Abstract

This work aims to develop a fan-beam tomographic sensor using tunable diode lasers that can simultaneously image temperature and gas concentration with both high spatial and temporal resolutions. The sensor features three key advantages. First, the sensor bases on a stationary fan-beam arrangement, by which a high spatial resolution is guaranteed because the distance between two neighboring detectors in a view is approximately reduced to the size of a photodiode. Second, fan-beam illumination from five views is simultaneously generated instead of rotating either the fanned beams or the target, which significantly enhances the temporal resolution. Third, a novel set of optics with the combination of anamorphic prism pair and cylindrical lens is designed, which greatly improves the uniformity of the planar beams, and hence improves the reconstruction fidelity. This paper reports the tomographic model, optics design, numerical simulation and experimental validation of the sensor. The sensor exhibits good applicability for flame monitoring and combustion diagnosis.

© 2015 Optical Society of America

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References

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2015 (5)

M. A. Bolshov, Y. A. Kuritsyn, and Y. V. Romanovskii, “Tunable diode laser spectroscopy as a technique for combustion diagnostics,” Spectrochim. Acta B At. Spectrosc. 106, 45–66 (2015).
[Crossref]

M. P. Wood and K. B. Ozanyan, “Simultaneous temperature, concentration, and pressure imaging of water vapor in a turbine engine,” IEEE Sens. J. 15(1), 545–551 (2015).
[Crossref]

S. Shakya, P. Munshi, A. Luke, and D. Mewes, “Computerized tomography application in oil industry using KT-2 signature,” Res. Nondestr. Eval. 26(2), 61–89 (2015).
[Crossref]

S. A. Tsekenis, N. Tait, and H. McCann, “Spatially resolved and observer-free experimental quantification of spatial resolution in tomographic images,” Rev. Sci. Instrum. 86(3), 035104 (2015).
[Crossref] [PubMed]

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

2014 (6)

L. Xu, C. Liu, D. Zheng, Z. Cao, and W. Cai, “Digital signal processor-based high-precision on-line Voigt lineshape fitting for direct absorption spectroscopy,” Rev. Sci. Instrum. 85(12), 123108 (2014).
[Crossref] [PubMed]

C. Liu, L. Xu, Z. Cao, and H. McCann, “Reconstruction of axisymmetric temperature and gas concentration distributions by combining fan-beam TDLAS with onion-peeling deconvolution,” IEEE Trans. Instrum. Meas. 63(12), 3067–3075 (2014).
[Crossref]

L. Xu, T. Wei, J. Zhou, and Z. Cao, “Modified Landweber algorithm for robust particle sizing by using Fraunhofer diffraction,” Appl. Opt. 53(27), 6185–6193 (2014).
[Crossref] [PubMed]

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

W. Cai and C. F. Kaminski, “A tomographic technique for the simultaneous imaging of temperature, chemical species, and pressure in reactive flows using absorption spectroscopy with frequency-agile lasers,” Appl. Phys. Lett. 104(3), 034101 (2014).
[Crossref]

R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014).
[Crossref]

2013 (4)

2012 (2)

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Med. 28(2), 94–108 (2012).
[Crossref] [PubMed]

M. G. Twynstra and K. J. Daun, “Laser-absorption tomography beam arrangement optimization using resolution matrices,” Appl. Opt. 51(29), 7059–7068 (2012).
[Crossref] [PubMed]

2011 (2)

V. Kasyutich and P. Martin, “Towards a two-dimensional concentration and temperature laser absorption tomography sensor system,” Appl. Phys. B 102(1), 149–162 (2011).
[Crossref]

F. Li, X. Yu, H. Gu, Z. Li, Y. Zhao, L. Ma, L. Chen, and X. Chang, “Simultaneous measurements of multiple flow parameters for scramjet characterization using tunable diode-laser sensors,” Appl. Opt. 50(36), 6697–6707 (2011).
[Crossref] [PubMed]

2010 (2)

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[Crossref]

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

2009 (1)

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 (2)

X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of Nonuniform Temperature Distributions Using Line-of-Sight Absorption Spectroscopy,” AIAA J. 45(2), 411–419 (2007).
[Crossref]

M. Lackner, “Tunable diode laser absorption spectroscopy (TDLAS) in the process industries - A review,” Rev. Chem. Eng. 23(2), 65–147 (2007).
[Crossref]

2005 (1)

2001 (1)

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

1999 (1)

W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. 10(11), 1065–1069 (1999).
[Crossref]

1993 (1)

1992 (1)

Beister, M.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Med. 28(2), 94–108 (2012).
[Crossref] [PubMed]

Beiting, E. J.

