Abstract

A series of previous studies, both numerical and experimental, have demonstrated the advantages of hyperspectral tomography (HT) as a promising technique to measure the two-dimensional distributions of temperature and species concentration in reacting flows. This paper intends to prepare the mathematical groundwork for extended use of the HT technique for three-dimensional and/or time-correlated measurements. Direct application of the methods developed previously encounters both experimental and computational difficulties. Numerical studies reported in this paper suggest that the use of proper orthogonal decomposition (POD) is effective to overcome these difficulties. The use of POD in HT significantly reduces the computational cost, enhances the fidelity of the tomographic reconstructions, and improves the stability of the reconstruction in the presence of measurement noise. Implications of these results for practical applications are also discussed.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2009 (1)

2008 (4)

2007 (2)

2005 (1)

2000 (3)

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808-817 (2000).

A. Levy and M. Lindenbaum, “Sequential Karhunen-Loeve basis extraction and its application to images,” IEEE Trans. Image Process. 9, 1371-1374 (2000).
[CrossRef]

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

1988 (1)

M. Ravichandran and F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption-measurements,” Combust. Sci. Technol. 60, 231-248 (1988).
[CrossRef]

1987 (1)

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous-variables with the simulated annealing algorithm,” ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Alden, M.

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

An, X.

Cai, W.

Candel, S.

N. Docquier and S. Candel, “Sensor requirements for combustion control,” in Applied Combustion Diagnostics, K. Kohse-Hoinghaus, and J. B. Jeffries, eds. (Taylor & Francis, 2002), Chap. 21.

Carey, S. J.

Caswell, A. W.

Chatterjee, A.

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808-817 (2000).

Colbourne, S. M.

Corana, A.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous-variables with the simulated annealing algorithm,” ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Crossley, S. D.

Dagel, D.

Docquier, N.

N. Docquier and S. Candel, “Sensor requirements for combustion control,” in Applied Combustion Diagnostics, K. Kohse-Hoinghaus, and J. B. Jeffries, eds. (Taylor & Francis, 2002), Chap. 21.

Dreizler, A.

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

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]

Flannery, B. P.

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

Fujimoto, J. G.

Garcia-Stewart, C. A.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Gluesenkamp, M.

Gord, J. R.

Gouldin, F. C.

M. Ravichandran and F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption-measurements,” Combust. Sci. Technol. 60, 231-248 (1988).
[CrossRef]

Herold, R. E.

Hindle, F. P.

Huber, R.

Hult, J.

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

Hurr, W. J.

Kaminski, C. F.

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

King, G. B.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Kraetschmer, T.

Kranendonk, L. A.

Levy, A.

A. Levy and M. Lindenbaum, “Sequential Karhunen-Loeve basis extraction and its application to images,” IEEE Trans. Image Process. 9, 1371-1374 (2000).
[CrossRef]

Lindenbaum, M.

A. Levy and M. Lindenbaum, “Sequential Karhunen-Loeve basis extraction and its application to images,” IEEE Trans. Image Process. 9, 1371-1374 (2000).
[CrossRef]

Lindenmaier, S.

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

Ma, L.

Maas, U.

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

Marchesi, M.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous-variables with the simulated annealing algorithm,” ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Martini, C.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous-variables with the simulated annealing algorithm,” ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

McCannet, H.

Meyer, T. R.

Murray, S. C.

Okura, Y.

Ozanyan, K. B.

Pegrum, S. H.

Press, W. H.

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

Ravichandran, M.

M. Ravichandran and F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption-measurements,” Combust. Sci. Technol. 60, 231-248 (1988).
[CrossRef]

Ridella, S.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous-variables with the simulated annealing algorithm,” ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Roy, S.

Sanders, S. T.

Teukolsky, S. A.

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

Turner, P. J.

Urata, Y.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Vetterling, W. T.

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

Wright, P.

ACM Trans. Math. Softw. (1)

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous-variables with the simulated annealing algorithm,” ACM Trans. Math. Softw. 13, 262-280 (1987).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

A. Dreizler, S. Lindenmaier, U. Maas, J. Hult, M. Alden, and C. F. Kaminski, “Characterisation of a spark ignition system by planar laser-induced fluorescence of OH at high repetition rates and comparison with chemical kinetic calculations,” Appl. Phys. B 70, 287-294 (2000).
[CrossRef]

Combust. Sci. Technol. (1)

M. Ravichandran and F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption-measurements,” Combust. Sci. Technol. 60, 231-248 (1988).
[CrossRef]

Comput. Phys. Commun. (1)

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

Curr. Sci. (1)

A. Chatterjee, “An introduction to the proper orthogonal decomposition,” Curr. Sci. 78, 808-817 (2000).

IEEE Trans. Image Process. (1)

A. Levy and M. Lindenbaum, “Sequential Karhunen-Loeve basis extraction and its application to images,” IEEE Trans. Image Process. 9, 1371-1374 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Other (4)

N. Docquier and S. Candel, “Sensor requirements for combustion control,” in Applied Combustion Diagnostics, K. Kohse-Hoinghaus, and J. B. Jeffries, eds. (Taylor & Francis, 2002), Chap. 21.

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

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

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

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Figures (9)

Fig. 1
Fig. 1

Definition of the coordinate system and the discretization configuration.

Fig. 2
Fig. 2

Block diagram illustrating the simulated annealing algorithm.

Fig. 3
Fig. 3

First 80 eigenvalues of the correlation matrices formed by the snapshots of T and X distributions.

Fig. 4
Fig. 4

Random T distribution approximated by the linear combination of the first 25 KL basis functions.

Fig. 5
Fig. 5

Relative computational times for a HT problem consisting of three 10 × 10 frames. Case A: direct reconstruction without using POD. The computational time of this case is used to normalize all cases. Case B: reconstruction using 25 KL basis functions. Cases C and D: reconstruction when these three frames are assumed to be correlated.

Fig. 6
Fig. 6

Comparison of the fidelity of direct reconstruction and reconstruction using POD. Two phantoms were used and the results are shown in Panels A and B.

Fig. 7
Fig. 7

Use of POD improves the control of the bounds. Panel A: the upper and lowers bounds of the weighting coefficients. Panel B: the distance between the upper and lower bounds.

Fig. 8
Fig. 8

Use of POD improves the fidelity of HT reconstruction in the presence of measurement noise.

Fig. 9
Fig. 9

Simulation of the reconstruction of time-correlated HT measurements. Twenty-five KL basis functions were used in the POD.

Equations (7)

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p ( L j , λ i ) = P a b S [ T ( x , y ) , λ i ] · X ( x , y ) · d l ,
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 ,
F ( T rec , X rec ) = D ( T rec , X rec ) + γ T · R T ( T rec ) + γ X · R X ( X rec ) ,
if    Δ F = F ( x New ) F ( x Old ) 0 , accept   x New ; else accept  x New   with probability  p SA = exp ( Δ F T SA ) ,
T ( x , y ) k = 1 K a k Φ k ( x , y ) ,
e T = m = 1 M n = 1 N | T m , n K L T m , n | m = 1 M n = 1 N | T m , n | ,
Φ k 2 ( x , y ) d x d y = constant for     k = 1 , 2 , , 25.

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