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  1. E. T. Whitaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1948).
  2. I. J. D. Craig, “The Inversion of Abel’s Integral Equation in Astrophysical Problems,” Astron. Astrophys. 79, 121 (1979).
  3. R. Gordon, R. Bender, G. Herman, “Algebraic Reconstruction Techniques for Three Dimensional Electron Microscopy and X-Ray Photography,” J. Theor. Biol. 29, 471 (1970); A. J. Jakeman, R. S. Anderssen, “Abel Type Integral Equations in Stereology. I: General Discussion,” J. Microsc. Oxford 105, 121 (1975).
    [CrossRef] [PubMed]
  4. M. Deutsch, I. Beniaminy, “Derivative-Free Inversion of Abel’s Integral Equation,” Appl. Phys. Lett. 41, 27 (1982).
    [CrossRef]
  5. A. Kuthy, “An Interferometer and Abel Inversion Procedure for the Measurement of the Electron Density Profile in a Cold Gas Blanket Experiment,” Nucl. Instrum. Methods 180, 7 (1981); A. M. Cormack, “Representation of a Function by its Line Integrals with Some Radiological Applications,” J. Appl. Phys. 34, 2722 (1963); G. H. Minerbo, E. M. Levy, “Inversion of Abel’s Integral Equation by Means of Orthogonal Polynomials,” SIAM J. Numer. Anal. 6, 598 (1969); E. W. Hanson, Phaih-Lanlaw, “Recursive Methods for Computing the Abel Transform and its Inverse,” J. Opt. Soc. Am. A 2, 510 (1985); R. S. Anderssen, “Stable Procedures for the Inversion of Abel’s Equation,” J. Inst. Math. Appl. 17, 329 (1976); C. J. Cremers, R. C. Birke-back, “Application of the Abel Integral Equation to Spectrographs Data,” Appl. Opt. 5, 1057 (1966).
    [CrossRef] [PubMed]
  6. M. Deutsch, “Abel Inversion with a Simple Analytic Representation for Experimental Data,” Appl. Phys. Lett. 42, 237 (1983).
    [CrossRef]
  7. M. Deutsch, I. Beniaminy, “Inversion of Abel’s Integral Equation for Experimental Data,” J. Appl. Phys. 54, 137 (1983).
    [CrossRef]
  8. A. Notea, “Resolving Power of Dynamic Radiation Gauges,” Nucl. Tech. 63, 121 (1983); A. Notea, “Evaluating Radiographic Systems Using the Resolving Power Function,” NDT Int. 16, 263 (1983); Y. Bushlin, D. Ingman, A. Notea, “Moments Analysis Method for the Determination of Dimensions from Radiographs,” Nucl. Tech. 74, 218 (1986).
    [CrossRef]
  9. A. Notea, D. Pal, M. Deutsch, “Density Distribution in Cylindrically Symmetric Objects from a Single Radiographic Image,” in Proceedings, Fourth European Conference on Non-Destructive Testing, London, Sept.1987 (in print).
  10. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1978).
  11. T. H. Newton, D. G. Potts, “Radiology of the Skull and Brain,” in Technical Aspects of Computed Tomography, Vol. 5 (Mosby, London, 1981).

1983 (3)

M. Deutsch, “Abel Inversion with a Simple Analytic Representation for Experimental Data,” Appl. Phys. Lett. 42, 237 (1983).
[CrossRef]

M. Deutsch, I. Beniaminy, “Inversion of Abel’s Integral Equation for Experimental Data,” J. Appl. Phys. 54, 137 (1983).
[CrossRef]

A. Notea, “Resolving Power of Dynamic Radiation Gauges,” Nucl. Tech. 63, 121 (1983); A. Notea, “Evaluating Radiographic Systems Using the Resolving Power Function,” NDT Int. 16, 263 (1983); Y. Bushlin, D. Ingman, A. Notea, “Moments Analysis Method for the Determination of Dimensions from Radiographs,” Nucl. Tech. 74, 218 (1986).
[CrossRef]

1982 (1)

M. Deutsch, I. Beniaminy, “Derivative-Free Inversion of Abel’s Integral Equation,” Appl. Phys. Lett. 41, 27 (1982).
[CrossRef]