Bolshov, M. A.

M. A. Bolshov, Y. A. Kuritsyn, and Y. V. Romanovskii, “Tunable diode laser spectroscopy as a technique for combustion diagnostics,” Spectrochim. Acta B At. Spectrosc. 106, 45–66 (2015).
[Crossref]

Cai, W.

W. Cai and C. F. Kaminski, “A tomographic technique for the simultaneous imaging of temperature, chemical species, and pressure in reactive flows using absorption spectroscopy with frequency-agile lasers,” Appl. Phys. Lett. 104(3), 034101 (2014).
[Crossref]

L. Xu, C. Liu, D. Zheng, Z. Cao, and W. Cai, “Digital signal processor-based high-precision on-line Voigt lineshape fitting for direct absorption spectroscopy,” Rev. Sci. Instrum. 85(12), 123108 (2014).
[Crossref] [PubMed]

L. Ma, W. Cai, A. W. Caswell, T. Kraetschmer, S. T. Sanders, S. Roy, and J. R. Gord, “Tomographic imaging of temperature and chemical species based on hyperspectral absorption spectroscopy,” Opt. Express 17(10), 8602–8613 (2009).
[Crossref] [PubMed]

Cao, Z.

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

C. Liu, L. Xu, Z. Cao, and H. McCann, “Reconstruction of axisymmetric temperature and gas concentration distributions by combining fan-beam TDLAS with onion-peeling deconvolution,” IEEE Trans. Instrum. Meas. 63(12), 3067–3075 (2014).
[Crossref]

L. Xu, C. Liu, D. Zheng, Z. Cao, and W. Cai, “Digital signal processor-based high-precision on-line Voigt lineshape fitting for direct absorption spectroscopy,” Rev. Sci. Instrum. 85(12), 123108 (2014).
[Crossref] [PubMed]

L. Xu, T. Wei, J. Zhou, and Z. Cao, “Modified Landweber algorithm for robust particle sizing by using Fraunhofer diffraction,” Appl. Opt. 53(27), 6185–6193 (2014).
[Crossref] [PubMed]

C. Liu, L. Xu, and Z. Cao, “Measurement of nonuniform temperature and concentration distributions by combiningline-of-sight TDLAS with regularization methods,” Appl. Opt. 52(20), 4827–4842 (2013).
[Crossref] [PubMed]

Carey, S. J.

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

Caswell, A. W.

Cen, K. F.

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

Chang, X.

Chen, L.

Chi, Y.

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

Clough, E.

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

Colbourne, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[Crossref]

Crossley, S. D.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[Crossref]

Daun, K. J.

Davidson, J. L.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 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(9), 094007 (2008).
[Crossref]

Ebert, V.

Garcia-Castillo, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[Crossref]

Garcia-Stewart, C.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[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]

Goldenstein, C. S.

R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014).
[Crossref]

Gord, J. R.

Gu, H.

Hanson, R.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Hanson, R. K.

R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014).
[Crossref]

X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of Nonuniform Temperature Distributions Using Line-of-Sight Absorption Spectroscopy,” AIAA J. 45(2), 411–419 (2007).
[Crossref]

Hindle, F. P.

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

Hong, Y.

J. Song, Y. Hong, G. Wang, and H. Pan, “Algebraic tomographic reconstruction of two-dimensional gas temperature based on tunable diode laser absorption spectroscopy,” Appl. Phys. B 112(4), 529–537 (2013).
[Crossref]

Huang, Q. X.

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

Jeffries, J.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Jeffries, J. B.

R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014).
[Crossref]

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of Nonuniform Temperature Distributions Using Line-of-Sight Absorption Spectroscopy,” AIAA J. 45(2), 411–419 (2007).
[Crossref]

Kaiser, S.

