Abstract

Gas distributions imaged by chemical species tomography (CST) vary in quality due to the discretization scheme, arrangement of optical paths, errors in the measurement model, and prior information included in reconstruction. There is currently no mathematically-rigorous framework for comparing the finite bases available to discretize a CST domain. Following from the Bayesian formulation of tomographic inversion, we show that Bayesian model selection can identify the mesh density, mode of interpolation, and prior information best-suited to reconstruct a set of measurement data. We validate this procedure with a simulated CST experiment, and generate accurate reconstructions despite limited measurement information. The flow field is represented using the finite element method, and Bayesian model selection is used to choose between three forms of polynomial support for a range of mesh resolutions, as well as four priors. We show that the model likelihood of Bayesian model selection is a good predictor of reconstruction accuracy.

© 2017 Optical Society of America

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References

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  1. W. Cai and C. F. Kaminski, “Tomographic absorption spectroscopy for the study of gas dynamics and reactive flows,” Pror. Energy Combust. Sci. 59, 1–31 (2017).
    [Crossref]
  2. Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
    [Crossref]
  3. S. J. Grauer, R. W. Tsang, and K. J. Daun, “Broadband chemical species tomography: Measurement theory and a proof-of-concept emission detection experiment,” J. Quant. Spectrosc. Radiat. Transf. 198, 145–154 (2017).
    [Crossref]
  4. F. Stritzke, S. van der Kley, A. Feiling, A. Dreizler, and S. Wagner, “Ammonia concentration distribution measurements in the exhaust of a heavy duty diesel engine based on limited data absorption tomography,” Opt. Express 25(7), 8180–8191 (2017).
    [Crossref] [PubMed]
  5. P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (SIAM, 2010).
  6. K. J. Daun, S. J. Grauer, and P. J. Hadwin, “Chemical species tomography of turbulent flows: Discrete ill-posed and rank deficient problems and the use of prior information,” J. Quant. Spectrosc. Radiat. Transf. 172, 58–74 (2016).
    [Crossref]
  7. D. Verhoeven, “Limited-data computed tomography algorithms for the physical sciences,” Appl. Opt. 32(20), 3736–3754 (1993).
    [Crossref] [PubMed]
  8. W. Verkruysse and L. A. Todd, “Novel algorithm for tomographic reconstruction of atmospheric chemicals with sparse sampling,” Environ. Sci. Technol. 39(7), 2247–2254 (2005).
    [Crossref] [PubMed]
  9. J. Floyd and A. M. Kempf, “Computed tomography of chemiluminescence (CTC): high resolution and instantaneous 3-D measurements of a matrix burner,” Proc. Combust. Inst. 33(1), 751–758 (2011).
    [Crossref]
  10. M. G. Twynstra, K. J. Daun, and S. L. Waslander, “Line-of-sight-attenuation chemical species tomography through the level set method,” J. Quant. Spectrosc. Radiat. Transf. 143, 25–34 (2014).
    [Crossref]
  11. 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(9), 094007 (2008).
    [Crossref]
  12. N. Terzija and H. McCann, “Wavelet-based image reconstruction for hard-field tomography with severely limited data,” IEEE Sens. J. 11(9), 1885–1893 (2011).
    [Crossref]
  13. S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Bayesian approach to the design of chemical species tomography experiments,” Appl. Opt. 55(21), 5772–5782 (2016).
    [Crossref] [PubMed]
  14. T. Yu, B. Tian, and W. Cai, “Development of a beam optimization method for absorption-based tomography,” Opt. Express 25(6), 5982–5999 (2017).
    [Crossref] [PubMed]
  15. N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
    [Crossref] [PubMed]
  16. M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
    [Crossref]
  17. M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
    [Crossref] [PubMed]
  18. C. Tape, Q. Liu, A. Maggi, and J. Tromp, “Seismic tomography of the southern California crust based on spectral-element and adjoint methods,” Geophys. J. Int. 180(1), 433–462 (2010).
    [Crossref]
  19. K. Mohri, S. Göers, J. Schöler, A. Rittler, T. Dreier, C. Schulz, and A. M. Kempf, “Instantaneous 3D-imaging of highly turbulent flames using Computed Tomography of Chemiluminescence (CTC),” Appl. Opt. 56(26), 7385–7395 (2017).
    [Crossref]
  20. W. D. Penny, “Comparing dynamic causal models using AIC, BIC and free energy,” Neuroimage 59(1), 319–330 (2012).
    [Crossref] [PubMed]
  21. J. R. Howell, M. P. Menguc, and R. Siegel, Thermal Radiation Heat Transfer (CRC Press, 2010).
  22. J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer, 2006).
  23. C. L. Lawson and R. J. Hanson, Solving Least-Squares Problems (Prentice Hall, 1974), Chap. 23.
  24. S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Improving chemical species tomography of turbulent flows using covariance estimation,” Appl. Opt. 56(13), 3900–3912 (2017).
    [Crossref] [PubMed]
  25. S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
    [Crossref]
  26. J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
    [Crossref]
  27. M. Sambridge, K. Gallagher, A. Jackson, and P. Rickwood, “Trans-dimensional inverse problems, model comparison, and the evidence,” Geophys. J. Int. 157(2), 528 (2006).
    [Crossref]
  28. R. E. Kass and A. E. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90(430), 773–795 (1995).
    [Crossref]
  29. M. Zöchbauer, H. Smith, and T. Lauer, “Advanced SCR flow modeling with a validated large eddy simulation,” SAE Technical Paper 2015–01–1046 (2015).
    [Crossref]
  30. P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46(2), 329–345 (2004).
    [Crossref]
  31. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
    [Crossref] [PubMed]

