Abstract

In this work, we describe an adaptation of Fourier–Hankel method to Abel inversion for the deflection tomographic reconstruction of axisymmetric temperature field. This technique is compared with existing methods to test the accuracy and error propagation using simulated Moiré stripes of natural convection flow above a heated horizontal disk in air. Simpson’s 1/3rd rule and one-point and two-point formulas are used in this comparison. The results showed that the proposed technique for Abel inversion is accurate and has the powerful capacity to control the smoothing degree of noise in the inversion process.

© 2013 Optical Society of America

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References

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  1. W. Hauf and U. Grigull, Optical Methods in Heat Transfer. Advances in Heat Transfer, 6th ed. (Academic, 1970).
  2. W. Merzkirch, Flow Visualization (Academic, 1974).
  3. C. M. Vest, Holographic Interferometry (Wiley, 1979).
  4. M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143 (1983).
    [CrossRef]
  5. M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27, 1956–1959 (1988).
    [CrossRef]
  6. J. D. Posner and D. Dunn-Rankin, “Temperature field measurements of small, nonpremixed flames with use of an Abel inversion of holographic interferograms,” Appl. Opt. 42, 952–959 (2003).
    [CrossRef]
  7. M. El Fagrich and H. Chehouani, “A simple Abel inversion method of interferometric data for temperature measurement in axisymmetric medium,” Opt. Lasers Eng. 50, 336–344 (2012).
    [CrossRef]
  8. R. Álvarez, A. Rodero, and M. C. Quintero, “An Abel inversion method for radially resolved measurements in the axial injection torch,” Spectrochim. Acta B 57, 1665–1680 (2002).
    [CrossRef]
  9. S. Ma, H. Gao, G. Zhang, and L. Wu, “Abel inversion using Legendre wavelets expansion,” J. Quant. Spectrosc. Radiat. Transfer 107, 61–71 (2007).
    [CrossRef]
  10. S. Ma, H. Gao H, and L. Wu, “Modified Fourier–Hankel method based on analysis of errors in Abel inversion using Fourier transform techniques,” Appl. Opt. 47, 1350–1357 (2008).
    [CrossRef]
  11. S. Ma, H. Gao, L. Wu, and G. Zhang, “Abel inversion using Legendre polynomials approximations,” J. Quant. Spectrosc. Radiat. Transfer 109, 1745–1757 (2008).
    [CrossRef]
  12. K. A. Agrawal, B. W. Albers, and W. D. W. Griffin, “Abel inversion of deflectometric measurements in dynamic flows,” Appl. Opt. 38, 3394–3398 (1999).
    [CrossRef]
  13. P. S. Kolhe and A. K. Agrawal, “Abel inversion of deflectometric data: comparison of accuracy and noise propagation of existing techniques,” Appl. Opt. 48, 3894–3902 (2009).
    [CrossRef]
  14. C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered back projection methods,” Appl. Opt. 31, 1146–1152 (1992).
    [CrossRef]
  15. O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200–207 (1960).
    [CrossRef]
  16. L. M. Smith, D. R. Keefer, and S. I. Sudharsanan, “Abel inversion using transform techniques,” J. Quant. Spectrosc. Radiat. Transfer 39, 367–373 (1988).
    [CrossRef]
  17. G. C.-Y. Chan and G. M. Hieftje, “A LabVIEW program for determining confidence intervals of Abel-inverted radial emission profiles,” Spectrochim. Acta B 60, 1486–1501 (2005).
    [CrossRef]
  18. H. Chehouani and A. Elmotassadeq, “Simulation and visualization of the refraction effects in the thermal boundary layer near a heated horizontal down-facing disk,” Opt. Lasers Eng. 47, 477–483 (2009).
    [CrossRef]
  19. M. Takeda, H. Kobayashi, and K. Ina, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  20. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).
  21. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).
  22. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1996).
  23. W. Yu, D. Yun, and L. Wang, “Talbot and Fourier moiré deflectometry and its applications in engineering evaluation,” Opt. Lasers Eng. 25, 163–177 (1996).
    [CrossRef]

2012 (1)

M. El Fagrich and H. Chehouani, “A simple Abel inversion method of interferometric data for temperature measurement in axisymmetric medium,” Opt. Lasers Eng. 50, 336–344 (2012).
[CrossRef]

