Abstract

We present a method based on Tikhonov regularization for solving one-dimensional inverse tomography problems that arise in combustion applications. In this technique, Tikhonov regularization transforms the ill-conditioned set of equations generated by onion-peeling deconvolution into a well-conditioned set that is less susceptible to measurement errors that arise in experimental settings. The performance of this method is compared to that of onion-peeling and Abel three-point deconvolution by solving for a known field variable distribution from projected data contaminated with an artificially generated error. The results show that Tikhonov deconvolution provides a more accurate field distribution than onion-peeling and Abel three-point deconvolution and is more stable than the other two methods as the distance between projected data points decreases.

© 2006 Optical Society of America

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References

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  1. R. J. Hall and P. A. Bonczyk, "Sooting flame thermometry using emission/absorption tomography," Appl. Opt. 29, 4590-4598 (1990).
    [CrossRef] [PubMed]
  2. A. Schwarz, "Multitomographic flame analysis with a Schlieren apparatus," Meas. Sci. Technol. 7, 406-413 (1996).
    [CrossRef]
  3. C. J. Dasch, "One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods," Appl. Opt. 31, 1146-1152 (1992).
    [CrossRef] [PubMed]
  4. R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991).
  5. G. N. Ramachandran and A. V. Lakshminarayanan, "Three-dimensional reconstruction from radiographs and electron micrographs: applications of convolutions instead of Fourier transforms," Proc. Natl. Acad. Sci. U.S.A. 69, 2236-2240 (1971).
    [CrossRef]
  6. L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from x-ray transmissions," IEEE Trans. Nucl. Sci. NS-21, 228-236 (1971).
  7. C. J. Cremers and R. C. Birkebak, "Application of the Abel integral equation to spectrographic data," Appl. Opt. 5, 1057-1063 (1966).
    [CrossRef] [PubMed]
  8. M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," J. Appl. Phys. 54, 137-143 (1983).
    [CrossRef]
  9. R. S. Anderssen, "Stable procedures for the inversion of Abel's equation," J. Inst. Math. Appl. 17, 329-342 (1976).
    [CrossRef]
  10. C. K. Chan and P. Lu, "On the stability of the solution of Abel's integral equation," J. Phys. A 14, 575-578 (1981).
    [CrossRef]
  11. M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," Appl. Phys. Lett. 41, 27-28 (1983).
    [CrossRef]
  12. J. Hadamard, Lectures on Cauchy's Problem in Linear Differential Equations (Yale U. Press, 1923).
  13. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981), pp. 28-30.
  14. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
    [CrossRef]
  15. A. N. Tikhonov, "Inverse problems in heat conduction," J. Eng. Phys. 29, 816-820 (1975).
    [CrossRef]
  16. K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
    [CrossRef]

2005 (1)

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

1996 (1)

A. Schwarz, "Multitomographic flame analysis with a Schlieren apparatus," Meas. Sci. Technol. 7, 406-413 (1996).
[CrossRef]

1992 (1)

1990 (1)

1983 (2)

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," J. Appl. Phys. 54, 137-143 (1983).
[CrossRef]

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," Appl. Phys. Lett. 41, 27-28 (1983).
[CrossRef]

1981 (1)

C. K. Chan and P. Lu, "On the stability of the solution of Abel's integral equation," J. Phys. A 14, 575-578 (1981).
[CrossRef]

1976 (1)

R. S. Anderssen, "Stable procedures for the inversion of Abel's equation," J. Inst. Math. Appl. 17, 329-342 (1976).
[CrossRef]

1975 (1)

A. N. Tikhonov, "Inverse problems in heat conduction," J. Eng. Phys. 29, 816-820 (1975).
[CrossRef]

1971 (2)

G. N. Ramachandran and A. V. Lakshminarayanan, "Three-dimensional reconstruction from radiographs and electron micrographs: applications of convolutions instead of Fourier transforms," Proc. Natl. Acad. Sci. U.S.A. 69, 2236-2240 (1971).
[CrossRef]

L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from x-ray transmissions," IEEE Trans. Nucl. Sci. NS-21, 228-236 (1971).