Kalender, W. A.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Med. 28(2), 94–108 (2012).
[Crossref] [PubMed]

Kaminski, C. F.

W. Cai and C. F. Kaminski, “A tomographic technique for the simultaneous imaging of temperature, chemical species, and pressure in reactive flows using absorption spectroscopy with frequency-agile lasers,” Appl. Phys. Lett. 104(3), 034101 (2014).
[Crossref]

Kasyutich, V.

V. Kasyutich and P. Martin, “Towards a two-dimensional concentration and temperature laser absorption tomography sensor system,” Appl. Phys. B 102(1), 149–162 (2011).
[Crossref]

Klein, A.

Kolditz, D.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Med. 28(2), 94–108 (2012).
[Crossref] [PubMed]

Kraetschmer, T.

Kuritsyn, Y. A.

M. A. Bolshov, Y. A. Kuritsyn, and Y. V. Romanovskii, “Tunable diode laser spectroscopy as a technique for combustion diagnostics,” Spectrochim. Acta B At. Spectrosc. 106, 45–66 (2015).
[Crossref]

Lackner, M.

M. Lackner, “Tunable diode laser absorption spectroscopy (TDLAS) in the process industries - A review,” Rev. Chem. Eng. 23(2), 65–147 (2007).
[Crossref]

Li, F.

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

F. Li, X. Yu, H. Gu, Z. Li, Y. Zhao, L. Ma, L. Chen, and X. Chang, “Simultaneous measurements of multiple flow parameters for scramjet characterization using tunable diode-laser sensors,” Appl. Opt. 50(36), 6697–6707 (2011).
[Crossref] [PubMed]

Li, N.

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

Li, X.

Li, Z.

Litt, T.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 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(9), 094007 (2008).
[Crossref]

Liu, C.

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

C. Liu, L. Xu, Z. Cao, and H. McCann, “Reconstruction of axisymmetric temperature and gas concentration distributions by combining fan-beam TDLAS with onion-peeling deconvolution,” IEEE Trans. Instrum. Meas. 63(12), 3067–3075 (2014).
[Crossref]

L. Xu, C. Liu, D. Zheng, Z. Cao, and W. Cai, “Digital signal processor-based high-precision on-line Voigt lineshape fitting for direct absorption spectroscopy,” Rev. Sci. Instrum. 85(12), 123108 (2014).
[Crossref] [PubMed]

C. Liu, L. Xu, and Z. Cao, “Measurement of nonuniform temperature and concentration distributions by combiningline-of-sight TDLAS with regularization methods,” Appl. Opt. 52(20), 4827–4842 (2013).
[Crossref] [PubMed]

Liu, X.

X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of Nonuniform Temperature Distributions Using Line-of-Sight Absorption Spectroscopy,” AIAA J. 45(2), 411–419 (2007).
[Crossref]

Luke, A.

S. Shakya, P. Munshi, A. Luke, and D. Mewes, “Computerized tomography application in oil industry using KT-2 signature,” Res. Nondestr. Eval. 26(2), 61–89 (2015).
[Crossref]

Ma, L.

Martin, P.

V. Kasyutich and P. Martin, “Towards a two-dimensional concentration and temperature laser absorption tomography sensor system,” Appl. Phys. B 102(1), 149–162 (2011).
[Crossref]

McCann, H.

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

S. A. Tsekenis, N. Tait, and H. McCann, “Spatially resolved and observer-free experimental quantification of spatial resolution in tomographic images,” Rev. Sci. Instrum. 86(3), 035104 (2015).
[Crossref] [PubMed]

C. Liu, L. Xu, Z. Cao, and H. McCann, “Reconstruction of axisymmetric temperature and gas concentration distributions by combining fan-beam TDLAS with onion-peeling deconvolution,” IEEE Trans. Instrum. Meas. 63(12), 3067–3075 (2014).
[Crossref]

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 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(9), 094007 (2008).
[Crossref]

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. 10(11), 1065–1069 (1999).
[Crossref]

Meffert, C.

Mewes, D.

S. Shakya, P. Munshi, A. Luke, and D. Mewes, “Computerized tomography application in oil industry using KT-2 signature,” Res. Nondestr. Eval. 26(2), 61–89 (2015).
[Crossref]

Munshi, P.

