Abstract

Errors in discrete Abel inversion methods using Fourier transform techniques have been analyzed. The Fourier expansion method is very accurate but sensitive to noise. The Fourier–Hankel method has a significant systematic negative deviation, which increases with the radius; inversion error of the method can be reduced by adjusting the value of a factor. With a decrease of the factor both methods show a noise filtering property. Based on the analysis, a modified Fourier–Hankel method that is accurate, computationally efficient, and has the ability to filter noise in the inversion process is proposed for applying to experimental data.

© 2008 Optical Society of America

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References

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    [CrossRef]
  2. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, Vol. 3 of Combustion Science and Technology Book Series, 2nd ed. (Gordon and Breach, 1996).
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    [CrossRef]
  8. K. Tatekura, “Determination of the index profile of optical fibers from transverse interferograms using Fourier theory,” Appl. Opt. 22, 460-463 (1983).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. R. Álvarez, A. Rodero, and M. C. Quintero, “An Abel inversion method for radially resolved measurements in the axial injection torch,” Spectrochim. Acta, Part B 57, 1665-1680 (2002).
    [CrossRef]
  14. A. Sáinz, A. Díaz, D. Casas, M. Pineda, F. Cubillo, and M. D. Calzada, “Abel inversion applied to a small set of emission data from a microwave plasma,” Appl. Spectrosc. 60, 229-236(2006).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2007

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

2006

2002

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]

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

1996

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. Transf. 55, 231-243 (1996).
[CrossRef]

1991

J. Dong and R. J. Kearney, “Symmetrizing, filtering and Abel inversion using Fourier transform techniques,” J. Quant. Spectrosc. Radiat. Transf. 46, 141-149 (1991).
[CrossRef]

1988

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

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

1986

Yu. E. Voskoboinikov and N. G. Preobrazhenskii, “Abel inversion with high accuracy in the problems of optics and spectroscopy,” Opt. Spectrosc. 60, 111-113 (1986).

1983

1981

S. M. Candel, “An algorithm for the Fourier-Bessel transform,” Comput. Phys. Commun. 23, 343-353 (1981).
[CrossRef]

1978

1969

G. N. Minerbo and M. E. Levy, “Inversion of Abel's integral equation by means of orthogonal polynomials,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 6, 598-616 (1969).

1966

Appl. Opt.

Appl. Spectrosc.

Comput. Phys. Commun.

S. M. Candel, “An algorithm for the Fourier-Bessel transform,” Comput. Phys. Commun. 23, 343-353 (1981).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

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

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

J. Dong and R. J. Kearney, “Symmetrizing, filtering and Abel inversion using Fourier transform techniques,” J. Quant. Spectrosc. Radiat. Transf. 46, 141-149 (1991).
[CrossRef]

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. Transf. 55, 231-243 (1996).
[CrossRef]

Opt. Spectrosc.

Yu. E. Voskoboinikov and N. G. Preobrazhenskii, “Abel inversion with high accuracy in the problems of optics and spectroscopy,” Opt. Spectrosc. 60, 111-113 (1986).

Rev. Sci. Instrum.

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]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

G. N. Minerbo and M. E. Levy, “Inversion of Abel's integral equation by means of orthogonal polynomials,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 6, 598-616 (1969).

Spectrochim. Acta, Part B

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

Other

H. R. Griem, Principles of Plasma Spectroscopy (Cambridge U. 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 and Breach, 1996).

B. J. Whitaker, Imaging in Molecular Dynamics (Cambridge U. Press, 2003).
[CrossRef]

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

D. H. Manzella, F. M. Curran, R. M. Myers, and D. M. Zube, Preliminary Plume Characteristics of an Arcjet Thruster, Tech. Rep. No. 103241 (NASA, 1990).

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

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Figures (7)

Fig. 1
Fig. 1

Radial variations of the emission coefficients and the corresponding absolute errors in the inversion performed with the FE and FH methods.

