Abstract

I have analyzed and compared the noise property and accuracy of three kinds of Abel inversion technique, i.e., the polynomial interpolation, versatile polynomial fitting (VPF) and modified Fourier–Hankel (MFH) methods. All these techniques will amplify noise due to the intrinsic property of Abel inversion. A technique that is more sensitive to noise also has a higher inversion accuracy for data without noise. Among the techniques without a noise resisting property, the third-degree polynomial interpolation and MFH methods have comparable performance and give higher inversion accuracies than other techniques. The VPF and MFH methods, which can be used without extra filtering of noise, yield markedly better results compared with those obtained by using noise filters in advance of inversion. Both of these two methods can be considered for applying to experimental data if there are no better smoothing techniques available.

© 2011 Optical Society of America

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References

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  1. H. R. Griem, Principles of Plasma Spectroscopy (Cambridge University Press, 1997).
    [CrossRef]
  2. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, Vol.  3 of Combustion Science and Technology Book Series, 2nd ed. (Gordon & Breach, 1996).
  3. O. H. Nestor and H. N. Olsen, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200–207(1960).
    [CrossRef]
  4. K. Bockasten, “Transformation of observed radiances into radial distribution of the emission of a plasma,” J. Opt. Soc. Am. 51, 943–947 (1961).
    [CrossRef]
  5. G. N. Minerbo and M. E. Levy, “Inversion of Abel’s integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598–616 (1969).
    [CrossRef]
  6. C. J. Cremers and R. C. Birkebak, “Application of the Abel integral equation to spectrographic data,” Appl. Opt. 5, 1057–1064 (1966).
    [CrossRef] [PubMed]
  7. 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]
  8. S. Ma, H. Gao, G. Zhang, and L. Wu, “A versatile analytical expression for the inverse Abel transform applied to experimental data with noise,” Appl. Spectrosc. 62, 701–707(2008).
    [CrossRef] [PubMed]
  9. M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27, 1956–1959 (1988).
    [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. S. Ma, H. Gao, 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] [PubMed]
  13. V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634–2642 (2002).
    [CrossRef]
  14. G. A. Garcia, L. Nahon, and I. Powis, “Two-dimensional charged particle image inversion using a polar basis function expansion,” Rev. Sci. Instrum. 75, 4989–4996 (2004).
    [CrossRef]
  15. R. Piche, “Noise-filtering properties of numerical methods for the inverse Abel transform,” IEEE Trans. Instrum. Meas. 41, 517–522 (1992).
    [CrossRef]
  16. C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. 31, 1146–1152 (1992).
    [CrossRef] [PubMed]
  17. 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] [PubMed]
  18. G. E. Andrews, R. Askey, and R. Roy, Special Functions(Cambridge University Press, 1999).
  19. G. N. Watson, Theory of Bessel Functions (Cambridge University Press, 1966).
  20. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, 1996).
  21. M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
    [CrossRef]
  22. L. M. Smith, “Nonstationary noise effects in the Abel inversion,” IEEE Trans. Inf. Theory 34, 158–161 (1988).
    [CrossRef]
  23. G. C.-Y. Chan and G. M. Hieftje, “Estimation of confidence intervals for radial emissivity and optimization of data treatment techniques in Abel inversion,” Spectrochim. Acta B 61, 31–41 (2006).
    [CrossRef]
  24. F. Magnus and J. T. Gudmundsson, “Digital smoothing of the Langmuir probe I-V characteristic,” Rev. Sci. Instrum. 79, 073503 (2008).
    [CrossRef] [PubMed]
  25. J. I. Fernández Palop, J. Ballesteros, V. Colomer, and M. A. Hernández, “A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I-V probe characteristic,” Rev. Sci. Instrum. 66, 4625–4636 (1995).
    [CrossRef]

2009 (1)

2008 (4)

S. Ma, H. Gao, 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] [PubMed]

S. Ma, H. Gao, G. Zhang, and L. Wu, “A versatile analytical expression for the inverse Abel transform applied to experimental data with noise,” Appl. Spectrosc. 62, 701–707(2008).
[CrossRef] [PubMed]

