller, C. E.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Moazzen-Ahmadi, N.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Morozov, V. A.

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417 (1966).

Murray, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

Murray, S. C.

Naumenko, O. V.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Nikitin, A. V.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Orphal, J.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Ozanyan, K. B.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

P. Wright, C. A. Garcia-Stewart, S. J. Carey, F. P. Hindle, S. H. Pegrum, S. M. Colbourne, P. J. Turner, W. J. Hurr, T. J. Litt, S. C. Murray, S. D. Crossley, K. B. Ozanyan, and H. McCann, “Toward in-cylinder absorption tomography in a production engine,” Appl. Opt. 44, 6578–6592 (2005).
[CrossRef] [PubMed]

C. A. Garcia-Stewart, N. Polydorides, K. B. Ozanyan, and H. McCann, “Image reconstruction algorithms for high-speed chemical species tomography,” in Third World Conference on Industrial Process Tomography (Virtual Centre for Industrial Process Tomography, 2003), pp. 80–86.

Pegrum, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

Pegrum, S. H.

Perevalov, V. I.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Perrin, A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Pirttilä, J.

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

Polydorides, N.

C. A. Garcia-Stewart, N. Polydorides, K. B. Ozanyan, and H. McCann, “Image reconstruction algorithms for high-speed chemical species tomography,” in Third World Conference on Industrial Process Tomography (Virtual Centre for Industrial Process Tomography, 2003), pp. 80–86.

Predoi-Cross, A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Rinsland, C. P.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Rotger, M.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Rothman, L. S.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Sadanandan, R.

R. Sadanandan, M. Stöhr, and W. Meir, “Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor,” Appl. Phys. B 90, 609–618 (2008).
[CrossRef]

Santoro, R. J.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
[CrossRef]

Semerjian, H. G.

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
[CrossRef]

Seppänen, A.

A. Seppänen, A. Voutilainen, and J. P. Kaipio, “State estimation in process tomography-reconstruction of velocity fields using EIT,” Inverse Probl. 25, 1–23 (2009).
[CrossRef]

A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
[CrossRef]

J. P. Kaipio, S. Duncan, and A. Seppänen, “State estimation,” in Process Imaging for Automatic Control, D.M.Scott and H.McCann, eds. (CRC, 2005), pp. 207–235.

Shepp, L. A.

L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1971).

Siltanen, S.

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

Šimecková, M.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Simon, D.

D. Simon, Optimal State Estimation (Wiley, 2006).
[CrossRef]

Smith, M. A. H.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Somersalo, E.

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
[CrossRef]

Stöhr, M.

R. Sadanandan, M. Stöhr, and W. Meir, “Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor,” Appl. Phys. B 90, 609–618 (2008).
[CrossRef]

Sung, K.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Tashkun, S. A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Taylor, A. M. K. P.

B. Gillet, Y. Hardalupas, C. Kavounides, and A. M. K. P. Taylor, “Infrared absorption for measurement of hydrocarbon concentration in fuel/air mixtures (MAST-B-LIQUID),” Appl. Therm. Eng. 24, 1633–1653 (2004).
[CrossRef]

Tennyson, J.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Terzija, N.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Toth, R. A.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Turner, P.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

Turner, P. J.

Vandaelè, A. C.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Vander Auwera, D.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

Vauhknonen, M.

K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
[CrossRef]

Vauhknonen, P. J.

A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
[CrossRef]

Vauhkonen, M.

A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
[CrossRef]

Voutilainen, A.

A. Seppänen, A. Voutilainen, and J. P. Kaipio, “State estimation in process tomography-reconstruction of velocity fields using EIT,” Inverse Probl. 25, 1–23 (2009).
[CrossRef]

Wahba, G.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Williams, J. J. R.

