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

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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]
  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]
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    [CrossRef]
  16. J. Hadamard, Lectures on Cauchy's Problem in Linear Differential Equations (Yale U. Press, 1923).
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    [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).

2006 (1)

2001 (2)

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]

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]

1999 (1)

1996 (3)

M. Hanke, "Limitations of the L-curve method in ill-posed problems," BIT 36, 287-301 (1996).
[CrossRef]

C. R. Vogel, "Non-convergence of the L-curve regularization parameter selection method," Inverse Probl. 12, 535-547 (1996).
[CrossRef]

T. Regenska, "A regularization parameter in discrete ill-posed problems," SIAM J. Sci. Comput. (USA) 17, 740-749 (1996).
[CrossRef]

1993 (1)

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]

1992 (1)

1990 (1)

1988 (1)

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]

1985 (1)

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]

1979 (1)

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]

1974 (1)

D. M. Allen, "The relationship between variable selection and data augmentation and a method for prediction," Technometrics 16, 125-127 (1974).
[CrossRef]

1966 (1)

V. A. Morozov, "On the solution of functional equations by the method of regularization," Sov. Math. Dokl. 7, 414-417 (1966).

Appl. Opt. (5)

BIT (1)

M. Hanke, "Limitations of the L-curve method in ill-posed problems," BIT 36, 287-301 (1996).
[CrossRef]

Inverse Probl. (1)

C. R. Vogel, "Non-convergence of the L-curve regularization parameter selection method," Inverse Probl. 12, 535-547 (1996).
[CrossRef]

J. Am. Stat. Assoc. (1)

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]

J. Quant. Spectrosc. Radiat. Transfer (1)

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]

SIAM J. Sci. Comput. (2)

T. Regenska, "A regularization parameter in discrete ill-posed problems," SIAM J. Sci. Comput. (USA) 17, 740-749 (1996).
[CrossRef]

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]

Sov. Math. Dokl. (1)

V. A. Morozov, "On the solution of functional equations by the method of regularization," Sov. Math. Dokl. 7, 414-417 (1966).

Technometrics (2)

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]

D. M. Allen, "The relationship between variable selection and data augmentation and a method for prediction," Technometrics 16, 125-127 (1974).
[CrossRef]

USSR Comput. Math. Math. Phys. (1)

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]

Other (9)

H. W. Engel, M. Hanke, and A. Neubauer, Regularization of Ill-Posed Problems (Kluwer, 1996).
[CrossRef]

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1986).

R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications (Springer, 1993).

G. Wahba, Spline Models for Observational Data (SIAM, 1990).
[CrossRef]

J. Hadamard, Lectures on Cauchy's Problem in Linear Differential Equations (Yale U. Press, 1923).

C. Wu, Department of Chemical Engineering, Tsinghua University, Beijing, China (personal communication, 2007).

A. N. Tikhonov and V. A. Arsenin, Solution of Ill-Posed Problems (Winston & Sons, 1977).

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
[CrossRef]

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).

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Figures (12)

Fig. 1
Fig. 1

(Color online) Axisymmetric flame deconvolution geometry.

Fig. 2
Fig. 2

(Color online) Singular values and condition numbers for OP and ATP matrices.

Fig. 3
Fig. 3

(Color online) L-curve.

Fig. 4
Fig. 4

(Color online) Field distributions of the absorption coefficient at 30 and 55   mm above the burner [3].

Fig. 5
Fig. 5

(Color online) Accuracy of unregularized versus regularized solutions found using parameter selection methods for the 30   mm profile.

Fig. 6
Fig. 6

(Color online) Accuracy of unregularized versus regularized solutions found using parameter selection methods for the 55   mm profile.

Fig. 7
Fig. 7

(Color online) L-curves and L-curve curvatures for problems having different condition numbers, for the 30   mm profile contaminated with Gaussian noise, σ = 0.01 p max . (Solid and dashed curves represent OP and ATP, respectively.)

Fig. 8
Fig. 8

(Color online) Accuracy of Tikhonov-regularized solutions found using parameter selection methods for the 30   mm profile.

Fig. 9
Fig. 9

(Color online) Accuracy of Tikhonov-regularized solutions found using parameter selection methods for the 55   mm profile.

Fig. 10
Fig. 10

(Color online) Scatterplots showing the root-mean-square errors of recovered solutions for different quadratures (left) and noise levels (right), relative to regularization parameters normalized by the optimal regularization parameter for the 30   mm profile. Numbers in parentheses denote the instances when a regularization parameter was not found.

Fig. 11
Fig. 11

(Color online) Comparison of Tikhonov solutions obtained using regularization and solutions obtained using unregularized ATP deconvolution with loess presmoothing of projection data for the 30   mm profile.

Fig. 12
Fig. 12

(Color online) Comparison of Tikhonov solutions obtained using regularization and solutions obtained using unregularized ATP deconvolution with loess presmoothing of projection data for the 55   mm profile.

Equations (18)

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P ( x ) = 2 x R F ( r ) r r 2 x 2 d r ,
F ( r ) = 1 π x R P ( x ) x 2 r 2 d x ,
A OP x = b ,
x = D ATP b ,
δ x x A · A 1 δ b b = Cond ( A ) δ b b .
[ A λ L ] x = [ b 0 ] ,
L = [ 1 1 0 0 0 1 1 0 0 0 1 1 ]
Min x ( A x b 2 2 + λ 2 L x 2 2 ) .
A = U Σ V T = i = 1 N u i σ i v i T ,
x = i = 1 N u i T b σ i v i = i = 1 N ( u i T b 0 σ i v i + u i T δ b σ i v i ) ,
ε tot, λ ε pert, λ + ε reg, λ = A λ 1 δ b 2 + ( A 1 A λ 1 ) b 0 2 ,
A x λ b 2 = δ δ b 2 ,
κ λ = ρ λ η λ ρ λ η λ [ ( ρ λ ) 2 + ( η λ ) 2 ] 3 / 2 ,
V ( λ ) = 1 N i = 1 N ( [ A x λ k ] k b k ) 2 ,
V ( λ ) = B λ ( I A A λ 1 ) b 2 2 ,
G ( λ ) = ( I A A λ 1 ) b 2 2 Tr ( I A A λ 1 ) 2 = A x λ b 2 2 Tr ( I A A λ 1 ) 2 .
P i = P ( x i ) = l n ( I λ I λ 0 ) = S i α λ a ( s ) d s = x R α λ a ( r ) r r 2 x i 2 d r
ε rms = f f exact 2 N · Max ( f exact ) ,

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