Abstract

A fast Fourier transform based Abel inversion technique is proposed. The method is faster than previously used techniques, potentially very accurate (even for a relatively small number of points), and capable of handling large data sets. The technique is discussed in the context of its use with 2-D digital interferogram analysis algorithms. Several examples are given.

© 1988 Optical Society of America

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References

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  1. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 262–265.
  2. W. J. Pearce, in Proceedings, Conference on Extremely High Temperature, Boston, 18–19 Mar. (Wiley, New York, 1958), pp. 123–124.
  3. R. W. Landenburg, W. Lewis, R. N. Phease, H. S. Taylor, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion, T. von Karman et al., Eds. (Oxford U. P., London, 1955).
  4. K. Bockasten, “Transformation of Observed Radiances into Radial Distribution of the Emission of a Plasma,” J. Opt. Soc. Am. 51, 943 (1961).
    [CrossRef]
  5. V. V. Pikalov, N. G. Preobrazhenskii, “Abel Transformation in the Interferometer Holography of a Point Explosion,” Combust. Explos. Shock Waves USSR 10, 827 (1975).
    [CrossRef]
  6. C. Fleurier, J. Chapelle, “Inversion of Abel’s Integral Equation: Application to Plasma Spectroscopy,” Comput. Phys. Commun. 7, 200 (1974).
    [CrossRef]
  7. L. S. Fan, W. Squire, “Inversion of Abel’s Integral Equation by a Direct Method,” Comput. Phys. Commun. 10, 98 (1975).
    [CrossRef]
  8. V. A. Gribkov, V. Ya. Nikulin, G. V. Sklizkov, “Double-Beam Interferometry Method for Investigating Axisymmetric Configurations of Dense Plasma,” Sov. J. Quantum. Electron. 1, 606 (1972).
    [CrossRef]
  9. D. W. Sweeney, D. T. Attwood, L. W. Coleman, “Interferometric Probing of Laser Produced Plasmas,” Appl. Opt. 15, 1126 (1976).
    [CrossRef] [PubMed]
  10. K. Tatekura, “Determination of the Index Profile of Optical Fibers from Transverse Interferograms Using Fourier Theory,” Appl. Opt. 22, 460 (1983).
    [CrossRef] [PubMed]
  11. M. Kalal, K. A. Nugent, B. Luther-Davies, “Phase-Amplitude Imaging: Its Application to Fully Automated Analysis of Magnetic Field Measurements in Laser-Produced Plasmas,” Appl. Opt. 26, 1674 (1987).
    [CrossRef] [PubMed]
  12. M. Takeda, H. Ina, S. Kobayashi, “Fourier-Transform Method of Fringe-Pattern Analysis for Computer-Based Topography and Interferometry,” J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  13. W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898 (1983).
    [CrossRef] [PubMed]
  14. G. A. Mastin, D. C. Ghiglia, “Digital Extraction of Interference Fringe Contours,” Appl. Opt. 24, 1727 (1985).
    [CrossRef] [PubMed]
  15. K. A. Nugent, “Interferogram Analysis Using an Accurate Fully Automatic Algorithm,” Appl. Opt. 24, 3101 (1985).
    [CrossRef] [PubMed]
  16. D. J. Bone, H-A. Bachor, R. J. Sandeman, “Fringe-Pattern Analysis Using a 2-D Fourier Transform,” Appl. Opt. 25, 1653 (1986).
    [CrossRef] [PubMed]
  17. T. Kreis, “Digital Holographic Interference-Phase Measurement Using the Fourier-Transform Method,” J. Opt. Soc. Am. A 3, 847 (1986).
    [CrossRef]

1987 (1)

1986 (2)

1985 (2)

1983 (2)

1982 (1)

1976 (1)

1975 (2)

V. V. Pikalov, N. G. Preobrazhenskii, “Abel Transformation in the Interferometer Holography of a Point Explosion,” Combust. Explos. Shock Waves USSR 10, 827 (1975).
[CrossRef]