S. Shakya, P. Munshi, A. Luke, and D. Mewes, “Computerized tomography application in oil industry using KT-2 signature,” Res. Nondestr. Eval. 26(2), 61–89 (2015).
[Crossref]

Murray, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[Crossref]

Ozanyan, K.

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

Ozanyan, K. B.

M. P. Wood and K. B. Ozanyan, “Simultaneous temperature, concentration, and pressure imaging of water vapor in a turbine engine,” IEEE Sens. J. 15(1), 545–551 (2015).
[Crossref]

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 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(9), 094007 (2008).
[Crossref]

Pan, H.

J. Song, Y. Hong, G. Wang, and H. Pan, “Algebraic tomographic reconstruction of two-dimensional gas temperature based on tunable diode laser absorption spectroscopy,” Appl. Phys. B 112(4), 529–537 (2013).
[Crossref]

Pegrum, S.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 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(9), 094007 (2008).
[Crossref]

Plemmons, D. H.

Pummill, R.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Romanovskii, Y. V.

M. A. Bolshov, Y. A. Kuritsyn, and Y. V. Romanovskii, “Tunable diode laser spectroscopy as a technique for combustion diagnostics,” Spectrochim. Acta B At. Spectrosc. 106, 45–66 (2015).
[Crossref]

Roy, S.

Sanders, S. T.

Schultz, I. A.

R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014).
[Crossref]

Schulz, C.

Shakya, S.

S. Shakya, P. Munshi, A. Luke, and D. Mewes, “Computerized tomography application in oil industry using KT-2 signature,” Res. Nondestr. Eval. 26(2), 61–89 (2015).
[Crossref]

Song, J.

J. Song, Y. Hong, G. Wang, and H. Pan, “Algebraic tomographic reconstruction of two-dimensional gas temperature based on tunable diode laser absorption spectroscopy,” Appl. Phys. B 112(4), 529–537 (2013).
[Crossref]

Spearrin, R. M.

R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014).
[Crossref]

Spink, D. M.

W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. 10(11), 1065–1069 (1999).
[Crossref]

Sun, K.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Sur, R.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Tait, N.

S. A. Tsekenis, N. Tait, and H. McCann, “Spatially resolved and observer-free experimental quantification of spatial resolution in tomographic images,” Rev. Sci. Instrum. 86(3), 035104 (2015).
[Crossref] [PubMed]

Terzija, N.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 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(9), 094007 (2008).
[Crossref]

Tsekenis, S.

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

Tsekenis, S. A.

S. A. Tsekenis, N. Tait, and H. McCann, “Spatially resolved and observer-free experimental quantification of spatial resolution in tomographic images,” Rev. Sci. Instrum. 86(3), 035104 (2015).
[Crossref] [PubMed]

Turner, P.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[Crossref]

Twynstra, M. G.

Varghese, P. L.

Verhoeven, D.

Villarreal, R.

Wagner, D.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Wagner, S.

Waind, T.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Wang, F.

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

Wang, G.

J. Song, Y. Hong, G. Wang, and H. Pan, “Algebraic tomographic reconstruction of two-dimensional gas temperature based on tunable diode laser absorption spectroscopy,” Appl. Phys. B 112(4), 529–537 (2013).
[Crossref]

Wei, T.

Whitty, K.

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Winterbone, D. E.

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

Witzel, O.

Wood, M. P.

M. P. Wood and K. B. Ozanyan, “Simultaneous temperature, concentration, and pressure imaging of water vapor in a turbine engine,” IEEE Sens. J. 15(1), 545–551 (2015).
[Crossref]

Wright, P.

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 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(9), 094007 (2008).
[Crossref]

Xu, L.