X. Zhou, K. H. Luo, and J. J. R. Williams, “Large-eddy simulation of a turbulent forced plume,” Eur. J. Mech. B, Fluids 20, 233–254 (2001).
[CrossRef]

Wright, P.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

P. Wright, C. A. Garcia-Stewart, S. J. Carey, F. P. Hindle, S. H. Pegrum, S. M. Colbourne, P. J. Turner, W. J. Hurr, T. J. Litt, S. C. Murray, S. D. Crossley, K. B. Ozanyan, and H. McCann, “Toward in-cylinder absorption tomography in a production engine,” Appl. Opt. 44, 6578–6592 (2005).
[CrossRef] [PubMed]

Yan, Z. H.

Z. H. Yan, “Large eddy simulations of a turbulent thermal plume,” Heat Mass Transfer 43, 503–514 (2007).
[CrossRef]

Z. H. Yan, “A numerical study of effect of initial condition on large eddy simulation of thermal plume,” Numer. Heat Transf. B 43, 167–178 (2003).
[CrossRef]

Zdunek, R.

R. Zdunek, “Multigrid regularized image reconstruction for limited-data tomography,” Comput. Methods Sci. Technol. 13, 67–77 (2007).

Zhou, X.

X. Zhou, K. H. Luo, and J. J. R. Williams, “Large-eddy simulation of a turbulent forced plume,” Eur. J. Mech. B, Fluids 20, 233–254 (2001).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

R. Sadanandan, M. Stöhr, and W. Meir, “Simultaneous OH-PLIF and PIV measurements in a gas turbine model combustor,” Appl. Phys. B 90, 609–618 (2008).
[CrossRef]

Appl. Therm. Eng. (1)

B. Gillet, Y. Hardalupas, C. Kavounides, and A. M. K. P. Taylor, “Infrared absorption for measurement of hydrocarbon concentration in fuel/air mixtures (MAST-B-LIQUID),” Appl. Therm. Eng. 24, 1633–1653 (2004).
[CrossRef]

Chem. Eng. J. (1)

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. A. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litc, S. Murray, K. B. Ozanyan, and H. McCann, “High speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158, 2–10 (2010).
[CrossRef]

Comput. Methods Sci. Technol. (1)

R. Zdunek, “Multigrid regularized image reconstruction for limited-data tomography,” Comput. Methods Sci. Technol. 13, 67–77 (2007).

Eur. J. Mech. B, Fluids (1)

X. Zhou, K. H. Luo, and J. J. R. Williams, “Large-eddy simulation of a turbulent forced plume,” Eur. J. Mech. B, Fluids 20, 233–254 (2001).
[CrossRef]

Heat Mass Transfer (1)

Z. H. Yan, “Large eddy simulations of a turbulent thermal plume,” Heat Mass Transfer 43, 503–514 (2007).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1971).

Int. J. Heat Mass Transfer (1)

R. J. Santoro, H. G. Semerjian, P. J. Emmerman, and R. Goulard, “Optical tomography for flow field diagnostics,” Int. J. Heat Mass Transfer 24, 1139–1150 (1981).
[CrossRef]

Inverse Probl. (2)

A. Seppänen, M. Vauhkonen, P. J. Vauhknonen, E. Somersalo, and J. P. Kaipio, “State estimation with fluid dynamical evolution models in process tomography: an application to impedance tomography,” Inverse Probl. 17, 467–483(2001).
[CrossRef]

A. Seppänen, A. Voutilainen, and J. P. Kaipio, “State estimation in process tomography-reconstruction of velocity fields using EIT,” Inverse Probl. 25, 1–23 (2009).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaelè, and D. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 110, 533–572 (2009).
[CrossRef]

K. J. Daun, “Infrared species limited data tomography through Tikhonov reconstruction,” J. Quant. Spectrosc. Radiat. Transfer 111, 105–115 (2010).
[CrossRef]

Meas. Sci. Technol. (3)

K. Y. Kim, B. S. Kim, M. C. Kim, Y. J. Lee, and M. Vauhknonen, “Image reconstruction in time-varying electrical impedance tomography based on the extended Kalman filter,” Meas. Sci. Technol. 12, 1032–1039 (2001).
[CrossRef]