2017 (7)

W. Cai and C. F. Kaminski, “Tomographic absorption spectroscopy for the study of gas dynamics and reactive flows,” Pror. Energy Combust. Sci. 59, 1–31 (2017).
[Crossref]

Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
[Crossref]

S. J. Grauer, R. W. Tsang, and K. J. Daun, “Broadband chemical species tomography: Measurement theory and a proof-of-concept emission detection experiment,” J. Quant. Spectrosc. Radiat. Transf. 198, 145–154 (2017).
[Crossref]

T. Yu, B. Tian, and W. Cai, “Development of a beam optimization method for absorption-based tomography,” Opt. Express 25(6), 5982–5999 (2017).
[Crossref] [PubMed]

F. Stritzke, S. van der Kley, A. Feiling, A. Dreizler, and S. Wagner, “Ammonia concentration distribution measurements in the exhaust of a heavy duty diesel engine based on limited data absorption tomography,” Opt. Express 25(7), 8180–8191 (2017).
[Crossref] [PubMed]

S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Improving chemical species tomography of turbulent flows using covariance estimation,” Appl. Opt. 56(13), 3900–3912 (2017).
[Crossref] [PubMed]

K. Mohri, S. Göers, J. Schöler, A. Rittler, T. Dreier, C. Schulz, and A. M. Kempf, “Instantaneous 3D-imaging of highly turbulent flames using Computed Tomography of Chemiluminescence (CTC),” Appl. Opt. 56(26), 7385–7395 (2017).
[Crossref]

2016 (3)

S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Bayesian approach to the design of chemical species tomography experiments,” Appl. Opt. 55(21), 5772–5782 (2016).
[Crossref] [PubMed]

N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
[Crossref] [PubMed]

K. J. Daun, S. J. Grauer, and P. J. Hadwin, “Chemical species tomography of turbulent flows: Discrete ill-posed and rank deficient problems and the use of prior information,” J. Quant. Spectrosc. Radiat. Transf. 172, 58–74 (2016).
[Crossref]

2014 (1)

M. G. Twynstra, K. J. Daun, and S. L. Waslander, “Line-of-sight-attenuation chemical species tomography through the level set method,” J. Quant. Spectrosc. Radiat. Transf. 143, 25–34 (2014).
[Crossref]

2012 (1)

W. D. Penny, “Comparing dynamic causal models using AIC, BIC and free energy,” Neuroimage 59(1), 319–330 (2012).
[Crossref] [PubMed]

2011 (2)

J. Floyd and A. M. Kempf, “Computed tomography of chemiluminescence (CTC): high resolution and instantaneous 3-D measurements of a matrix burner,” Proc. Combust. Inst. 33(1), 751–758 (2011).
[Crossref]