Fig. 2
Fig. 2

Variations of the standard deviation σ for the FH method calculated with various values of the number of data n as a function of α 2 .

Fig. 3
Fig. 3

Radial distributions of the difference f ( k π , α r ) between J 0 ( α ω r ) and g ( ω , αr ) for various values of k calculated with intensities in Eq. (13).

Fig. 4
Fig. 4

Variations of k G ( α k ) / n , f ( k π , α r n ) , and their product k G ( α k ) f ( k π , α r n ) / n as functions of k calculated using the inten sity data in (a) Eq. (13) with off-axis peaks and (b) Eq. (15) with a Gaussian type.

Fig. 5
Fig. 5

Comparison of distributions of the theoretical emission coefficients and the values inverted using the FE and FH methods with (a) the value of I 1 ( 1 / 2 ) decreased to 0.95 I 1 ( 1 / 2 ) to simulate the noise in the experimental data and (b) intensities of I 2 ( x ) added normally distributed random noise with a standard deviation S = 0.005 . The values of α for the FE and FH methods are 1 and 0.1, respectively. Results of the FH method are offset from the original positions for clarity.

Fig. 6
Fig. 6

Comparison of distributions of the theoretical emission coefficients and the values inverted with the FH method and its modified form of Eq. (23).

Fig. 7
Fig. 7

Distribution of functions ε k ( r ) = ( 1 r 2 ) k for a range of k.

Tables (1)

Tables Icon

Table 1 Comparison of Standard Deviations σ for the Reconstruction of Functions ε k ( r ) = ( 1 r 2 ) k for Different k Using the FE, FH, and MFH Methods

Equations (24)

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I ( x ) = 2 x R ( r 2 x 2 ) 1 / 2 ε ( r ) r d r ,
ε ( r ) = 1 π r R ( x 2 r 2 ) 1 / 2 I ( x ) d x ,
ε ( r i ) = j = 0 n P i j I ( x j ) ,
I ( x ) = a 0 + k = 1 a k cos ( k π e 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 ( x 2 r 2 ) 1 / 2 sin ( ω x ) d x .
ε ( r i ) = π 2 n R k = 1 n k a ( k ) g ( k π , i / n ) ,
a ( k ) = j = n n 1 I ( x j ) cos ( j k π n ) .
ε ( r ) = 1 2 π 0 G ( ω ) ω J 0 ( ω r ) d ω ,
J 0 ( ω r ) = 2 π r ( x 2 r 2 ) 1 / 2 sin ( ω x ) d x
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 ) .
I 1 ( x ) = 16 105 ( 1 x 2 ) 5 / 2 ( 19 + 72 x 2 ) , 0 x 1 ,
ε 1 ( r ) = ( 1 r 2 ) 2 ( 1 + 12 r 2 ) , 0 r 1.
I 2 ( x ) = exp ( x 2 / 400 ) , 0 x 100 ,
ε 2 ( r ) = 1 20 π exp ( r 2 / 400 ) , 0 r 100.
ε ( r i ) = α 2 π 2 n R k = 1 n k G ( α k ) g ( k π , α i / n ) .
g ( ω , r i ) = 2 π i n ( t 2 i 2 ) 1 / 2 sin ( ω t n ) d t ,
J 0 ( ω r i ) = 2 π i ( t 2 i 2 ) 1 / 2 sin ( ω t n ) d t .
f ( ω , r , R ) = J 0 ( ω r ) g ( ω , r , R ) = 2 π R ( x 2 r 2 ) 1 / 2 sin ( ω x ) d x .
σ = { 1 / ( n + 1 ) n [ ε c ( r i ) ε t ( r i ) ] 2 } 1 / 2 ,
δ ( α , r ) = α 2 π 2 n R k = 1 n k G ( α k ) f ( k π , α r ) .
ε ( r i ) = α 2 π 2 n R j = n n 1 I ( x j ) k = 1 n / α k J 0 ( α i k π n ) cos ( α j k π n ) ,
ε k ( r ) = ( 1 r 2 ) k , 0 r 1.

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