Abstract

In a previous paper, we described a novel technique to exploit hyperspectral absorption spectroscopy to retrieve tomographic imaging of temperature and species concentration simultaneously. This technique casts the tomographic inversion into a nonlinear minimization problem with regularizations. Here a simple and effective method is developed to determine the optimal regularization parameters in the nonlinear optimization problem. This method, combined with the minimization method described previously, provides a robust algorithm for hyperspectral tomography. This method takes advantage of an inherent feature of absorption and is therefore expected to be useful for other sensing techniques based on absorption spectroscopy.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. 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]

2008

W. Cai and L. Ma, “Numerical investigation of hyperspectral tomography for simultaneous temperature and concentration imaging,” Appl. Opt. 47, 3751-3759 (2008).
[CrossRef] [PubMed]

W. Cai, D. J. Ewing, and L. Ma, “Application of simulated annealing for multispectral tomography,” Comput. Phys. Commun. 179, 250-255 (2008).
[CrossRef]

2005

2004

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

X. Zhou, X. Liu, J. B. Jeffries, and R. K. Hanson, “Development of a sensor for temperature and water concentration in combustion gases using a single tunable diode laser,” Meas. Sci. Technol. 14, 1459-1468 (2003).
[CrossRef]

1999

E. L. Piccolomini, and F. Zama, “The conjugate gradient regularization method in computed tomography problems,” Appl. Math. Comp. 102, 87-99 (1999).
[CrossRef]

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microw. Theory Tech. 47, 336-344 (1999).
[CrossRef]

1997

A. Franchois and C. Pichot, “Microwave imaging--complex permittivity reconstruction with a Levenberg-Marquardt method,” IEEE Trans. Antennas Propag. 45, 203-215 (1997).
[CrossRef]

1992

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral-equations of the 1st kind,” Inverse Probl. 8, 849-872 (1992).
[CrossRef]

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561-580 (1992).
[CrossRef]

1979

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]

1967

Cai, W.

W. Cai and L. Ma, “Numerical investigation of hyperspectral tomography for simultaneous temperature and concentration imaging,” Appl. Opt. 47, 3751-3759 (2008).
[CrossRef] [PubMed]

W. Cai, D. J. Ewing, and L. Ma, “Application of simulated annealing for multispectral tomography,” Comput. Phys. Commun. 179, 250-255 (2008).
[CrossRef]

Carey, S. J.

Colbourne, S. M.

Crossley, S. D.

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Gordon and Breach, 1996).

Ewing, D. J.

W. Cai, D. J. Ewing, and L. Ma, “Application of simulated annealing for multispectral tomography,” Comput. Phys. Commun. 179, 250-255 (2008).
[CrossRef]

Feldmann, O.

Optical Measurements: Techniques and Applications, F. Mayinger and O. Feldmann, eds. (Springer, 2001).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University, 1992).

Franchois, A.

A. Franchois and C. Pichot, “Microwave imaging--complex permittivity reconstruction with a Levenberg-Marquardt method,” IEEE Trans. Antennas Propag. 45, 203-215 (1997).
[CrossRef]

Garcia-Stewart, C. A.

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]

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]

Hansen, P. C.

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral-equations of the 1st kind,” Inverse Probl. 8, 849-872 (1992).
[CrossRef]

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561-580 (1992).
[CrossRef]

Hanson, R. K.

X. Zhou, X. Liu, J. B. Jeffries, and R. K. Hanson, “Development of a sensor for temperature and water concentration in combustion gases using a single tunable diode laser,” Meas. Sci. Technol. 14, 1459-1468 (2003).
[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]

Herman, C. T.

C. T. Herman, Image Reconstruction from Projections--The Fundamentals of Computerized Tomography (Academic Press, 1980).

Hindle, F. P.

Howell, H. B.

Hurr, W. J.

Jeffries, J. B.

X. Zhou, X. Liu, J. B. Jeffries, and R. K. Hanson, “Development of a sensor for temperature and water concentration in combustion gases using a single tunable diode laser,” Meas. Sci. Technol. 14, 1459-1468 (2003).
[CrossRef]

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]

Litt, T. J.

Liu, X.

X. Zhou, X. Liu, J. B. Jeffries, and R. K. Hanson, “Development of a sensor for temperature and water concentration in combustion gases using a single tunable diode laser,” Meas. Sci. Technol. 14, 1459-1468 (2003).
[CrossRef]

Ma, L.

W. Cai, D. J. Ewing, and L. Ma, “Application of simulated annealing for multispectral tomography,” Comput. Phys. Commun. 179, 250-255 (2008).
[CrossRef]

W. Cai and L. Ma, “Numerical investigation of hyperspectral tomography for simultaneous temperature and concentration imaging,” Appl. Opt. 47, 3751-3759 (2008).
[CrossRef] [PubMed]

McCann, H.

Murray, S. C.

Ozanyan, K. B.

Pegrum, S. H.

Piccolomini, E. L.

E. L. Piccolomini, and F. Zama, “The conjugate gradient regularization method in computed tomography problems,” Appl. Math. Comp. 102, 87-99 (1999).
[CrossRef]

Pichot, C.

A. Franchois and C. Pichot, “Microwave imaging--complex permittivity reconstruction with a Levenberg-Marquardt method,” IEEE Trans. Antennas Propag. 45, 203-215 (1997).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University, 1992).