N. Terzija and H. McCann, “Wavelet-based image reconstruction for hard-field tomography with severely limited data,” IEEE Sens. J. 11(9), 1885–1893 (2011).
[Crossref]

2010 (1)

C. Tape, Q. Liu, A. Maggi, and J. Tromp, “Seismic tomography of the southern California crust based on spectral-element and adjoint methods,” Geophys. J. Int. 180(1), 433–462 (2010).
[Crossref]

2008 (1)

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(9), 094007 (2008).
[Crossref]

2007 (1)

J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
[Crossref]

2006 (2)

M. Sambridge, K. Gallagher, A. Jackson, and P. Rickwood, “Trans-dimensional inverse problems, model comparison, and the evidence,” Geophys. J. Int. 157(2), 528 (2006).
[Crossref]

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
[Crossref]

2005 (1)

W. Verkruysse and L. A. Todd, “Novel algorithm for tomographic reconstruction of atmospheric chemicals with sparse sampling,” Environ. Sci. Technol. 39(7), 2247–2254 (2005).
[Crossref] [PubMed]

2004 (2)

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46(2), 329–345 (2004).
[Crossref]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

1998 (1)

M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
[Crossref] [PubMed]

1995 (1)

R. E. Kass and A. E. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90(430), 773–795 (1995).
[Crossref]

1993 (2)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

D. Verhoeven, “Limited-data computed tomography algorithms for the physical sciences,” Appl. Opt. 32(20), 3736–3754 (1993).
[Crossref] [PubMed]

Arridge, S. R.

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Cai, W.

T. Yu, B. Tian, and W. Cai, “Development of a beam optimization method for absorption-based tomography,” Opt. Express 25(6), 5982–5999 (2017).
[Crossref] [PubMed]

W. Cai and C. F. Kaminski, “Tomographic absorption spectroscopy for the study of gas dynamics and reactive flows,” Pror. Energy Combust. Sci. 59, 1–31 (2017).
[Crossref]

Chen, L.

Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
[Crossref]

Daun, K. J.

S. J. Grauer, R. W. Tsang, and K. J. Daun, “Broadband chemical species tomography: Measurement theory and a proof-of-concept emission detection experiment,” J. Quant. Spectrosc. Radiat. Transf. 198, 145–154 (2017).
[Crossref]

S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Improving chemical species tomography of turbulent flows using covariance estimation,” Appl. Opt. 56(13), 3900–3912 (2017).
[Crossref] [PubMed]

K. J. Daun, S. J. Grauer, and P. J. Hadwin, “Chemical species tomography of turbulent flows: Discrete ill-posed and rank deficient problems and the use of prior information,” J. Quant. Spectrosc. Radiat. Transf. 172, 58–74 (2016).
[Crossref]

S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Bayesian approach to the design of chemical species tomography experiments,” Appl. Opt. 55(21), 5772–5782 (2016).
[Crossref] [PubMed]

M. G. Twynstra, K. J. Daun, and S. L. Waslander, “Line-of-sight-attenuation chemical species tomography through the level set method,” J. Quant. Spectrosc. Radiat. Transf. 143, 25–34 (2014).
[Crossref]

Davidson, J. L.

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(9), 094007 (2008).
[Crossref]

Delpy, D. T.

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

Dreier, T.

Dreizler, A.

Feiling, A.

Floyd, J.

J. Floyd and A. M. Kempf, “Computed tomography of chemiluminescence (CTC): high resolution and instantaneous 3-D measurements of a matrix burner,” Proc. Combust. Inst. 33(1), 751–758 (2011).
[Crossref]

Gallagher, K.

M. Sambridge, K. Gallagher, A. Jackson, and P. Rickwood, “Trans-dimensional inverse problems, model comparison, and the evidence,” Geophys. J. Int. 157(2), 528 (2006).
[Crossref]

Garcia-Stewart, C. A.

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(9), 094007 (2008).
[Crossref]

Goedecke, G. H.

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
[Crossref]

Göers, S.

Grauer, S. J.