2009 (2)

H. Chehouani and A. Elmotassadeq, “Simulation and visualization of the refraction effects in the thermal boundary layer near a heated horizontal down-facing disk,” Opt. Lasers Eng. 47, 477–483 (2009).
[CrossRef]

P. S. Kolhe and A. K. Agrawal, “Abel inversion of deflectometric data: comparison of accuracy and noise propagation of existing techniques,” Appl. Opt. 48, 3894–3902 (2009).
[CrossRef]

2008 (2)

S. Ma, H. Gao H, and L. Wu, “Modified Fourier–Hankel method based on analysis of errors in Abel inversion using Fourier transform techniques,” Appl. Opt. 47, 1350–1357 (2008).
[CrossRef]

S. Ma, H. Gao, L. Wu, and G. Zhang, “Abel inversion using Legendre polynomials approximations,” J. Quant. Spectrosc. Radiat. Transfer 109, 1745–1757 (2008).
[CrossRef]

2007 (1)

S. Ma, H. Gao, G. Zhang, and L. Wu, “Abel inversion using Legendre wavelets expansion,” J. Quant. Spectrosc. Radiat. Transfer 107, 61–71 (2007).
[CrossRef]

2005 (1)

G. C.-Y. Chan and G. M. Hieftje, “A LabVIEW program for determining confidence intervals of Abel-inverted radial emission profiles,” Spectrochim. Acta B 60, 1486–1501 (2005).
[CrossRef]

2003 (1)

2002 (1)

R. Álvarez, A. Rodero, and M. C. Quintero, “An Abel inversion method for radially resolved measurements in the axial injection torch,” Spectrochim. Acta B 57, 1665–1680 (2002).
[CrossRef]

1999 (1)

1996 (1)

W. Yu, D. Yun, and L. Wang, “Talbot and Fourier moiré deflectometry and its applications in engineering evaluation,” Opt. Lasers Eng. 25, 163–177 (1996).
[CrossRef]

1992 (1)

1988 (2)

M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27, 1956–1959 (1988).
[CrossRef]

L. M. Smith, D. R. Keefer, and S. I. Sudharsanan, “Abel inversion using transform techniques,” J. Quant. Spectrosc. Radiat. Transfer 39, 367–373 (1988).
[CrossRef]

1983 (1)

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143 (1983).
[CrossRef]

1982 (1)

1960 (1)

O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200–207 (1960).
[CrossRef]

Agrawal, A. K.

Agrawal, K. A.

Albers, B. W.

Álvarez, R.

R. Álvarez, A. Rodero, and M. C. Quintero, “An Abel inversion method for radially resolved measurements in the axial injection torch,” Spectrochim. Acta B 57, 1665–1680 (2002).
[CrossRef]

Beniaminy, I.

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143 (1983).
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).

Chan, G. C.-Y.

G. C.-Y. Chan and G. M. Hieftje, “A LabVIEW program for determining confidence intervals of Abel-inverted radial emission profiles,” Spectrochim. Acta B 60, 1486–1501 (2005).
[CrossRef]

Chehouani, H.

M. El Fagrich and H. Chehouani, “A simple Abel inversion method of interferometric data for temperature measurement in axisymmetric medium,” Opt. Lasers Eng. 50, 336–344 (2012).
[CrossRef]

H. Chehouani and A. Elmotassadeq, “Simulation and visualization of the refraction effects in the thermal boundary layer near a heated horizontal down-facing disk,” Opt. Lasers Eng. 47, 477–483 (2009).
[CrossRef]

Dasch, C. J.

Deutsch, M.

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143 (1983).
[CrossRef]

Dunn-Rankin, D.

El Fagrich, M.

M. El Fagrich and H. Chehouani, “A simple Abel inversion method of interferometric data for temperature measurement in axisymmetric medium,” Opt. Lasers Eng. 50, 336–344 (2012).
[CrossRef]

Elmotassadeq, A.

H. Chehouani and A. Elmotassadeq, “Simulation and visualization of the refraction effects in the thermal boundary layer near a heated horizontal down-facing disk,” Opt. Lasers Eng. 47, 477–483 (2009).
[CrossRef]

Gao, H.