Rekanos, I. T.

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microw. Theory Tech. 47, 336-344 (1999).
[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]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University, 1992).

Tsiboukis, T. D.

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microw. Theory Tech. 47, 336-344 (1999).
[CrossRef]

Turner, P. J.

Twomey, S.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University, 1992).

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]

Wright, P.

Yioultsis, T. V.

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microw. Theory Tech. 47, 336-344 (1999).
[CrossRef]

Zama, F.

E. L. Piccolomini, and F. Zama, “The conjugate gradient regularization method in computed tomography problems,” Appl. Math. Comp. 102, 87-99 (1999).
[CrossRef]

Zhou, X.

X. Zhou, X. Liu, J. B. Jeffries, and R. K. Hanson, “Development of a sensor for temperature and water concentration in combustion gases using a single tunable diode laser,” Meas. Sci. Technol. 14, 1459-1468 (2003).
[CrossRef]

Appl. Math. Comp.

E. L. Piccolomini, and F. Zama, “The conjugate gradient regularization method in computed tomography problems,” Appl. Math. Comp. 102, 87-99 (1999).
[CrossRef]

Appl. Opt.

Appl. Therm. Eng.

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]

Comput. Phys. Commun.

W. Cai, D. J. Ewing, and L. Ma, “Application of simulated annealing for multispectral tomography,” Comput. Phys. Commun. 179, 250-255 (2008).
[CrossRef]

IEEE Trans. Antennas Propag.

A. Franchois and C. Pichot, “Microwave imaging--complex permittivity reconstruction with a Levenberg-Marquardt method,” IEEE Trans. Antennas Propag. 45, 203-215 (1997).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

I. T. Rekanos, T. V. Yioultsis, and T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microw. Theory Tech. 47, 336-344 (1999).
[CrossRef]

Inverse Probl.

P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral-equations of the 1st kind,” Inverse Probl. 8, 849-872 (1992).
[CrossRef]

Meas. Sci. Technol.

X. Zhou, X. Liu, J. B. Jeffries, and R. K. Hanson, “Development of a sensor for temperature and water concentration in combustion gases using a single tunable diode laser,” Meas. Sci. Technol. 14, 1459-1468 (2003).
[CrossRef]

SIAM Rev.

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561-580 (1992).
[CrossRef]

Technometrics

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

C. T. Herman, Image Reconstruction from Projections--The Fundamentals of Computerized Tomography (Academic Press, 1980).

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Gordon and Breach, 1996).

Optical Measurements: Techniques and Applications, F. Mayinger and O. Feldmann, eds. (Springer, 2001).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University, 1992).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the HT problem and relevant notations.

Fig. 2
Fig. 2

Normalized mean absolute distance measure ( e T ) of T reconstruction at different combinations of γ T and γ X , illustrating the insensitivity of T reconstruction with respect to γ T and γ X over a wide range.

Fig. 3
Fig. 3

Contribution from D and the regularization factor to the master function, F, at different values of γ T , with γ X = 1 × 10 8 γ T .

Fig. 4
Fig. 4

Calculations shown in Fig. 2 performed with γ X = 0 , illustrating the method to estimate the optimal γ T and the insensitivity of T reconstruction with respect to γ X .

Fig. 5
Fig. 5

Block diagram summarizing the determination of the optimal regularization parameters in the HT technique.

Fig. 6
Fig. 6

Plot of the residual versus the regularization factor with g ranging from 10 9 to 10 3 , resulting in an L-shaped curve with a distinct corner.

Fig. 7
Fig. 7

Values of e X at different values of g, illustrating the minimum e X is achieved at a g corresponding to the corner of the L-curve.

Fig. 8
Fig. 8

Values of e X at different combinations of γ T and γ X , illustrating the insensitivity of the X reconstruction to the selection of γ T and γ X .

Equations (10)

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

p ( L j , λ i ) = P a b S [ T ( x , y ) , λ i ] X ( x , y ) d l ,
S ( T , λ i ) = S ( T 0 , λ i ) Q ( T ) Q ( T 0 ) exp [ h c E k ( 1 T - 1 T 0 ) ] × 1 exp ( h c 2 k T λ i ) 1 exp ( h c 2 k T 0 λ i ) ,
min D ( T rec , X rec ) = j = 1 J i = 1 I [ p m ( L j , λ i ) p c ( L j , λ i ) ] 2 p m ( L j , λ i ) 2 ,
R T ( T ) = m = 1 M n = 1 N [ T m , n 1 8 ( T m 1 , n 1 + T m 1 , n + T m 1 , n + 1 + T m , n 1 + T m , n + 1 + T m + 1 , n 1 + T m + 1 , n + T m + 1 , n + 1 ) ] 2 .
F ( T rec , X rec ) = D ( T rec , X rec ) + γ T R T ( T rec ) + γ X R X ( X rec ) ,
e T = m = 1 M n = 1 N | T m , n rec T m , n | m = 1 M n = 1 N | T m , n | ,
e X = m = 1 M n = 1 N | X m , n rec X m , n | m = 1 M n = 1 N | X m , n | .
p ¯ m = S ¯ ( T rec , λ ) · X ¯ ,
min p ¯ m S ¯ ( T rec , λ ) · X ¯ + g · H ¯ · X ¯ ,
g 0 = T r ( S T · S ) T r ( H T · H ) ,

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