S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Improving chemical species tomography of turbulent flows using covariance estimation,” Appl. Opt. 56(13), 3900–3912 (2017).
[Crossref] [PubMed]

S. J. Grauer, R. W. Tsang, and K. J. Daun, “Broadband chemical species tomography: Measurement theory and a proof-of-concept emission detection experiment,” J. Quant. Spectrosc. Radiat. Transf. 198, 145–154 (2017).
[Crossref]

K. J. Daun, S. J. Grauer, and P. J. Hadwin, “Chemical species tomography of turbulent flows: Discrete ill-posed and rank deficient problems and the use of prior information,” J. Quant. Spectrosc. Radiat. Transf. 172, 58–74 (2016).
[Crossref]

S. J. Grauer, P. J. Hadwin, and K. J. Daun, “Bayesian approach to the design of chemical species tomography experiments,” Appl. Opt. 55(21), 5772–5782 (2016).
[Crossref] [PubMed]

Guo, Z.

Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
[Crossref]

Hadwin, P. J.

Jackson, A.

M. Sambridge, K. Gallagher, A. Jackson, and P. Rickwood, “Trans-dimensional inverse problems, model comparison, and the evidence,” Geophys. J. Int. 157(2), 528 (2006).
[Crossref]

Kaipio, J. P.

J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
[Crossref]

M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
[Crossref] [PubMed]

Kaminski, C. F.

W. Cai and C. F. Kaminski, “Tomographic absorption spectroscopy for the study of gas dynamics and reactive flows,” Pror. Energy Combust. Sci. 59, 1–31 (2017).
[Crossref]

Karjalainen, P. A.

M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
[Crossref] [PubMed]

Kass, R. E.

R. E. Kass and A. E. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90(430), 773–795 (1995).
[Crossref]

Kempf, A. M.

K. Mohri, S. Göers, J. Schöler, A. Rittler, T. Dreier, C. Schulz, and A. M. Kempf, “Instantaneous 3D-imaging of highly turbulent flames using Computed Tomography of Chemiluminescence (CTC),” Appl. Opt. 56(26), 7385–7395 (2017).
[Crossref]

J. Floyd and A. M. Kempf, “Computed tomography of chemiluminescence (CTC): high resolution and instantaneous 3-D measurements of a matrix burner,” Proc. Combust. Inst. 33(1), 751–758 (2011).
[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(9), 094007 (2008).
[Crossref]

Liu, Q.

C. Tape, Q. Liu, A. Maggi, and J. Tromp, “Seismic tomography of the southern California crust based on spectral-element and adjoint methods,” Geophys. J. Int. 180(1), 433–462 (2010).
[Crossref]

Maggi, A.

C. Tape, Q. Liu, A. Maggi, and J. Tromp, “Seismic tomography of the southern California crust based on spectral-element and adjoint methods,” Geophys. J. Int. 180(1), 433–462 (2010).
[Crossref]

McCann, H.

N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
[Crossref] [PubMed]

N. Terzija and H. McCann, “Wavelet-based image reconstruction for hard-field tomography with severely limited data,” IEEE Sens. J. 11(9), 1885–1893 (2011).
[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(9), 094007 (2008).
[Crossref]

Mohri, K.

Ostashev, V. E.

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
[Crossref]

Ozanyan, K. B.

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(9), 094007 (2008).
[Crossref]

Pegrum, S.

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(9), 094007 (2008).
[Crossref]

Penny, W. D.

W. D. Penny, “Comparing dynamic causal models using AIC, BIC and free energy,” Neuroimage 59(1), 319–330 (2012).
[Crossref] [PubMed]

Persson, P.-O.

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46(2), 329–345 (2004).
[Crossref]

Polydorides, N.

N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
[Crossref] [PubMed]

Prat, V. A.

N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
[Crossref] [PubMed]

Raftery, A. E.

R. E. Kass and A. E. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90(430), 773–795 (1995).
[Crossref]

Rickwood, P.

M. Sambridge, K. Gallagher, A. Jackson, and P. Rickwood, “Trans-dimensional inverse problems, model comparison, and the evidence,” Geophys. J. Int. 157(2), 528 (2006).
[Crossref]

Rittler, A.

Sambridge, M.

M. Sambridge, K. Gallagher, A. Jackson, and P. Rickwood, “Trans-dimensional inverse problems, model comparison, and the evidence,” Geophys. J. Int. 157(2), 528 (2006).
[Crossref]

Schöler, J.