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

C. Liu, L. Xu, Z. Cao, and H. McCann, “Reconstruction of axisymmetric temperature and gas concentration distributions by combining fan-beam TDLAS with onion-peeling deconvolution,” IEEE Trans. Instrum. Meas. 63(12), 3067–3075 (2014).
[Crossref]

L. Xu, T. Wei, J. Zhou, and Z. Cao, “Modified Landweber algorithm for robust particle sizing by using Fraunhofer diffraction,” Appl. Opt. 53(27), 6185–6193 (2014).
[Crossref] [PubMed]

L. Xu, C. Liu, D. Zheng, Z. Cao, and W. Cai, “Digital signal processor-based high-precision on-line Voigt lineshape fitting for direct absorption spectroscopy,” Rev. Sci. Instrum. 85(12), 123108 (2014).
[Crossref] [PubMed]

C. Liu, L. Xu, and Z. Cao, “Measurement of nonuniform temperature and concentration distributions by combiningline-of-sight TDLAS with regularization methods,” Appl. Opt. 52(20), 4827–4842 (2013).
[Crossref] [PubMed]

Yan, J. H.

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

Yang, W. Q.

W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. 10(11), 1065–1069 (1999).
[Crossref]

York, T. A.

W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. 10(11), 1065–1069 (1999).
[Crossref]

Yu, X.

Zhao, Y.

Zheng, D.

L. Xu, C. Liu, D. Zheng, Z. Cao, and W. Cai, “Digital signal processor-based high-precision on-line Voigt lineshape fitting for direct absorption spectroscopy,” Rev. Sci. Instrum. 85(12), 123108 (2014).
[Crossref] [PubMed]

Zhou, J.

AIAA J. (1)

X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of Nonuniform Temperature Distributions Using Line-of-Sight Absorption Spectroscopy,” AIAA J. 45(2), 411–419 (2007).
[Crossref]

Appl. Opt. (7)

Appl. Phys. B (5)

R. M. Spearrin, C. S. Goldenstein, I. A. Schultz, J. B. Jeffries, and R. K. Hanson, “Simultaneous sensing of temperature, CO, and CO2 in a scramjet combustor using quantum cascade laser absorption spectroscopy,” Appl. Phys. B 117(2), 689–698 (2014).
[Crossref]

J. Song, Y. Hong, G. Wang, and H. Pan, “Algebraic tomographic reconstruction of two-dimensional gas temperature based on tunable diode laser absorption spectroscopy,” Appl. Phys. B 112(4), 529–537 (2013).
[Crossref]

V. Kasyutich and P. Martin, “Towards a two-dimensional concentration and temperature laser absorption tomography sensor system,” Appl. Phys. B 102(1), 149–162 (2011).
[Crossref]

C. Liu, L. Xu, F. Li, Z. Cao, S. Tsekenis, and H. McCann, “Resolution-doubled one-dimensional wavelength modulation spectroscopy tomography for flame flatness validation of a flat-flame burner,” Appl. Phys. B 120(3), 407-416 (2015).

R. Sur, K. Sun, J. Jeffries, R. Hanson, R. Pummill, T. Waind, D. Wagner, and K. Whitty, “TDLAS-based sensors for in situ measurement of syngas composition in a pressurized, oxygen-blown, entrained flow coal gasifier,” Appl. Phys. B 116(1), 33–42 (2014).
[Crossref]

Appl. Phys. Lett. (1)

W. Cai and C. F. Kaminski, “A tomographic technique for the simultaneous imaging of temperature, chemical species, and pressure in reactive flows using absorption spectroscopy with frequency-agile lasers,” Appl. Phys. Lett. 104(3), 034101 (2014).
[Crossref]

Chem. Eng. J. (1)

P. Wright, N. Terzija, J. L. Davidson, S. Garcia-Castillo, C. Garcia-Stewart, S. Pegrum, S. Colbourne, P. Turner, S. D. Crossley, T. Litt, S. Murray, K. B. Ozanyan, and H. McCann, “High-speed chemical species tomography in a multi-cylinder automotive engine,” Chem. Eng. J. 158(1), 2–10 (2010).
[Crossref]

IEEE Sens. J. (1)

M. P. Wood and K. B. Ozanyan, “Simultaneous temperature, concentration, and pressure imaging of water vapor in a turbine engine,” IEEE Sens. J. 15(1), 545–551 (2015).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

C. Liu, L. Xu, Z. Cao, and H. McCann, “Reconstruction of axisymmetric temperature and gas concentration distributions by combining fan-beam TDLAS with onion-peeling deconvolution,” IEEE Trans. Instrum. Meas. 63(12), 3067–3075 (2014).
[Crossref]