S. Ma, H. Gao, L. Wu, and G. Zhang, “Abel inversion using Legendre polynomials approximations,” J. Quant. Spectrosc. Radiat. Transfer 109, 1745–1757 (2008).
[CrossRef]

S. Ma, H. Gao, G. Zhang, and L. Wu, “Abel inversion using Legendre wavelets expansion,” J. Quant. Spectrosc. Radiat. Transfer 107, 61–71 (2007).
[CrossRef]

Gao H, H.

Griffin, W. D. W.

Grigull, U.

W. Hauf and U. Grigull, Optical Methods in Heat Transfer. Advances in Heat Transfer, 6th ed. (Academic, 1970).

Hauf, W.

W. Hauf and U. Grigull, Optical Methods in Heat Transfer. Advances in Heat Transfer, 6th ed. (Academic, 1970).

Hieftje, G. M.

G. C.-Y. Chan and G. M. Hieftje, “A LabVIEW program for determining confidence intervals of Abel-inverted radial emission profiles,” Spectrochim. Acta B 60, 1486–1501 (2005).
[CrossRef]

Ina, K.

Kalal, M.

Keefer, D. R.

L. M. Smith, D. R. Keefer, and S. I. Sudharsanan, “Abel inversion using transform techniques,” J. Quant. Spectrosc. Radiat. Transfer 39, 367–373 (1988).
[CrossRef]

Kobayashi, H.

Kolhe, P. S.

Ma, S.

S. Ma, H. Gao H, and L. Wu, “Modified Fourier–Hankel method based on analysis of errors in Abel inversion using Fourier transform techniques,” Appl. Opt. 47, 1350–1357 (2008).
[CrossRef]

S. Ma, H. Gao, L. Wu, and G. Zhang, “Abel inversion using Legendre polynomials approximations,” J. Quant. Spectrosc. Radiat. Transfer 109, 1745–1757 (2008).
[CrossRef]

S. Ma, H. Gao, G. Zhang, and L. Wu, “Abel inversion using Legendre wavelets expansion,” J. Quant. Spectrosc. Radiat. Transfer 107, 61–71 (2007).
[CrossRef]

Merzkirch, W.

W. Merzkirch, Flow Visualization (Academic, 1974).

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1996).

Nestor, O. H.

O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200–207 (1960).
[CrossRef]

Nugent, K. A.

Olson, H. N.

O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200–207 (1960).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1996).

Posner, J. D.

Quintero, M. C.

R. Álvarez, A. Rodero, and M. C. Quintero, “An Abel inversion method for radially resolved measurements in the axial injection torch,” Spectrochim. Acta B 57, 1665–1680 (2002).
[CrossRef]

Rodero, A.

R. Álvarez, A. Rodero, and M. C. Quintero, “An Abel inversion method for radially resolved measurements in the axial injection torch,” Spectrochim. Acta B 57, 1665–1680 (2002).
[CrossRef]

Smith, L. M.

L. M. Smith, D. R. Keefer, and S. I. Sudharsanan, “Abel inversion using transform techniques,” J. Quant. Spectrosc. Radiat. Transfer 39, 367–373 (1988).
[CrossRef]

Sudharsanan, S. I.

L. M. Smith, D. R. Keefer, and S. I. Sudharsanan, “Abel inversion using transform techniques,” J. Quant. Spectrosc. Radiat. Transfer 39, 367–373 (1988).
[CrossRef]

Takeda, M.

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, 1979).

Wang, L.

W. Yu, D. Yun, and L. Wang, “Talbot and Fourier moiré deflectometry and its applications in engineering evaluation,” Opt. Lasers Eng. 25, 163–177 (1996).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1996).

Wu, L.

S. Ma, H. Gao H, and L. Wu, “Modified Fourier–Hankel method based on analysis of errors in Abel inversion using Fourier transform techniques,” Appl. Opt. 47, 1350–1357 (2008).
[CrossRef]

S. Ma, H. Gao, L. Wu, and G. Zhang, “Abel inversion using Legendre polynomials approximations,” J. Quant. Spectrosc. Radiat. Transfer 109, 1745–1757 (2008).
[CrossRef]

S. Ma, H. Gao, G. Zhang, and L. Wu, “Abel inversion using Legendre wavelets expansion,” J. Quant. Spectrosc. Radiat. Transfer 107, 61–71 (2007).
[CrossRef]

Yu, W.