Schulz, C.

Schweiger, M.

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Somersalo, E.

J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
[Crossref]

M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
[Crossref] [PubMed]

Song, Y.

Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
[Crossref]

Strang, G.

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46(2), 329–345 (2004).
[Crossref]

Stritzke, F.

Tape, C.

C. Tape, Q. Liu, A. Maggi, and J. Tromp, “Seismic tomography of the southern California crust based on spectral-element and adjoint methods,” Geophys. J. Int. 180(1), 433–462 (2010).
[Crossref]

Terzija, N.

N. Terzija and H. McCann, “Wavelet-based image reconstruction for hard-field tomography with severely limited data,” IEEE Sens. J. 11(9), 1885–1893 (2011).
[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(9), 094007 (2008).
[Crossref]

Tian, B.

Todd, L. A.

W. Verkruysse and L. A. Todd, “Novel algorithm for tomographic reconstruction of atmospheric chemicals with sparse sampling,” Environ. Sci. Technol. 39(7), 2247–2254 (2005).
[Crossref] [PubMed]

Tromp, J.

C. Tape, Q. Liu, A. Maggi, and J. Tromp, “Seismic tomography of the southern California crust based on spectral-element and adjoint methods,” Geophys. J. Int. 180(1), 433–462 (2010).
[Crossref]

Tsang, R. W.

S. J. Grauer, R. W. Tsang, and K. J. Daun, “Broadband chemical species tomography: Measurement theory and a proof-of-concept emission detection experiment,” J. Quant. Spectrosc. Radiat. Transf. 198, 145–154 (2017).
[Crossref]

Tsekenis, S.-A.

N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
[Crossref] [PubMed]

Twynstra, M. G.

M. G. Twynstra, K. J. Daun, and S. L. Waslander, “Line-of-sight-attenuation chemical species tomography through the level set method,” J. Quant. Spectrosc. Radiat. Transf. 143, 25–34 (2014).
[Crossref]

Vadász, D.

M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
[Crossref] [PubMed]

van der Kley, S.

Vauhkonen, M.

M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
[Crossref] [PubMed]

Vecherin, S. N.

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
[Crossref]

Verhoeven, D.

Verkruysse, W.

W. Verkruysse and L. A. Todd, “Novel algorithm for tomographic reconstruction of atmospheric chemicals with sparse sampling,” Environ. Sci. Technol. 39(7), 2247–2254 (2005).
[Crossref] [PubMed]

Voronovich, A. G.

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
[Crossref]

Wagner, S.

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Waslander, S. L.

M. G. Twynstra, K. J. Daun, and S. L. Waslander, “Line-of-sight-attenuation chemical species tomography through the level set method,” J. Quant. Spectrosc. Radiat. Transf. 143, 25–34 (2014).
[Crossref]

Wilson, D. K.

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
[Crossref]

Wright, P.

N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
[Crossref] [PubMed]

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(9), 094007 (2008).
[Crossref]

Wulan, T.

Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
[Crossref]

Yu, T.

Yuan, Q.

Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
[Crossref]

Appl. Opt. (4)

Environ. Sci. Technol. (1)

W. Verkruysse and L. A. Todd, “Novel algorithm for tomographic reconstruction of atmospheric chemicals with sparse sampling,” Environ. Sci. Technol. 39(7), 2247–2254 (2005).
[Crossref] [PubMed]

Geophys. J. Int. (2)

M. Sambridge, K. Gallagher, A. Jackson, and P. Rickwood, “Trans-dimensional inverse problems, model comparison, and the evidence,” Geophys. J. Int. 157(2), 528 (2006).
[Crossref]

C. Tape, Q. Liu, A. Maggi, and J. Tromp, “Seismic tomography of the southern California crust based on spectral-element and adjoint methods,” Geophys. J. Int. 180(1), 433–462 (2010).
[Crossref]

IEEE Sens. J. (1)

N. Terzija and H. McCann, “Wavelet-based image reconstruction for hard-field tomography with severely limited data,” IEEE Sens. J. 11(9), 1885–1893 (2011).
[Crossref]

IEEE Trans. Image Process. (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (1)