1981 (1)

A. Kuthy, “An Interferometer and Abel Inversion Procedure for the Measurement of the Electron Density Profile in a Cold Gas Blanket Experiment,” Nucl. Instrum. Methods 180, 7 (1981); A. M. Cormack, “Representation of a Function by its Line Integrals with Some Radiological Applications,” J. Appl. Phys. 34, 2722 (1963); G. H. Minerbo, E. M. Levy, “Inversion of Abel’s Integral Equation by Means of Orthogonal Polynomials,” SIAM J. Numer. Anal. 6, 598 (1969); E. W. Hanson, Phaih-Lanlaw, “Recursive Methods for Computing the Abel Transform and its Inverse,” J. Opt. Soc. Am. A 2, 510 (1985); R. S. Anderssen, “Stable Procedures for the Inversion of Abel’s Equation,” J. Inst. Math. Appl. 17, 329 (1976); C. J. Cremers, R. C. Birke-back, “Application of the Abel Integral Equation to Spectrographs Data,” Appl. Opt. 5, 1057 (1966).
[CrossRef] [PubMed]

1979 (1)

I. J. D. Craig, “The Inversion of Abel’s Integral Equation in Astrophysical Problems,” Astron. Astrophys. 79, 121 (1979).

1970 (1)

R. Gordon, R. Bender, G. Herman, “Algebraic Reconstruction Techniques for Three Dimensional Electron Microscopy and X-Ray Photography,” J. Theor. Biol. 29, 471 (1970); A. J. Jakeman, R. S. Anderssen, “Abel Type Integral Equations in Stereology. I: General Discussion,” J. Microsc. Oxford 105, 121 (1975).
[CrossRef] [PubMed]

Bender, R.

R. Gordon, R. Bender, G. Herman, “Algebraic Reconstruction Techniques for Three Dimensional Electron Microscopy and X-Ray Photography,” J. Theor. Biol. 29, 471 (1970); A. J. Jakeman, R. S. Anderssen, “Abel Type Integral Equations in Stereology. I: General Discussion,” J. Microsc. Oxford 105, 121 (1975).
[CrossRef] [PubMed]

Beniaminy, I.

M. Deutsch, I. Beniaminy, “Inversion of Abel’s Integral Equation for Experimental Data,” J. Appl. Phys. 54, 137 (1983).
[CrossRef]

M. Deutsch, I. Beniaminy, “Derivative-Free Inversion of Abel’s Integral Equation,” Appl. Phys. Lett. 41, 27 (1982).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1978).

Craig, I. J. D.

I. J. D. Craig, “The Inversion of Abel’s Integral Equation in Astrophysical Problems,” Astron. Astrophys. 79, 121 (1979).

Deutsch, M.

M. Deutsch, “Abel Inversion with a Simple Analytic Representation for Experimental Data,” Appl. Phys. Lett. 42, 237 (1983).
[CrossRef]

M. Deutsch, I. Beniaminy, “Inversion of Abel’s Integral Equation for Experimental Data,” J. Appl. Phys. 54, 137 (1983).
[CrossRef]

M. Deutsch, I. Beniaminy, “Derivative-Free Inversion of Abel’s Integral Equation,” Appl. Phys. Lett. 41, 27 (1982).
[CrossRef]

A. Notea, D. Pal, M. Deutsch, “Density Distribution in Cylindrically Symmetric Objects from a Single Radiographic Image,” in Proceedings, Fourth European Conference on Non-Destructive Testing, London, Sept.1987 (in print).

Gordon, R.

R. Gordon, R. Bender, G. Herman, “Algebraic Reconstruction Techniques for Three Dimensional Electron Microscopy and X-Ray Photography,” J. Theor. Biol. 29, 471 (1970); A. J. Jakeman, R. S. Anderssen, “Abel Type Integral Equations in Stereology. I: General Discussion,” J. Microsc. Oxford 105, 121 (1975).
[CrossRef] [PubMed]

Herman, G.