W. Yu, D. Yun, and L. Wang, “Talbot and Fourier moiré deflectometry and its applications in engineering evaluation,” Opt. Lasers Eng. 25, 163–177 (1996).
[CrossRef]

Yun, D.

W. Yu, D. Yun, and L. Wang, “Talbot and Fourier moiré deflectometry and its applications in engineering evaluation,” Opt. Lasers Eng. 25, 163–177 (1996).
[CrossRef]

Zhang, G.

S. Ma, H. Gao, L. Wu, and G. Zhang, “Abel inversion using Legendre polynomials approximations,” J. Quant. Spectrosc. Radiat. Transfer 109, 1745–1757 (2008).
[CrossRef]

S. Ma, H. Gao, G. Zhang, and L. Wu, “Abel inversion using Legendre wavelets expansion,” J. Quant. Spectrosc. Radiat. Transfer 107, 61–71 (2007).
[CrossRef]

Appl. Opt. (6)

J. Appl. Phys. (1)

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (3)

S. Ma, H. Gao, L. Wu, and G. Zhang, “Abel inversion using Legendre polynomials approximations,” J. Quant. Spectrosc. Radiat. Transfer 109, 1745–1757 (2008).
[CrossRef]

S. Ma, H. Gao, G. Zhang, and L. Wu, “Abel inversion using Legendre wavelets expansion,” J. Quant. Spectrosc. Radiat. Transfer 107, 61–71 (2007).
[CrossRef]

L. M. Smith, D. R. Keefer, and S. I. Sudharsanan, “Abel inversion using transform techniques,” J. Quant. Spectrosc. Radiat. Transfer 39, 367–373 (1988).
[CrossRef]

Opt. Lasers Eng. (3)

M. El Fagrich and H. Chehouani, “A simple Abel inversion method of interferometric data for temperature measurement in axisymmetric medium,” Opt. Lasers Eng. 50, 336–344 (2012).
[CrossRef]

W. Yu, D. Yun, and L. Wang, “Talbot and Fourier moiré deflectometry and its applications in engineering evaluation,” Opt. Lasers Eng. 25, 163–177 (1996).
[CrossRef]

H. Chehouani and A. Elmotassadeq, “Simulation and visualization of the refraction effects in the thermal boundary layer near a heated horizontal down-facing disk,” Opt. Lasers Eng. 47, 477–483 (2009).
[CrossRef]

SIAM Rev. (1)

O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200–207 (1960).
[CrossRef]

Spectrochim. Acta B (2)

G. C.-Y. Chan and G. M. Hieftje, “A LabVIEW program for determining confidence intervals of Abel-inverted radial emission profiles,” Spectrochim. Acta B 60, 1486–1501 (2005).
[CrossRef]

R. Álvarez, A. Rodero, and M. C. Quintero, “An Abel inversion method for radially resolved measurements in the axial injection torch,” Spectrochim. Acta B 57, 1665–1680 (2002).
[CrossRef]

Other (6)

W. Hauf and U. Grigull, Optical Methods in Heat Transfer. Advances in Heat Transfer, 6th ed. (Academic, 1970).

W. Merzkirch, Flow Visualization (Academic, 1974).

C. M. Vest, Holographic Interferometry (Wiley, 1979).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1922).

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965).

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1996).

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Figures (15)

Fig. 1.
Fig. 1.

Schematic diagram illustrating axisymmetric temperature field above a heated disc.

Fig. 2.
Fig. 2.

Isotherms above the heated disc.

Fig. 3.
Fig. 3.

Radial temperature profile at z=1mm above the disc.

Fig. 4.
Fig. 4.

Relative refractive index function profile at z=1mm above the disc.

Fig. 5.
Fig. 5.

Deflection angle test profile used for algorithm verification.

Fig. 6.
Fig. 6.

Simulated Moiré pattern using Eq. (8) with Im=100; V=100; v=5; G=1100.

Fig. 7.
Fig. 7.

Comparison of inversion coefficient (D9j) for different algorithms (N=25).