M. Vauhkonen, D. Vadász, P. A. Karjalainen, E. Somersalo, and J. P. Kaipio, “Tikhonov regularization and prior information in electrical impedance tomography,” IEEE Trans. Med. Imaging 17(2), 285–293 (1998).
[Crossref] [PubMed]

J. Acoust. Soc. Am. (1)

S. N. Vecherin, V. E. Ostashev, G. H. Goedecke, D. K. Wilson, and A. G. Voronovich, “Time-dependent stochastic inversion in acoustic travel-time tomography of the atmosphere,” J. Acoust. Soc. Am. 119(5), 2579–2588 (2006).
[Crossref]

J. Am. Stat. Assoc. (1)

R. E. Kass and A. E. Raftery, “Bayes factors,” J. Am. Stat. Assoc. 90(430), 773–795 (1995).
[Crossref]

J. Comput. Appl. Math. (1)

J. P. Kaipio and E. Somersalo, “Statistical inverse problems: discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
[Crossref]

J. Math. Imaging Vis. (1)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vis. 3(3), 263–283 (1993).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (3)

K. J. Daun, S. J. Grauer, and P. J. Hadwin, “Chemical species tomography of turbulent flows: Discrete ill-posed and rank deficient problems and the use of prior information,” J. Quant. Spectrosc. Radiat. Transf. 172, 58–74 (2016).
[Crossref]

M. G. Twynstra, K. J. Daun, and S. L. Waslander, “Line-of-sight-attenuation chemical species tomography through the level set method,” J. Quant. Spectrosc. Radiat. Transf. 143, 25–34 (2014).
[Crossref]

S. J. Grauer, R. W. Tsang, and K. J. Daun, “Broadband chemical species tomography: Measurement theory and a proof-of-concept emission detection experiment,” J. Quant. Spectrosc. Radiat. Transf. 198, 145–154 (2017).
[Crossref]

Meas. Sci. Technol. (1)

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(9), 094007 (2008).
[Crossref]

Neuroimage (1)

W. D. Penny, “Comparing dynamic causal models using AIC, BIC and free energy,” Neuroimage 59(1), 319–330 (2012).
[Crossref] [PubMed]

Opt. Commun. (1)

Z. Guo, Y. Song, Q. Yuan, T. Wulan, and L. Chen, “Simultaneous reconstruction of 3D refractive index, temperature, and intensity distribution of combustion flame by double computed tomography technologies based on spatial phase-shifting method,” Opt. Commun. 393, 123–130 (2017).
[Crossref]

Opt. Express (2)

Proc. Combust. Inst. (1)

J. Floyd and A. M. Kempf, “Computed tomography of chemiluminescence (CTC): high resolution and instantaneous 3-D measurements of a matrix burner,” Proc. Combust. Inst. 33(1), 751–758 (2011).
[Crossref]

Proc. Math. Phys. Eng. Sci. (1)

N. Polydorides, S.-A. Tsekenis, H. McCann, V. A. Prat, and P. Wright, “An efficient approach for limited-data chemical species tomography and its error bounds,” Proc. Math. Phys. Eng. Sci. 472(2187), 20150875 (2016).
[Crossref] [PubMed]

Pror. Energy Combust. Sci. (1)

W. Cai and C. F. Kaminski, “Tomographic absorption spectroscopy for the study of gas dynamics and reactive flows,” Pror. Energy Combust. Sci. 59, 1–31 (2017).
[Crossref]

SIAM Rev. (1)

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46(2), 329–345 (2004).
[Crossref]

Other (5)

M. Zöchbauer, H. Smith, and T. Lauer, “Advanced SCR flow modeling with a validated large eddy simulation,” SAE Technical Paper 2015–01–1046 (2015).
[Crossref]

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (SIAM, 2010).

J. R. Howell, M. P. Menguc, and R. Siegel, Thermal Radiation Heat Transfer (CRC Press, 2010).

J. P. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer, 2006).

C. L. Lawson and R. J. Hanson, Solving Least-Squares Problems (Prentice Hall, 1974), Chap. 23.

Supplementary Material (1)

NameDescription
» Visualization 1       The visualization depicts chemical species tomography of ammonia: a) LES data projected onto a ground truth mesh, b) reconstructions on a high-resolution mesh, and c) reconstructions on a coarse mesh.