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]

F. Magnus and J. T. Gudmundsson, “Digital smoothing of the Langmuir probe I-V characteristic,” Rev. Sci. Instrum. 79, 073503 (2008).
[CrossRef] [PubMed]

2006 (1)

G. C.-Y. Chan and G. M. Hieftje, “Estimation of confidence intervals for radial emissivity and optimization of data treatment techniques in Abel inversion,” Spectrochim. Acta B 61, 31–41 (2006).
[CrossRef]

2004 (1)

G. A. Garcia, L. Nahon, and I. Powis, “Two-dimensional charged particle image inversion using a polar basis function expansion,” Rev. Sci. Instrum. 75, 4989–4996 (2004).
[CrossRef]

2002 (2)

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]

V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634–2642 (2002).
[CrossRef]

1996 (1)

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[CrossRef]

1995 (1)

J. I. Fernández Palop, J. Ballesteros, V. Colomer, and M. A. Hernández, “A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I-V probe characteristic,” Rev. Sci. Instrum. 66, 4625–4636 (1995).
[CrossRef]

1992 (2)

C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. 31, 1146–1152 (1992).
[CrossRef] [PubMed]

R. Piche, “Noise-filtering properties of numerical methods for the inverse Abel transform,” IEEE Trans. Instrum. Meas. 41, 517–522 (1992).
[CrossRef]

1988 (3)

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]

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

L. M. Smith, “Nonstationary noise effects in the Abel inversion,” IEEE Trans. Inf. Theory 34, 158–161 (1988).
[CrossRef]

1969 (1)

G. N. Minerbo and M. E. Levy, “Inversion of Abel’s integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598–616 (1969).
[CrossRef]

1966 (1)

1961 (1)

1960 (1)

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

Agrawal, A. K.

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]

Andrews, G. E.

G. E. Andrews, R. Askey, and R. Roy, Special Functions(Cambridge University Press, 1999).

Askey, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions(Cambridge University Press, 1999).

Ballesteros, J.

J. I. Fernández Palop, J. Ballesteros, V. Colomer, and M. A. Hernández, “A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I-V probe characteristic,” Rev. Sci. Instrum. 66, 4625–4636 (1995).
[CrossRef]

Birkebak, R. C.

Bockasten, K.

Brake, M. L.

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[CrossRef]

Buie, M. J.

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[CrossRef]

Chan, G. C.-Y.

G. C.-Y. Chan and G. M. Hieftje, “Estimation of confidence intervals for radial emissivity and optimization of data treatment techniques in Abel inversion,” Spectrochim. Acta B 61, 31–41 (2006).
[CrossRef]

Colomer, V.

J. I. Fernández Palop, J. Ballesteros, V. Colomer, and M. A. Hernández, “A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I-V probe characteristic,” Rev. Sci. Instrum. 66, 4625–4636 (1995).
[CrossRef]

Cremers, C. J.

Dasch, C. J.

Dribinski, V.

V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634–2642 (2002).
[CrossRef]

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, Vol.  3 of Combustion Science and Technology Book Series, 2nd ed. (Gordon & Breach, 1996).

Gao, H.

Garcia, G. A.

G. A. Garcia, L. Nahon, and I. Powis, “Two-dimensional charged particle image inversion using a polar basis function expansion,” Rev. Sci. Instrum. 75, 4989–4996 (2004).
[CrossRef]

Griem, H. R.

H. R. Griem, Principles of Plasma Spectroscopy (Cambridge University Press, 1997).
[CrossRef]

Gudmundsson, J. T.

F. Magnus and J. T. Gudmundsson, “Digital smoothing of the Langmuir probe I-V characteristic,” Rev. Sci. Instrum. 79, 073503 (2008).
[CrossRef] [PubMed]

Hernández, M. A.

J. I. Fernández Palop, J. Ballesteros, V. Colomer, and M. A. Hernández, “A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I-V probe characteristic,” Rev. Sci. Instrum. 66, 4625–4636 (1995).
[CrossRef]

Hieftje, G. M.