L. S. Fan, W. Squire, “Inversion of Abel’s Integral Equation by a Direct Method,” Comput. Phys. Commun. 10, 98 (1975).
[CrossRef]

1974 (1)

C. Fleurier, J. Chapelle, “Inversion of Abel’s Integral Equation: Application to Plasma Spectroscopy,” Comput. Phys. Commun. 7, 200 (1974).
[CrossRef]

1972 (1)

V. A. Gribkov, V. Ya. Nikulin, G. V. Sklizkov, “Double-Beam Interferometry Method for Investigating Axisymmetric Configurations of Dense Plasma,” Sov. J. Quantum. Electron. 1, 606 (1972).
[CrossRef]

1961 (1)

Attwood, D. T.

Bachor, H-A.

Bockasten, K.

Bone, D. J.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 262–265.

Chapelle, J.

C. Fleurier, J. Chapelle, “Inversion of Abel’s Integral Equation: Application to Plasma Spectroscopy,” Comput. Phys. Commun. 7, 200 (1974).
[CrossRef]

Coleman, L. W.

Fan, L. S.

L. S. Fan, W. Squire, “Inversion of Abel’s Integral Equation by a Direct Method,” Comput. Phys. Commun. 10, 98 (1975).
[CrossRef]

Fleurier, C.

C. Fleurier, J. Chapelle, “Inversion of Abel’s Integral Equation: Application to Plasma Spectroscopy,” Comput. Phys. Commun. 7, 200 (1974).
[CrossRef]

Ghiglia, D. C.

Gribkov, V. A.

V. A. Gribkov, V. Ya. Nikulin, G. V. Sklizkov, “Double-Beam Interferometry Method for Investigating Axisymmetric Configurations of Dense Plasma,” Sov. J. Quantum. Electron. 1, 606 (1972).
[CrossRef]

Ina, H.

Kalal, M.

Kobayashi, S.

Kreis, T.

Landenburg, R. W.

R. W. Landenburg, W. Lewis, R. N. Phease, H. S. Taylor, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion, T. von Karman et al., Eds. (Oxford U. P., London, 1955).

Lewis, W.

R. W. Landenburg, W. Lewis, R. N. Phease, H. S. Taylor, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion, T. von Karman et al., Eds. (Oxford U. P., London, 1955).

Luther-Davies, B.

Macy, W. W.

Mastin, G. A.

Nikulin, V. Ya.

V. A. Gribkov, V. Ya. Nikulin, G. V. Sklizkov, “Double-Beam Interferometry Method for Investigating Axisymmetric Configurations of Dense Plasma,” Sov. J. Quantum. Electron. 1, 606 (1972).
[CrossRef]

Nugent, K. A.

Pearce, W. J.

W. J. Pearce, in Proceedings, Conference on Extremely High Temperature, Boston, 18–19 Mar. (Wiley, New York, 1958), pp. 123–124.

Phease, R. N.

R. W. Landenburg, W. Lewis, R. N. Phease, H. S. Taylor, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion, T. von Karman et al., Eds. (Oxford U. P., London, 1955).

Pikalov, V. V.

V. V. Pikalov, N. G. Preobrazhenskii, “Abel Transformation in the Interferometer Holography of a Point Explosion,” Combust. Explos. Shock Waves USSR 10, 827 (1975).
[CrossRef]

Preobrazhenskii, N. G.

V. V. Pikalov, N. G. Preobrazhenskii, “Abel Transformation in the Interferometer Holography of a Point Explosion,” Combust. Explos. Shock Waves USSR 10, 827 (1975).
[CrossRef]

Sandeman, R. J.

Sklizkov, G. V.

V. A. Gribkov, V. Ya. Nikulin, G. V. Sklizkov, “Double-Beam Interferometry Method for Investigating Axisymmetric Configurations of Dense Plasma,” Sov. J. Quantum. Electron. 1, 606 (1972).
[CrossRef]

Squire, W.

L. S. Fan, W. Squire, “Inversion of Abel’s Integral Equation by a Direct Method,” Comput. Phys. Commun. 10, 98 (1975).
[CrossRef]

Sweeney, D. W.

