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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: James C. Wyant
  • Vol. 47, Iss. 3 — Jan. 20, 2008
  • pp: 407–416

Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization

Emil O. Åkesson and Kyle J. Daun  »View Author Affiliations


Applied Optics, Vol. 47, Issue 3, pp. 407-416 (2008)
http://dx.doi.org/10.1364/AO.47.000407


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Abstract

Deconvolution of optically collected axisymmetric flame data is equivalent to solving an ill-posed problem subject to severe error amplification. Tikhonov regularization has recently been shown to be well suited for stabilizing this deconvolution, although the success of this method hinges on choosing a suitable regularization parameter. Incorporating a parameter selection scheme transforms this technique into a reliable automatic algorithm that outperforms unregularized deconvolution of a smoothed data set, which is currently the most popular way to analyze axisymmetric data. We review the discrepancy principle, L-curve curvature, and generalized cross-validation parameter selection schemes and conclude that the L-curve curvature algorithm is best suited to this problem.

© 2008 Optical Society of America

OCIS Codes
(120.1740) Instrumentation, measurement, and metrology : Combustion diagnostics
(120.7000) Instrumentation, measurement, and metrology : Transmission
(280.1740) Remote sensing and sensors : Combustion diagnostics
(280.2470) Remote sensing and sensors : Flames

ToC Category:
Remote Sensing and Sensors

History
Original Manuscript: August 22, 2007
Manuscript Accepted: November 24, 2007
Published: January 17, 2008

Citation
Emil O. Åkesson and Kyle J. Daun, "Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization," Appl. Opt. 47, 407-416 (2008)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-3-407


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References

  1. C. J. Dasch, "One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods," Appl. Opt. 31, 1146-1152 (1992). [CrossRef] [PubMed]
  2. R. J. Hall and P. A. Bonczyk, "Sooting flame thermometry using emission/absorption tomography," Appl. Opt. 29, 4590-4598 (1990). [CrossRef] [PubMed]
  3. D. R. Snelling, K. A. Thomson, G. J. Smallwood, and Ö. L. Gülder, "Two-dimensional imaging of soot volume fraction in laminar diffusion flames," Appl. Opt. 38, 2478-2485 (1999). [CrossRef]
  4. F. Cignoli, S. De Luliis, V. Manta, and G. Zizak, "Two-dimensional two-wavelength emission technique for soot diagnostics," Appl. Opt. 40, 5370-5378 (2001). [CrossRef]
  5. R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications (Springer, 1993).
  6. J. P. Holloway, S. Shannon, S. M. Sepke, and M. L. Brake, "A reconstruction algorithm for a spatially resolved plasma optical emission spectroscopy sensor," J. Quant. Spectrosc. Radiat. Transfer 68, 101-115 (2001). [CrossRef]
  7. K. J. Daun, K. A. Thomson, F. Liu, and G. J. Smallwood, "Deconvolution of axisymmetric flame properties using Tikhonov regularization," Appl. Opt. 45, 4638-4646 (2006). [CrossRef] [PubMed]
  8. C. Wu, Department of Chemical Engineering, Tsinghua University, Beijing, China (personal communication, 2007).
  9. A. N. Tikhonov and V. A. Arsenin, Solution of Ill-Posed Problems (Winston & Sons, 1977).
  10. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998). [CrossRef]
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).
  12. V. A. Morozov, "On the solution of functional equations by the method of regularization," Sov. Math. Dokl. 7, 414-417 (1966).
  13. P. C. Hansen and D. P. O'Leary, "The use of the L-curve in the regularization of discrete ill-posed problems," SIAM J. Sci. Comput. (USA) 14, 1487-1503 (1993). [CrossRef]
  14. 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]
  15. G. Wahba, Spline Models for Observational Data (SIAM, 1990). [CrossRef]
  16. J. Hadamard, Lectures on Cauchy's Problem in Linear Differential Equations (Yale U. Press, 1923).
  17. W. S. Cleveland and S. J. Devlin, "Locally-weighted regression: an approach to regression analysis by local fitting," J. Am. Stat. Assoc. 83, 596-610 (1988). [CrossRef]
  18. H. W. Engel, M. Hanke, and A. Neubauer, Regularization of Ill-Posed Problems (Kluwer, 1996). [CrossRef]
  19. M. Hanke, "Limitations of the L-curve method in ill-posed problems," BIT 36, 287-301 (1996). [CrossRef]
  20. A. B. Bakushinskii, "Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion," USSR Comput. Math. Math. Phys. 24, 181-182 (1985). [CrossRef]
  21. C. R. Vogel, "Non-convergence of the L-curve regularization parameter selection method," Inverse Probl. 12, 535-547 (1996). [CrossRef]
  22. T. Regenska, "A regularization parameter in discrete ill-posed problems," SIAM J. Sci. Comput. (USA) 17, 740-749 (1996). [CrossRef]
  23. D. M. Allen, "The relationship between variable selection and data augmentation and a method for prediction," Technometrics 16, 125-127 (1974). [CrossRef]
  24. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1986).

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