J. Electron. Imaging (1)

F. P. Hindle, S. J. Carey, K. Ozanyan, D. E. Winterbone, E. Clough, and H. McCann, “Measurement of gaseous hydrocarbon distribution by a near-infrared absorption tomography system,” J. Electron. Imaging 10(3), 593–600 (2001).
[Crossref]

Meas. Sci. Technol. (3)

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]

F. Wang, K. F. Cen, N. Li, J. B. Jeffries, Q. X. Huang, J. H. Yan, and Y. Chi, “Two-dimensional tomography for gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy,” Meas. Sci. Technol. 21(4), 045301 (2010).
[Crossref]

W. Q. Yang, D. M. Spink, T. A. York, and H. McCann, “An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography,” Meas. Sci. Technol. 10(11), 1065–1069 (1999).
[Crossref]

Opt. Express (3)

Phys. Med. (1)

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Med. 28(2), 94–108 (2012).
[Crossref] [PubMed]

Res. Nondestr. Eval. (1)

S. Shakya, P. Munshi, A. Luke, and D. Mewes, “Computerized tomography application in oil industry using KT-2 signature,” Res. Nondestr. Eval. 26(2), 61–89 (2015).
[Crossref]

Rev. Chem. Eng. (1)

M. Lackner, “Tunable diode laser absorption spectroscopy (TDLAS) in the process industries - A review,” Rev. Chem. Eng. 23(2), 65–147 (2007).
[Crossref]

Rev. Sci. Instrum. (2)

S. A. Tsekenis, N. Tait, and H. McCann, “Spatially resolved and observer-free experimental quantification of spatial resolution in tomographic images,” Rev. Sci. Instrum. 86(3), 035104 (2015).
[Crossref] [PubMed]

L. Xu, C. Liu, D. Zheng, Z. Cao, and W. Cai, “Digital signal processor-based high-precision on-line Voigt lineshape fitting for direct absorption spectroscopy,” Rev. Sci. Instrum. 85(12), 123108 (2014).
[Crossref] [PubMed]

Spectrochim. Acta B At. Spectrosc. (1)

M. A. Bolshov, Y. A. Kuritsyn, and Y. V. Romanovskii, “Tunable diode laser spectroscopy as a technique for combustion diagnostics,” Spectrochim. Acta B At. Spectrosc. 106, 45–66 (2015).
[Crossref]

Other (1)

M. Buchmann and D. Mewes, “Tomographic Measurement and Reconstruction Techniques,” in Optical Measurements, F. Mayinger and O. Feldmann eds. (Springer Berlin Heidelberg, 2001), pp. 301–339.

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Figures (16)