N. Terzija, J. L. Davidson, C. A. Garcia-Stewart, P. Wright, K. B. Ozanyan, S. Pegrum, T. J. Litt, and H. McCann, “Image optimization for chemical species tomography with an irregular and sparse beam array,” Meas. Sci. Technol. 19, 094007(2008).
[CrossRef]

A. E. Klingbeil, J. B. Jeffries, and R. K. Hanson, “Temperature- and pressure-dependent absorption cross sections of gaseous hydrocarbons at 3.39 μm,” Meas. Sci. Technol. 17, 1950–1957(2006).
[CrossRef]

Numer. Heat Transf. B (1)

Z. H. Yan, “A numerical study of effect of initial condition on large eddy simulation of thermal plume,” Numer. Heat Transf. B 43, 167–178 (2003).
[CrossRef]

Phys. Med. Biol. (1)

S. Siltanen, V. Kolehmainen, S. Järvenpää, J. P. Kaipio, P. Koistinen, M. Lassas, J. Pirttilä, and E. Somersalo, “Statistical inversion for medical x-ray tomography with few radiographs: I. General theory,” Phys. Med. Biol. 48, 1437–1463(2003).
[CrossRef] [PubMed]

SIAM J. Sci. Comput. (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Sov. Math. Dokl. (1)

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417 (1966).

Technometrics (1)

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Other (10)

J. P. Kaipio, S. Duncan, and A. Seppänen, “State estimation,” in Process Imaging for Automatic Control, D.M.Scott and H.McCann, eds. (CRC, 2005), pp. 207–235.

D. Simon, Optimal State Estimation (Wiley, 2006).
[CrossRef]

FDS-SMV, “Fire dynamics simulator,” Software Package, Version 5.5, National Institute for Standards and Technology, 2010.

K. McGrattan and G. Forney, Fire Dynamics Simulator (Version 5) Technical Reference Guide (U. S. Government Printing Office, 2009).

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Overseas Publishers Association, 1996), Chap. 7.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 191–200.

E. Bryner, G. S. Diskin, C. P. Goyne, and J. C. McDaniel, “Development of an infrared laser absorption tomography system for a scramjet combustor,” in AIAA Aerodynamics Measurement Technology and Ground Testing Conference (AIAA, 2006), paper CP3445.

C. A. Garcia-Stewart, N. Polydorides, K. B. Ozanyan, and H. McCann, “Image reconstruction algorithms for high-speed chemical species tomography,” in Third World Conference on Industrial Process Tomography (Virtual Centre for Industrial Process Tomography, 2003), pp. 80–86.

P. C. Hansen, Rank Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
[CrossRef]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

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Figures (11)

Fig. 1
Fig. 1

Geometry of a beam transecting the tomography field.

Fig. 2
Fig. 2

Example tomography problem. (a) The A matrix is rank-deficient since some elements are not transected by the beams, so (b) solving A x = b directly through Eq. (6) results in x LS , a physically unrealistic solution. (c) Using MAP with Gibbs smoothness and nonnegativity priors generates a more accurate reconstruction.

Fig. 3
Fig. 3

LES simulation of a mixed buoyant methane plume.

Fig. 4
Fig. 4

Beam configuration in the tomography plane, and methane molar concentration at 13 s . The tomography domain is defined by the 1 m × 1 m white box.

Fig. 5
Fig. 5

Sample beam attenuation data, corresponding to the highlighted beams in Fig. 4.

Fig. 6
Fig. 6

Singular values from the augmented matrix equation, for various values of λ.

Fig. 7
Fig. 7

Error of static reconstructions at 13 s . Setting λ too small fails to fill the null space of A, while setting λ too large oversmooths the solution. The optimal value is around λ = 0.1 , consistent with the singular values shown in Fig. 6.

Fig. 8
Fig. 8

Comparison of the static MAP and Kalman filter reconstructions at various process times, for σ m = 10 3 .

Fig. 9
Fig. 9

Average errors produced for the static MAP and Kalman filter solutions for different measurement noise and state variable covariance.