R. Gordon, R. Bender, G. Herman, “Algebraic Reconstruction Techniques for Three Dimensional Electron Microscopy and X-Ray Photography,” J. Theor. Biol. 29, 471 (1970); A. J. Jakeman, R. S. Anderssen, “Abel Type Integral Equations in Stereology. I: General Discussion,” J. Microsc. Oxford 105, 121 (1975).
[CrossRef] [PubMed]

Kuthy, A.

A. Kuthy, “An Interferometer and Abel Inversion Procedure for the Measurement of the Electron Density Profile in a Cold Gas Blanket Experiment,” Nucl. Instrum. Methods 180, 7 (1981); A. M. Cormack, “Representation of a Function by its Line Integrals with Some Radiological Applications,” J. Appl. Phys. 34, 2722 (1963); G. H. Minerbo, E. M. Levy, “Inversion of Abel’s Integral Equation by Means of Orthogonal Polynomials,” SIAM J. Numer. Anal. 6, 598 (1969); E. W. Hanson, Phaih-Lanlaw, “Recursive Methods for Computing the Abel Transform and its Inverse,” J. Opt. Soc. Am. A 2, 510 (1985); R. S. Anderssen, “Stable Procedures for the Inversion of Abel’s Equation,” J. Inst. Math. Appl. 17, 329 (1976); C. J. Cremers, R. C. Birke-back, “Application of the Abel Integral Equation to Spectrographs Data,” Appl. Opt. 5, 1057 (1966).
[CrossRef] [PubMed]

Newton, T. H.

T. H. Newton, D. G. Potts, “Radiology of the Skull and Brain,” in Technical Aspects of Computed Tomography, Vol. 5 (Mosby, London, 1981).

Notea, A.

A. Notea, “Resolving Power of Dynamic Radiation Gauges,” Nucl. Tech. 63, 121 (1983); A. Notea, “Evaluating Radiographic Systems Using the Resolving Power Function,” NDT Int. 16, 263 (1983); Y. Bushlin, D. Ingman, A. Notea, “Moments Analysis Method for the Determination of Dimensions from Radiographs,” Nucl. Tech. 74, 218 (1986).
[CrossRef]

A. Notea, D. Pal, M. Deutsch, “Density Distribution in Cylindrically Symmetric Objects from a Single Radiographic Image,” in Proceedings, Fourth European Conference on Non-Destructive Testing, London, Sept.1987 (in print).

Pal, D.

A. Notea, D. Pal, M. Deutsch, “Density Distribution in Cylindrically Symmetric Objects from a Single Radiographic Image,” in Proceedings, Fourth European Conference on Non-Destructive Testing, London, Sept.1987 (in print).

Potts, D. G.

T. H. Newton, D. G. Potts, “Radiology of the Skull and Brain,” in Technical Aspects of Computed Tomography, Vol. 5 (Mosby, London, 1981).

Watson, G. N.

E. T. Whitaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1948).

Whitaker, E. T.

E. T. Whitaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1948).

Appl. Phys. Lett. (2)

M. Deutsch, I. Beniaminy, “Derivative-Free Inversion of Abel’s Integral Equation,” Appl. Phys. Lett. 41, 27 (1982).
[CrossRef]

M. Deutsch, “Abel Inversion with a Simple Analytic Representation for Experimental Data,” Appl. Phys. Lett. 42, 237 (1983).
[CrossRef]

Astron. Astrophys. (1)

I. J. D. Craig, “The Inversion of Abel’s Integral Equation in Astrophysical Problems,” Astron. Astrophys. 79, 121 (1979).

J. Appl. Phys. (1)

M. Deutsch, I. Beniaminy, “Inversion of Abel’s Integral Equation for Experimental Data,” J. Appl. Phys. 54, 137 (1983).
[CrossRef]

J. Theor. Biol. (1)

R. Gordon, R. Bender, G. Herman, “Algebraic Reconstruction Techniques for Three Dimensional Electron Microscopy and X-Ray Photography,” J. Theor. Biol. 29, 471 (1970); A. J. Jakeman, R. S. Anderssen, “Abel Type Integral Equations in Stereology. I: General Discussion,” J. Microsc. Oxford 105, 121 (1975).
[CrossRef] [PubMed]