Fig. 1
Fig. 1 Definitions of the coordinate system and the discretization configuration.
Fig. 2
Fig. 2 Schematic of the fan-beam generator. Panel (a): the optics design. Panel (b): the circular and elliptical beam at location 1 and 2 in Panel (a). Panel (c): uniformity of the fan-shaped planar illumination obtained with and without the use of the anamorphic prism pair (APP).
Fig. 3
Fig. 3 (a) Schematic and (b) layout of the five-view stationary fan-beam TDLAS-based tomographic sensor.
Fig. 4
Fig. 4 Qualification of the spatial resolution of the tomographic image. Panel (a): expected distribution of H2O concentration with a rectangular sharp-edged feature. Panel (b): the reconstructed distribution of H2O concentration and two sampled tracks to calculate the spatial resolution. Panel (c): edge spread functions and corresponding spatial resolution in the two sampled tracks.
Fig. 5
Fig. 5 Phantoms of asymmetric distributions of (a) temperature and (b) H2O concentration in the simulation.
Fig. 6
Fig. 6 Reconstructed distributions of (a) temperature and (b) H2O concentration with noise-free projections in the simulation.
Fig. 7
Fig. 7 Dependence of cT and cX on the number of grids with the designed tomographic sensor.
Fig. 8
Fig. 8 Reconstructed temperature distributions with noise-free projections when the number of grids equals (a) 124, (b) 332 and (c) 560, respectively.
Fig. 9
Fig. 9 Evaluation of the accuracy and robustness of the fan-beam TDLAS-based tomography sensor at different noise levels.
Fig. 10
Fig. 10 Reconstructed distributions of (a) temperature and (b) H2O concentration with 5% random noise adding on the noise-free projections in the simulation.
Fig. 11
Fig. 11 Flame generated by using a McKenna flat flame burner. In the experiment, the flow rates of methane, air and shrouding nitrogen were set to 1.2, 15.25 and 22.5 L/min, respectively.
Fig. 12
Fig. 12 Reconstructed distributions of (a) temperature and (b) H2O concentration at the height of 3 cm above the burner plug when the equivalence was set to 0.749.
Fig. 13
Fig. 13 Sampled TDLAS data obtained by a photodiode in room air and in the premixed flame for transitions at (a) v 1 = 7444.36 cm−1 and (b) v 2 = 7185.6 cm−1, respectively.
Fig. 14
Fig. 14 Experimental setup and tomographic results by putting a cube on the center of the burner plug. Panel (a) and Panel (b) show the schematic diagram and photo of the experimental setup, respectively. Panel (c) and Panel (d) show the reconstructed distributions of temperature and H2O concentrations, respectively.
Fig. 15
Fig. 15 Experimental setup and tomographic results by putting a cube on one side of the burner plug. Panel (a) and Panel (b) show the schematic diagram and photo of the experimental setup, respectively. Panel (c) and Panel (d) show the reconstructed distributions of temperature and H2O concentrations, respectively.
Fig. 16
Fig. 16 Experimental setup and tomographic results by putting two cubes on the burner plug. Panel (a) and Panel (b) show the schematic diagram and photo of the experimental setup, respectively. Panel (c) and Panel (d) show the reconstructed distributions of temperature and H2O concentrations, respectively.

Tables (3)

Tables Icon

Table 1 Dependence of cT on the number of views, noted as N v, and the number of laser beams in each view, noted as N l.

Tables Icon

Table 2 Dependence of cX on the number of views, noted as N v, and the number of laser beams in each view, noted as N l.

Tables Icon

Table 3 List of values of cT and convergence time of the modified Landweber algorithm, ART and SMART, respectively.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

α v = 0 L P ( x ) X ( x ) S [ T ( x ) ] ϕ d l ,
S ( T ) = S ( T 0 ) Q ( T 0 ) Q ( T ) ( T 0 T ) exp [ h c E ' ' k ( 1 T 1 T 0 ) ] × 1 exp ( h c v 0 / k T ) 1 exp ( h c v 0 / k T 0 ) ,
A v = α v d v = 0 L P ( x ) X a b s ( x ) S [ T ( x ) ] d l .
A v , i = j = 1 N a v , j L i j = j = 1 N [ P S ( T ) X ] v , j L i j ,
L a v = A v ,
L = [ L 11 L 12 L 1 N L 21 L 22 L 2 N L M 1 L M 2 L M N ] ,
a v k + 1 = a v k + λ k L T ( A v L a v k ) ,
Δ = j = 1 N | a v , j k + 1 a v , j k | / j = 1 N ( a v , j k ) .
R j = a v 1 , j a v 2 , j = S 1 ( T j ) S 2 ( T j ) = S 1 ( T 0 ) S 2 ( T 0 ) exp [ h c ( E 1 '' E 2 ' ' ) k ( 1 T j 1 T 0 ) ] .
T j = h c k ( E 2 ' ' E 1 ' ' ) / [ ln a v 1 , j a v 2 , j + ln S 2 ( T 0 ) S 1 ( T 0 ) + h c k ( E 2 ' ' E 1 ' ' ) T 0 ] .
X j = a v 1 , j / S 1 ( T j ) .
c T = j = 1 N p ( | T j r e c T j e x p | / T j e x p ) / N p ,
c X = j = 1 N p ( | X j r e c X j e x p | / X j e x p ) / N p ,
e T = j = 1 N ( | T j r e c T j M _ r e c | / T j M _ r e c ) / N ,
e X = j = 1 N ( | X j r e c X j M _ r e c | / X j M _ r e c ) / N ,

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