G. C.-Y. Chan and G. M. Hieftje, “Estimation of confidence intervals for radial emissivity and optimization of data treatment techniques in Abel inversion,” Spectrochim. Acta B 61, 31–41 (2006).
[CrossRef]

Holloway, J. P.

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[CrossRef]

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]

Kolhe, P. S.

Levy, M. E.

G. N. Minerbo and M. E. Levy, “Inversion of Abel’s integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598–616 (1969).
[CrossRef]

Ma, S.

Magnus, F.

F. Magnus and J. T. Gudmundsson, “Digital smoothing of the Langmuir probe I-V characteristic,” Rev. Sci. Instrum. 79, 073503 (2008).
[CrossRef] [PubMed]

Mandelshtam, V. A.

V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634–2642 (2002).
[CrossRef]

Minerbo, G. N.

G. N. Minerbo and M. E. Levy, “Inversion of Abel’s integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598–616 (1969).
[CrossRef]

Nahon, L.

G. A. Garcia, L. Nahon, and I. Powis, “Two-dimensional charged particle image inversion using a polar basis function expansion,” Rev. Sci. Instrum. 75, 4989–4996 (2004).
[CrossRef]

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, 1996).

Nestor, O. H.

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

Nugent, K. A.

Olsen, H. N.

O. H. Nestor and H. N. Olsen, “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, 2nd ed. (Prentice-Hall, 1996).

Ossadtchi, A.

V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634–2642 (2002).
[CrossRef]

Palop, J. I. Fernández

J. I. Fernández Palop, J. Ballesteros, V. Colomer, and M. A. Hernández, “A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I-V probe characteristic,” Rev. Sci. Instrum. 66, 4625–4636 (1995).
[CrossRef]

Pender, J. T. P.

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[CrossRef]

Piche, R.

R. Piche, “Noise-filtering properties of numerical methods for the inverse Abel transform,” IEEE Trans. Instrum. Meas. 41, 517–522 (1992).
[CrossRef]

Powis, I.

G. A. Garcia, L. Nahon, and I. Powis, “Two-dimensional charged particle image inversion using a polar basis function expansion,” Rev. Sci. Instrum. 75, 4989–4996 (2004).
[CrossRef]

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]

Reisler, H.

V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634–2642 (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]

Roy, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions(Cambridge University Press, 1999).

Smith, L. M.

L. M. Smith, “Nonstationary noise effects in the Abel inversion,” IEEE Trans. Inf. Theory 34, 158–161 (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]

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]

Ventzek, P. L. G.

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[CrossRef]

Vincent, T.

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[CrossRef]

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, 1966).

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, 1996).

Wu, L.

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, “A versatile analytical expression for the inverse Abel transform applied to experimental data with noise,” Appl. Spectrosc. 62, 701–707(2008).
[CrossRef] [PubMed]

Appl. Opt. (5)

Appl. Spectrosc. (1)

IEEE Trans. Inf. Theory (1)

L. M. Smith, “Nonstationary noise effects in the Abel inversion,” IEEE Trans. Inf. Theory 34, 158–161 (1988).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

R. Piche, “Noise-filtering properties of numerical methods for the inverse Abel transform,” IEEE Trans. Instrum. Meas. 41, 517–522 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to experimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231–243 (1996).
[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]

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]

Rev. Sci. Instrum. (4)

F. Magnus and J. T. Gudmundsson, “Digital smoothing of the Langmuir probe I-V characteristic,” Rev. Sci. Instrum. 79, 073503 (2008).
[CrossRef] [PubMed]

J. I. Fernández Palop, J. Ballesteros, V. Colomer, and M. A. Hernández, “A new smoothing method for obtaining the electron energy distribution function in plasmas by the numerical differentiation of the I-V probe characteristic,” Rev. Sci. Instrum. 66, 4625–4636 (1995).
[CrossRef]

V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634–2642 (2002).
[CrossRef]

G. A. Garcia, L. Nahon, and I. Powis, “Two-dimensional charged particle image inversion using a polar basis function expansion,” Rev. Sci. Instrum. 75, 4989–4996 (2004).
[CrossRef]