1966 (1)

Anderssen, R. S.

R. S. Anderssen, "Stable procedures for the inversion of Abel's equation," J. Inst. Math. Appl. 17, 329-342 (1976).
[CrossRef]

Beniaminy, I.

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," Appl. Phys. Lett. 41, 27-28 (1983).
[CrossRef]

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," J. Appl. Phys. 54, 137-143 (1983).
[CrossRef]

Birkebak, R. C.

Bonczyk, P. A.

Chan, C. K.

C. K. Chan and P. Lu, "On the stability of the solution of Abel's integral equation," J. Phys. A 14, 575-578 (1981).
[CrossRef]

Cremers, C. J.

Dasch, C. J.

Deutsch, M.

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," J. Appl. Phys. 54, 137-143 (1983).
[CrossRef]

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," Appl. Phys. Lett. 41, 27-28 (1983).
[CrossRef]

Fraser, R. A.

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981), pp. 28-30.

Gorenflo, R.

R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991).

Gülder, Ö. L.

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

Hadamard, J.

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

Hall, R. J.

Hansen, P. C.

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

Lakshminarayanan, A. V.

G. N. Ramachandran and A. V. Lakshminarayanan, "Three-dimensional reconstruction from radiographs and electron micrographs: applications of convolutions instead of Fourier transforms," Proc. Natl. Acad. Sci. U.S.A. 69, 2236-2240 (1971).
[CrossRef]

Logan, B. F.

L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from x-ray transmissions," IEEE Trans. Nucl. Sci. NS-21, 228-236 (1971).

Lu, P.

C. K. Chan and P. Lu, "On the stability of the solution of Abel's integral equation," J. Phys. A 14, 575-578 (1981).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981), pp. 28-30.

Ramachandran, G. N.

G. N. Ramachandran and A. V. Lakshminarayanan, "Three-dimensional reconstruction from radiographs and electron micrographs: applications of convolutions instead of Fourier transforms," Proc. Natl. Acad. Sci. U.S.A. 69, 2236-2240 (1971).
[CrossRef]

Schwarz, A.

A. Schwarz, "Multitomographic flame analysis with a Schlieren apparatus," Meas. Sci. Technol. 7, 406-413 (1996).
[CrossRef]

Shepp, L. A.

L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from x-ray transmissions," IEEE Trans. Nucl. Sci. NS-21, 228-236 (1971).

Smallwood, G. J.

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

Snelling, D. R.

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

Thomson, K. A.

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, "Inverse problems in heat conduction," J. Eng. Phys. 29, 816-820 (1975).
[CrossRef]

Vessella, S.

R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991).

Weckman, E. J.

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981), pp. 28-30.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," Appl. Phys. Lett. 41, 27-28 (1983).
[CrossRef]

Combust. Flame (1)

K. A. Thomson, Ö. L. Gülder, E. J. Weckman, R. A. Fraser, G. J. Smallwood, and D. R. Snelling, "Soot concentration and temperature measurements in coannular, nonpremixed CH4/air laminar flames at pressures up to 4 MPa," Combust. Flame 140, 222-232 (2005).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from x-ray transmissions," IEEE Trans. Nucl. Sci. NS-21, 228-236 (1971).

J. Appl. Phys. (1)

M. Deutsch and I. Beniaminy, "Derivative-free inversion of Abel's integral equation," J. Appl. Phys. 54, 137-143 (1983).
[CrossRef]

J. Eng. Phys. (1)

A. N. Tikhonov, "Inverse problems in heat conduction," J. Eng. Phys. 29, 816-820 (1975).
[CrossRef]

J. Inst. Math. Appl. (1)

R. S. Anderssen, "Stable procedures for the inversion of Abel's equation," J. Inst. Math. Appl. 17, 329-342 (1976).
[CrossRef]

J. Phys. A (1)

C. K. Chan and P. Lu, "On the stability of the solution of Abel's integral equation," J. Phys. A 14, 575-578 (1981).
[CrossRef]

Meas. Sci. Technol. (1)

A. Schwarz, "Multitomographic flame analysis with a Schlieren apparatus," Meas. Sci. Technol. 7, 406-413 (1996).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

G. N. Ramachandran and A. V. Lakshminarayanan, "Three-dimensional reconstruction from radiographs and electron micrographs: applications of convolutions instead of Fourier transforms," Proc. Natl. Acad. Sci. U.S.A. 69, 2236-2240 (1971).
[CrossRef]

Other (4)

R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics (Springer, 1991).