Fig. 8.
Fig. 8.

Variation of the average deviation as function of α.

Fig. 9.
Fig. 9.

Effect of the number of input data points on the accuracy of the AFH method.

Fig. 10.
Fig. 10.

Comparison between the initial profile of temperature at z=1mm and those reconstructed using AFH algorithm with α=1 and α=0.2, respectively.

Fig. 11.
Fig. 11.

Deviation σ versus the number of data N.

Fig. 12.
Fig. 12.

Relative error between the initial and reconstructed temperature profiles.

Fig. 13.
Fig. 13.

Noisy simulated Moiré patterns using random noise of magnitude 10%.

Fig. 14.
Fig. 14.

Noisy test profile of the deflection angle.

Fig. 15.
Fig. 15.

Variation of the average deviation as function of the parameter α for the two proposed versions of AFH method.

Equations (39)

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S(y)=2λyR0f(r)rdrr2y2,
f(r)=λπrR0dSdyy2r2dy.
ε(y)=2yyR0δrr2y2dr.
δ(r)=n(r)n01,
δ(r)=1πrR0ε(y)y2r2dy.
n=1+KPMRT,
δ(r)=T0T(r)11+RKPMT0.
I(y,z)=Im+Vcos[2πGε(y,z)+2πvy],
δ(ri)=j=0NDij·εj.
Dij=13π[2+(1+(1)(ji+1))]j2i2ifj>iandN=13π[2+(1+(1)(ji+1))]j2i2ifj=N=0ifj<i.
Dii=Dii+1foriN1=0fori=N.
Dij=12πln[j+1+(j+1)2i2(j1)+(j1)2i2]ifj>iandj1=12π[2+ln(j+1+(j+1)2i2j+j2i2)]ifj>iandj=1=12πln[j+1+(j+1)2i2(j1)+(j1)2i2]ifj=iandj0=0ifj<iorj=i=0.
Dij=1π[AijAij1(j+1)Bij+(j1)Bij1]ifj>iandj1=1π[Aij(j+1)Bij1]ifj>iandj=1=1π[Aij(j+1)Bij]ifj=iandi0=0ifj=i=0orj<i.
Aij=(j+1)2i2j2i2Bij=ln[j+1+(j+1)2i2j+j2i2].
ε(y)=+δ(x,y)ydx.
FT[ε](ω)=+ε(y)exp(i^ωy)dy,
FT[ε](ω)=++δ(x,y)yexp(i^ωy)dydx=i^ω++δ(x,y)exp(i^ωy)dydx.
FT[ε](ω)=i^ω0+02πrδ(r)exp(i^ωrsinθ)dθdr.
J0(ωr)=12π02πexp(i^ωrsinθ)dθ.
FT[ε](ω)=i^2πω0+δ(r)J0(ωr)rdr.
HT[δ](ω)=0+δ(r)J0(ωr)rdr,
FT[ε](ω)=i^2πωHT[δ](ω).
δ(r)=1i^2πHT1[FT[ε](ω)ω].
δ(r)=1i^2π0+FT[ε](ω)J0(ωr)dω.
FT[ε](ω)=2i^0+ε(y)sin(yω)dy.
FT[ε](ωk)=2i^0R0ε(y)sin(yωk)dy.
FT[ε](ωk)=2i^j=0Nε(yj)sin(yjωk)Δy.
ωmaxπΔy.
ΔωπR0.
Δω=απR0.
FT[ε](ωk)=2i^j=0Nε(yj)sin(απjkN)Δy.
δ(r)=1π0ωmaxj=0Nε(yj)sin(απjkN)J0(ωr)Δydω.
δ(ri)=αNj=0Nε(yj)k=0Nαsin(απjkN)J0(απkiN).
δ(ri)=j=0NDij·εj.
Dij=αNBikFkj
Bik=J0(απikN)if0iNand0kNαFkj=sin(απkjN)if0jNand0kNα.
σ=i=0N|Tcal(ri)Tinitial(ri)|2N+1.
Δ(ri)=Tcal(ri)Tinitial(ri)Tinitial(ri).
δ(ri)=αNj=0Nε(yj)k=0Nsin(απjkN)J0(απkiN).

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