SIAM J. Numer. Anal. (1)

G. N. Minerbo and M. E. Levy, “Inversion of Abel’s integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598–616 (1969).
[CrossRef]

SIAM Rev. (1)

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

Spectrochim. Acta B (2)

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]

G. C.-Y. Chan and G. M. Hieftje, “Estimation of confidence intervals for radial emissivity and optimization of data treatment techniques in Abel inversion,” Spectrochim. Acta B 61, 31–41 (2006).
[CrossRef]

Other (5)

H. R. Griem, Principles of Plasma Spectroscopy (Cambridge University Press, 1997).
[CrossRef]

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, Vol.  3 of Combustion Science and Technology Book Series, 2nd ed. (Gordon & Breach, 1996).

G. E. Andrews, R. Askey, and R. Roy, Special Functions(Cambridge University Press, 1999).

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, 1966).

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice-Hall, 1996).

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Figures (9)

Fig. 1
Fig. 1

Radial distribution of inversion uncertainties due to the singularity and derivative in Abel inversion. The uncertainty increases rapidly towards the source center, but only the most inner 10% region is very sensitive to noise.

Fig. 2
Fig. 2

(a–c) Inversion coefficients with N = 10 and (d–f) noise coefficients with N = 100 for different Abel inversion techniques: (a,d) inversion techniques without noise resisting property, for the VPF method K = N , M = 3 , and ρ = 3 / N and for the MFH method α = 0.01 and η = 1 , (b,e) the VPF method with K = 5 , M = 6 , and several overlapping factors ρ, and (c,f) the MFH method with α = 0.01 and different smoothing factors η.

Fig. 3
Fig. 3

Radial distributions of the emission coefficient for four test profiles.

Fig. 4
Fig. 4

Standard deviations for different inversion techniques with various values of N (5, 10, 20, 50, 100, and 200) for I 4 ( x ) : (a–c) noise free data and (d–f) noisy data with S = 0.03 . Other conditions are the same as Fig. 2.

Fig. 5
Fig. 5

(a) Intensity profiles of I 4 ( x ) without noise, added noise with S = 0.05 , and smoothed with the Gaussian filter ( σ = 4.8 ) and the Blackman filter ( w = 30 ). (b) Corresponding theoretical emission coefficient profile and the profiles inverted with Bockasten’s method for I 4 ( x ) smoothed by the Gaussian and Blackman filters and those inverted with the VPF ( K = 5 , M = 6 , and ρ = 1.1 ) and MFH ( α = 0.01 and η = 0.05 ) methods. The number of data points is 100. Each profile is offset from the nearest one for clarity.

Fig. 6
Fig. 6

Radial distributions of the absolute inversion error for I 4 ( x ) with noise of S = 0.05 : (a) Bockasten’s method with the Gaussian filter, (b) Bockasten’s method with the Blackman filter, (c) the VPF method, and (d) the MFH method. Other conditions are the same as Fig. 5.

Fig. 7
Fig. 7

Optimized overlapping factors ρ for the VPF method applied to I 1 ( x ) to I 4 ( x ) with noise of S = 0.005 and 0.02 and several values of N.

Fig. 8
Fig. 8

Optimized smoothing factor η for the MFH method applied to I 1 ( x ) to I 4 ( x ) with noise of S = 0.005 and 0.02 and several values of N.

Fig. 9
Fig. 9

Distributions of (a) experimental intensities measured from different layers of an arc plasma and (b) the corresponding radial emission coefficients inverted using Bockasten’s method [4], the VPF method with K = 5 , M = 6 , and ρ = 0.35 , and the MFH method with α = 0.01 and η = 0.25 .