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

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981), pp. 28-30.

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

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Figures (9)

Fig. 1
Fig. 1

(a) Evaluation of the spectral absorption coefficient distribution within an axisymmetric flame, and (b) discretization of the problem domain.

Fig. 2
Fig. 2

Geometric terms for 2D filtered backprojection.

Fig. 3
Fig. 3

Matrix condition numbers from traditional deconvolution methods.

Fig. 4
Fig. 4

Plots of F(x) = F 1(x) + F 2(x) for different amounts of regularization, corresponding to Eq. (16). The analytical solution is marked with a cross, and diamonds show solutions obtained by perturbing b with randomly generated δ b vectors with ‖δ b ‖ < 0.283.

Fig. 5
Fig. 5

Field variable distribution and normalized soot-volume fraction data.

Fig. 6
Fig. 6

L curve for N = 20.

Fig. 7
Fig. 7

Tikhonov solution obtained using different levels of regularization at N = 20.

Fig. 8
Fig. 8

Field distributions obtained from perturbed projected data for N = 20 and 100.

Fig. 9
Fig. 9

Accuracy of field distribution obtained from perturbed projected data.

Equations (23)

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P ( y ) = 2 y R f ( r ) r r 2 y 2  d r ,
f ( r ) = 1 π y R P ( y ) y 2 r 2  d y ,
P i = 2 j = i N 1 f j r i , j = i r j Δ r / 2 , j > i r j + Δ r / 2 r r 2 y i 2  d r ,
A OP, i j = { 0 , j < i 2 Δ r [ ( j + ½ ) 2 i 2 ] 1 / 2 , j = i 2 Δ r [ ( j + ½ ) 2 i 2 ] 1 / 2 [ ( j ½ ) 2 i 2 ] 1 / 2 , j > i .
f ( r , θ ) = 1 Δ r 2 0 π P ( y , ϕ ) ψ [ y r cos ( θ ϕ ) Δ r ] d r d ϕ ,
ψ ( n ) = 4 π ( 4 n 2 1 ) , n I,
ξ ( i , j ) = 0 π ψ ( j i cos ϕ ) d ϕ ,
D FBP i j = { ξ ( i , j ) / Δ r j = 0 2 ξ ( i , j ) / Δ r j > 0 ,
f i = 1 π j = 1 N 1 y j , j = i y j Δ r / 2 , j > i y j + Δ r / 2 P ˜ ( y , P j 1 , P j , P j + 1 ) y 2 r i 2  d y .
f ( r ) = 1 π { P ( R ) P ( r ) 1 r 2 + r R [ P ( y ) P ( r ) ] y ( y 2 r 2 ) 3 / 2  d y } ,
A ( x + δ x ) = b + δ b ,
δ x A 1 δ b .
δ x x A A 1 δ b b = Cond ( A ) δ b b ,
Cond ( A ) = w max / w min ,
A = U W V T ,
F 1 ( x ) = 1 2 x T A T A x x T A T b
A x = [ 4 2.5 2 1 ] [ x 1 x 2 ] = [ 4 2 ] = b ,
F 2 ( x ) = p = 0 n α p x T L p T L p x ,
L 0 = [ 1 - 1 0 0 0 1 - 1 0 0 0 1 - 1 ] ,
A Tik x ˜ * = b Tik ,
A Tik = ( A T A + α 0 A L 0     T L 0 )
P ˜ i = P i + ε ( μ , σ ) , i = 0 , 1 , , N 1 ,
ε rms ( N ) = 1 N { i = 1 N [ f ˜ i f ( r i ) ] 2 } 1 / 2 .

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