Fig. 10
Fig. 10

Sample beam attenuation measurements for low- and high-noise scenarios.

Fig. 11
Fig. 11

Time histories of estimation errors for static and Kalman filter methods, using all measurements and subsets of eight measurements. The Kalman filter shows almost no degradation in performance, while the static approach is greatly affected.

Equations (30)

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I η ( s , θ ) = I η 0 exp { 0 a η [ r ( s , θ ) ( u ) ] d u } ,
ln [ I η 0 / I η ( s , θ ) ] = b ( s , θ ) = 0 a η [ r ( s , θ ) ( u ) ] d u ,
b i = 0 a η [ r ( s , θ ) ( u ) ] d u j = 1 n A i j x j ,
Null ( A ) = { x n R n : Ax n = 0 } .
A = USV T = U · [ σ 1 σ 2 σ m 0 0 ] · V T ,
x LS = i = 1 m u i T b σ i v i ,
p ( x | b ) = p ( b | x ) · p p r ( x ) p ( b ) ,
p ( b | x ) = 1 2 π σ m 2 i = 1 m exp [ ( b i A i x ) 2 2 σ m 2 ] exp [ b Ax 2 2 2 σ m 2 ] .
p non-neg ( x ) = j = 1 n H ( x j ) ,
p Gibbs ( x ) exp [ β U ( x ) ] .
L i j = { 1 1 / n i 0 if     i = j if     j   neighbors   i otherwise ,
p ( x | b ) = p ( b | x ) · p Gibbs ( x ) · p non-neg ( x ) p ( b ) exp [ Ax b 2 2 2 σ m 2 β Lx 2 2 ] i = 1 n H ( x i ) .
ln p ( x | b ) ( Ax b 2 2 + λ 2 Lx 2 2 )
x * = arg min [ ( [ A L ] x [ b 0 ] ) T W ( [ A L ] x [ b 0 ] ) ] ,   s.t.   x 0 ,
W = [ I 0 0 λ 2 I ] ,
x k + 1 = F k x k + ε k ,
y k = G k x k + δ k ,
G k = G = [ A L ] ,
R k = σ m 2 [ I 0 0 1 / λ 2 I ] .
μ ¯ k + 1 = F k μ k ,
Σ ¯ k + 1 = F k Σ k ( F k ) T + Q k .
μ k + 1 = μ ¯ k + 1 + K k + 1 ( y k + 1 G μ ¯ k + 1 ) .
K k + 1 = Σ ¯ k + 1 ( G k + 1 ) T [ G k + 1 Σ ¯ k + 1 ( G k + 1 ) T + R k + 1 ] 1 .
Σ k + 1 = ( I K k + 1 G k + 1 ) Σ ¯ k + 1 ,
p ( x k | y k ) = p ( y k | x k ) · p ( x k | x k 1 ) p ( y k ) exp [ 1 2 ( x k x ¯ k ) T ( Σ ¯ k ) 1 ( x k x ¯ k ) 1 2 ( y k G k x k ) T ( R k ) 1 ( y k G k x k ) ] = exp [ 1 2 ( x k F x k 1 ) T ( F Σ ¯ k F T + Q k ) 1 ( x k F x k 1 ) 1 2 ( y k G k x k ) T ( R k ) 1 ( y k G k x k ) ] .
p ( x k | y k ) exp [ 1 2 ( x k Fx k 1 ) T ( F Σ k 1 F T + σ t 2 I ) 1 ( x k Fx k 1 ) 1 2 ( b k Ax k ) T ( σ m 2 I ) 1 ( b k Ax k ) 1 2 ( Lx k ) T [ σ m 2 λ 2 I ] 1 ( Lx k ) ] .
a η ( r ) = σ η N ( r ) = σ η χ CH 4 ( r ) P R ° T ,
b ˜ i ( k ) = b i ( k ) + δ b k ,
ε ( x λ ) = x λ x exact / x exact ave ,
x = x exact + δ x = j = 1 n u T b exact σ i v i + j = 1 n u T δ b σ i v ,

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