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Figures (7)

Fig. 1
Fig. 1 Finite element domain for CST: a) circular domain in global coordinates (x, y) with a single LOS, and b) single element in local coordinates (ξ, η) with piecewise constant c, linear li, and quadratic qi node placement. Linear nodes are also the first three quadratic nodes.
Fig. 2
Fig. 2 a) Geometry of a LES domain for a SCR simulation by Zöchbauer et al. [29] and the measurement array from Stritzke et al. [4], and b) the 4422-node ground truth mesh.
Fig. 3
Fig. 3 Number of basis functions for meshes in a CST model selection study.
Fig. 4
Fig. 4 a) Ground truth NH3 distribution at t = 0. 125 s, b) reconstructed NH3 on Φ(25) with a piecewise quadratic basis, and c) reconstructed NH3 on Φ(5) with piecewise constant basis.
Fig. 5
Fig. 5 Model selection applied to prior information; trends in a) the log model likelihood, and b) the SSIM index vs. the number of basis functions.
Fig. 6
Fig. 6 SSIM index vs. log model likelihood for a) different forms of prior information, and b) uniform, linear, and quadratic bases.
Fig. 7
Fig. 7 Comparison of μ0 and μest for increasing mesh density in terms of a) the log model likelihood components, and b) the Bayes factor.

Equations (25)

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I ηL = I η0 exp{ 0 L κ η du }+ 0 L κ η I bη exp{ u L κ η du }du ,
b i ln[ I η0 I ηL ] 0 L κ η du .
κ η (x,y) j=1 n α j ϕ j (x,y) ,
b i j=1 n α j 0 L ϕ j [ r i (u) ]du ,
[ ξ(x,y) η(x,y) ]= [ x l2 x l1 x l3 x l1 y l2 y l1 y l3 y l1 ] 1 [ x x l1 y y l1 ].
ϕ j (ξ,η)={ N j (η,ξ), 0, ξ,η[0,1],ξ+η1 otherwise ,
N l1 =1ξη, N l2 =ξ, N l3 =η,
N l1 =(1ξη)(12ξ2η), N l2 =ξ(2ξ1), N l3 =η(2η1), N q4 =4ξ(1ξη), N q5 =4ξη, N q6 =4η(1ξη) .
A ij = δΩΩ C δΩ S δΩ 0 L ϕ i [ r i (u) ]du .
S δΩ = pδΩ 0 L N p [ r i (u) ]du
π(x|b)= π(b|x) π pr (x) π(b) π(b|x) π pr (x),
π(b|x)= 1 (2π) m det( Γ e ) exp{ 1 2 Axb L e 2 },
π pr (x)= 1 (2π) n det( Γ x ) exp{ 1 2 xμ L x 2 },
π(x|b)exp{ 1 2 [ L e A L x ]x[ L e b L x μ ] 2 2 }.
x MAP = argmin x0 { [ L e A L x ]x[ L e b L x μ ] 2 2 }.
π(x|b)= 1 (2π) n det( Γ x|b ) exp{ 1 2 x x MAP L x|b 2 },
π( M i |b)= π(b| M i ) π pr ( M i ) π(b) π(b| M i ),
π(x|b, M i )= π(b|x, M i ) π pr (x| M i ) π(b| M i ) .
π(b| M i )= π(b|x, M i ) π pr (x| M i ) π(x|b, M i ) π( M i |b).
π(b| M i )= det( Γ x|b ) (2π) m det( Γ e )det( Γ x ) exp{ 1 2 [ L e A L x ] x MAP [ L e b L x μ ] 2 2 }.
B ij = π(b| M i ) π(b| M j ) .
L i =ln[ π(b| M i ) ] = 1 2 ln[ det( Γ x|b ) ] 1 2 ln[ det( Γ e ) ] 1 2 ln[ det( Γ x ) ]. 1 2 A x MAP b L e 2 1 2 x MAP μ L x 2
V= 1 2 k=1 n { ln[ λ x|b (k) λ x (k) ] } ,
D= 1 2 A x MAP b L e 2
P= 1 2 x MAP μ L x 2

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