Nucl. Instrum. Methods (1)

A. Kuthy, “An Interferometer and Abel Inversion Procedure for the Measurement of the Electron Density Profile in a Cold Gas Blanket Experiment,” Nucl. Instrum. Methods 180, 7 (1981); A. M. Cormack, “Representation of a Function by its Line Integrals with Some Radiological Applications,” J. Appl. Phys. 34, 2722 (1963); G. H. Minerbo, E. M. Levy, “Inversion of Abel’s Integral Equation by Means of Orthogonal Polynomials,” SIAM J. Numer. Anal. 6, 598 (1969); E. W. Hanson, Phaih-Lanlaw, “Recursive Methods for Computing the Abel Transform and its Inverse,” J. Opt. Soc. Am. A 2, 510 (1985); R. S. Anderssen, “Stable Procedures for the Inversion of Abel’s Equation,” J. Inst. Math. Appl. 17, 329 (1976); C. J. Cremers, R. C. Birke-back, “Application of the Abel Integral Equation to Spectrographs Data,” Appl. Opt. 5, 1057 (1966).
[CrossRef] [PubMed]

Nucl. Tech. (1)

A. Notea, “Resolving Power of Dynamic Radiation Gauges,” Nucl. Tech. 63, 121 (1983); A. Notea, “Evaluating Radiographic Systems Using the Resolving Power Function,” NDT Int. 16, 263 (1983); Y. Bushlin, D. Ingman, A. Notea, “Moments Analysis Method for the Determination of Dimensions from Radiographs,” Nucl. Tech. 74, 218 (1986).
[CrossRef]

Other (4)

A. Notea, D. Pal, M. Deutsch, “Density Distribution in Cylindrically Symmetric Objects from a Single Radiographic Image,” in Proceedings, Fourth European Conference on Non-Destructive Testing, London, Sept.1987 (in print).

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1978).

T. H. Newton, D. G. Potts, “Radiology of the Skull and Brain,” in Technical Aspects of Computed Tomography, Vol. 5 (Mosby, London, 1981).

E. T. Whitaker, G. N. Watson, A Course of Modern Analysis (Macmillan, New York, 1948).

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Figures (3)

Fig. 1
Fig. 1

Ceramic fuse housing and its radiograph. The arrow points to the region which was digitized to obtain the data of Fig. 2.

Fig. 2
Fig. 2

I(y) intensity profile: □, data digitized from the radiograph of Fig. 1 and averaged over twenty lines. For clarity only every second point is shown. —, Eq. (4) fitted to the data.

Fig. 3
Fig. 3

X-ray optical density profile g(r) calculated from the data of Fig. 2 using the present method (—) and the spline-based Abel inversion method of Ref. 7 (- - - -). Note the oscillations caused in the last method by the Gibbs phenomenon and their complete absence in the present one.

Equations (7)

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I ( y ) = 2 y R g ( r ) r ( r 2 - y 2 ) - 1 / 2 d r ,
g ( r ) = i = 1 N g i ( r ) ,
0 r < R i - 1 , g i ( r ) = k = 0 K a k i r k R i - 1 < r < R i , 0 R i < r .
I ( y ) = 2 i = j N - 1 k = 0 K a k i + 1 J k + 1 ( R i , R i + 1 ) + 2 k = 0 K a k j J k + 1 ( y , R j ) ,
J n ( a , b ) = a b r n ( r 2 - y 2 ) - 1 / 2 d r .
r n ( r 2 - y 2 ) - 1 / 2 d r = A { ( r 2 - y 2 ) 1 / 2 m = 1 n / 2 B y n - 2 m ( 2 r ) 2 m - 1 + y 2 ln [ r + ( r 2 - y 2 ) 1 / 2 ] } n - even , ( r 2 - y 2 ) 1 / 2 m = 0 ( n - 1 ) / 2 C ( 2 y ) n - 2 m - 1 r 2 m n - odd ,
A = n ! / [ 2 n · ( n / 2 ) ! ] , B = m ! ( m - 1 ) ! / ( 2 m ) ! , C = ( 2 m ) ! [ ( n - 1 2 ) ! / m ] 2 n ! .

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