Tables (1)

Tables Icon

Table 1 Standard Deviations of the Inversion Using the VPF ( K = 5 , M = 6 ), and MFH ( α = 0.01 ) Methods Obtained for I 1 ( x ) to I 4 ( x ) with Noise in the Level of S = 0.005 and 0.02 a

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

I ( x ) = 2 x R ε ( r ) r r 2 x 2 d r .
ε ( r ) = 1 π r R I ( x ) x 2 r 2 d x ,
ε ( r i ) = 1 Δ r j = 0 N D i j I ( x j ) ,
D i j = { 0 , 2 i N 1 , 0 j i 2 , U 0 ( i , j ) , 1 i N 1 , j = i 1 , U 1 ( i , j ) , 0 i N 1 , j = i , U 2 ( i , j ) , 0 i N 2 , j = i + 1 , U 3 ( i , j ) , 0 i N 3 , i + 2 j N 1 ,
U m ( i , j ) = 1 π l = 0 m k = 0 2 θ l k j l + 1 j l + 1 j l + 2 k t d t t 2 i 2 ,
Θ j = ( 3 j 2 + 6 j + 2 ) / 6 j + 1 1 / 2 ( 3 j 2 + 4 j 1 ) / 2 3 j 2 3 / 2 ( 3 j 2 + 2 j 2 ) / 2 3 j + 1 3 / 2 ( 3 j 2 1 ) / 6 j 1 / 2
Θ 0 = 0 0 0 0 7 / 2 9 / 4 0 4 3 0 1 / 2 3 / 4 , Θ N 1 = N + 1 / 2 1 0 2 N 2 2 0 0 0 0 0 0 0 .
I ( x ) n = 1 K ω n ( x ) P n M ( x ) = n = 1 K ω n ( x ) m = 0 M a n m x m ,
ε k ( r ) = 1 π R n = k K m = 1 M m a n m [ δ n k r x n x m 1 x 2 r 2 d x + ( 1 δ n k ) x n 1 x n x m 1 x 2 r 2 d x ] ,
P n M ( x ) = l = 0 M c n l f n l ( x ) = l = 0 M m = 0 l c n l d n l m x m ,
c n l = x n a x n b f n l ( x ) I ( x ) d x ,
f n l ( x ) = 2 l + 1 x n b x n a P l ( 2 x x n b x n a x n b x n a ) ,
a n m = l = m M c n l d n l m .
I ( x ) = a 0 + k = 1 a k cos ( k π x R ) ,
ε ( r ) = π 2 R k = 1 k a k g ( k π / R , r , R ) ,
g ( ω , r , R ) = 2 π r R sin ( ω x ) x 2 r 2 d x .
ε ( r ) = 1 2 π 0 G ( ω ) ω J 0 ( ω r ) d ω ,
ε ( r i ) = α 2 π 2 N R k = 1 N k G ( α k ) J 0 ( α i k π N ) ,
G ( α k ) = j = N N 1 I ( x j ) cos ( α j k π N ) .
ε ( r i ) = α 2 π N R j = 0 N ω j I ( x j ) k = 1 η N / α k J 0 ( α i k π N ) cos ( α j k π N ) ,
ε ( r ) 0 1 r 2 d t t 2 + r 2 = asinh r 2 1 .
ε ( 0 ) = 2 k 0 k sin ( 2 π ζ ) ζ d ζ π k , ζ = k x .
β i = ( j = 0 N D i j 2 ) 1 / 2 .
ε 1 ( r ) = exp [ ( π r ) 2 ] , 0 r 1 ,
ε 2 ( r ) = ( 1 r ) 2 ( 1 + 2 r ) , 0 r 1 ,
ε 3 ( r ) = 3 4 + 12 r 2 32 r 3 , 0 r 0.25 , = 16 27 ( 1 r ) 2 ( 1 + 8 r ) , 0.25 < r 1 ,
ε 4 ( r ) = ( 1 r 2 ) 2 ( 1 + 12 r 2 ) / ( 2197 / 972 ) , 0 r 1.
Δ ε ( r i ) = ε t ( r i ) ε c ( r i ) ,
σ = { 1 N i = 0 N 1 [ ε t ( r i ) ε c ( r i ) ] 2 } 1 / 2 .
g n ( x ) = k = 1 n ( n k ) ( 1 ) k + 1 1 σ 2 π k exp ( x 2 2 σ 2 k 2 ) ,
f B ( n ) = 0.42 0.5 cos ( 2 π n w ) + 0.08 cos ( 4 π n w ) ,

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