Takeda, M.

Tatekura, K.

Taylor, H. S.

R. W. Landenburg, W. Lewis, R. N. Phease, H. S. Taylor, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion, T. von Karman et al., Eds. (Oxford U. P., London, 1955).

Appl. Opt. (7)

Combust. Explos. Shock Waves USSR (1)

V. V. Pikalov, N. G. Preobrazhenskii, “Abel Transformation in the Interferometer Holography of a Point Explosion,” Combust. Explos. Shock Waves USSR 10, 827 (1975).
[CrossRef]

Comput. Phys. Commun. (2)

C. Fleurier, J. Chapelle, “Inversion of Abel’s Integral Equation: Application to Plasma Spectroscopy,” Comput. Phys. Commun. 7, 200 (1974).
[CrossRef]

L. S. Fan, W. Squire, “Inversion of Abel’s Integral Equation by a Direct Method,” Comput. Phys. Commun. 10, 98 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Sov. J. Quantum. Electron. (1)

V. A. Gribkov, V. Ya. Nikulin, G. V. Sklizkov, “Double-Beam Interferometry Method for Investigating Axisymmetric Configurations of Dense Plasma,” Sov. J. Quantum. Electron. 1, 606 (1972).
[CrossRef]

Other (3)

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 262–265.

W. J. Pearce, in Proceedings, Conference on Extremely High Temperature, Boston, 18–19 Mar. (Wiley, New York, 1958), pp. 123–124.

R. W. Landenburg, W. Lewis, R. N. Phease, H. S. Taylor, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion, T. von Karman et al., Eds. (Oxford U. P., London, 1955).

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Figures (4)

Fig. 1
Fig. 1

Comparison of graphs of functions gk(r) (solid lines) and J0(kπr) (dashed lines): (a) k = 1; (b) k = 2; (c) k = 12.

Fig. 2
Fig. 2

Functions fk(r) for a range of k. As k becomes larger these functions closely approximate a Gaussian.

Fig. 3
Fig. 3

Test function with a deep central dip (solid line) and the inversion of the Abel transform of this curve (open circles).

Fig. 4
Fig. 4

(a) Phase shift recovered from an experimental interferogram obtained during a study of the density structure in a laser-produced plasma, (b) Plasma density profile obtained with the Abel inversion technique.

Tables (1)

Tables Icon

Table I Example of Standard Deviations (SD) for the Reconstruction of Functions fk(r) = (1 − r2)k for Different k Using the Abel Inversion Technique Described in This Paper

Equations (17)

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S ( y ) = 2 y R f ( r ) ( r 2 - y 2 ) - 1 / 2 r d r ,
f ( r ) = - 1 π r R d S ( y ) d y ( y 2 - r 2 ) - 1 / 2 d y .
S ( y ) = a 0 + k = 1 + a k cos k π y R ,
f ( r ) = k = 1 + k a k R r R ( y 2 - r 2 ) - 1 / 2 sin k π y R d y .
t = ( y 2 - r 2 ) 1 / 2 R ,
f ( r ) = π 2 R k = 1 + k a k g k ( r R ) ,
g k ( ρ ) = 2 π 0 ( 1 - ρ 2 ) 1 / 2 ( t 2 + ρ 2 ) - 1 / 2 sin k π ( t 2 + ρ 2 ) 1 / 2 d t .
f ( r ) = 1 2 0 a ( ω ) J 0 ( ω r ) ω d ω ,
a ( ω ) = 2 π 0 cos ( ω y ) S ( y ) d y
g k ( ± 1 ) = 0 ,
d g k d ρ | ρ = 0 = 0 ,
d g k d ρ | ρ = ± 1 = 0 ,
lim k + g k ( 0 ) = 0.
lim k + k α k = 0
f k ( r ) = ( 1 - r 2 ) k ,             r 1 ,
y c = - R R y S ( y ) d y - R R S ( y ) d y
S ( 0 ) = π R k = 1 + k b k ,

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