Abstract

A new method has been developed to perform an Abel inversion. As smoothing is an essential step when one deals with experimental data, a controlled smoothing procedure using spline functions is used. This gives better results for test functions used by previous authors on data with or without scattering.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Stanisavljevic, N. Konjevic, Fizika 4, 13 (1972).
  2. C. J. Cremers, R. C. Birkebak, Appl. Opt. 5, 1057 (1966).
    [CrossRef] [PubMed]
  3. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  4. J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).
  5. W. Frie, Ann. Phys. N. Y. 10, 332 (1963).
  6. H. Edels, K. R. Hearne, A. Young, J. Math. Phys. N.Y. 41, 62 (1962).
  7. O. E. Berge, J. Richter, AFSC Rep. 61(052), 797 (1966).
  8. W. L. Barr, J. Opt. Soc. Am. 52, 885 (1962).
    [CrossRef]
  9. D. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
    [CrossRef]
  10. H. Maecker, Z. Phys. 136, 119 (1953).
    [CrossRef]
  11. R. Ladenburg, J. A. Winckler, C. C. Van Voorhis, Phys. Rev. 73, 1359 (1948).
    [CrossRef]
  12. K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
    [CrossRef]

1972 (1)

M. Stanisavljevic, N. Konjevic, Fizika 4, 13 (1972).

1966 (2)

C. J. Cremers, R. C. Birkebak, Appl. Opt. 5, 1057 (1966).
[CrossRef] [PubMed]

O. E. Berge, J. Richter, AFSC Rep. 61(052), 797 (1966).

1963 (1)

W. Frie, Ann. Phys. N. Y. 10, 332 (1963).

1962 (2)

H. Edels, K. R. Hearne, A. Young, J. Math. Phys. N.Y. 41, 62 (1962).

W. L. Barr, J. Opt. Soc. Am. 52, 885 (1962).
[CrossRef]

1961 (1)

1960 (1)

D. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

1953 (1)

H. Maecker, Z. Phys. 136, 119 (1953).
[CrossRef]

1948 (1)

R. Ladenburg, J. A. Winckler, C. C. Van Voorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

Ahlberg, J. H.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Barr, W. L.

Berge, O. E.

O. E. Berge, J. Richter, AFSC Rep. 61(052), 797 (1966).

Birkebak, R. C.

Bockasten, K.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Cremers, C. J.

Edels, H.

H. Edels, K. R. Hearne, A. Young, J. Math. Phys. N.Y. 41, 62 (1962).

Frie, W.

W. Frie, Ann. Phys. N. Y. 10, 332 (1963).

Hearne, K. R.

H. Edels, K. R. Hearne, A. Young, J. Math. Phys. N.Y. 41, 62 (1962).

Konjevic, N.

M. Stanisavljevic, N. Konjevic, Fizika 4, 13 (1972).

Ladenburg, R.

R. Ladenburg, J. A. Winckler, C. C. Van Voorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

Maecker, H.

H. Maecker, Z. Phys. 136, 119 (1953).
[CrossRef]

Nestor, D. H.

D. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Nilson, E. N.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Olsen, H. N.

D. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Richter, J.

O. E. Berge, J. Richter, AFSC Rep. 61(052), 797 (1966).

Stanisavljevic, M.

M. Stanisavljevic, N. Konjevic, Fizika 4, 13 (1972).

Van Voorhis, C. C.

R. Ladenburg, J. A. Winckler, C. C. Van Voorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

Walsh, J. L.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Winckler, J. A.

R. Ladenburg, J. A. Winckler, C. C. Van Voorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

Young, A.

H. Edels, K. R. Hearne, A. Young, J. Math. Phys. N.Y. 41, 62 (1962).

AFSC Rep. (1)

O. E. Berge, J. Richter, AFSC Rep. 61(052), 797 (1966).

Ann. Phys. N. Y. (1)

W. Frie, Ann. Phys. N. Y. 10, 332 (1963).

Appl. Opt. (1)

Fizika (1)

M. Stanisavljevic, N. Konjevic, Fizika 4, 13 (1972).

J. Math. Phys. N.Y. (1)

H. Edels, K. R. Hearne, A. Young, J. Math. Phys. N.Y. 41, 62 (1962).

J. Opt. Soc. Am. (2)

Phys. Rev. (1)

R. Ladenburg, J. A. Winckler, C. C. Van Voorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

SIAM Rev. (1)

D. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Z. Phys. (1)

H. Maecker, Z. Phys. 136, 119 (1953).
[CrossRef]

Other (2)

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

The effect of a too strong smoothing on the derivative of the function: ○— ○— represents the raw data; - - - - - - represents optimal smoothing; and — ·· —represents too strong smoothing.

Fig. 2
Fig. 2

The effect of smoothing on data affected by a statistic dispersion of standard deviation δ: — ● — represents the standard deviation without smoothing; and — ★ — represents the standard deviation with smoothing.

Tables (5)

Tables Icon

Table I Standard Deviations Obtained with N = 15 Points with Exact Values

Tables Icon

Table II Standard Deviations Obtained with Test Function Given by Eq. (7) at N = 11 and N = 21 with Exact Values

Tables Icon

Table III Standard Deviations Obtained with Data Good to One Decimal Place

Tables Icon

Table IV Standard Deviations Obtained with Test Function Defined by Eq. (7) with Data Good to Two Decimal Places

Tables Icon

Table V Standard Deviations Obtained with Data Good to Three Decimal Places

Equations (12)

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

E ( r ) = 1 π r R I ( x ) ( x 2 r 2 ) 1 / 2 d x ,
I ( x ) = 2 x R E ( r ) r ( r 2 x 2 ) 1 / 2 d x .
E = i = 1 i = N 1 x i x i [ S ( x ) ] 2 d x + i = 1 i = N [ y ( y i ) S ( x i ) ] 2 / σ i 2 S 0 ,
E ( r ) = 1 / π { F K ( x K + 1 , r ) F K ( r , r ) + i = K + 1 i = N 1 [ F i ( x i + 1 , r ) F i ( x i , r ) ] } ,
F i ( x , r ) = { ( x 2 r 2 ) 1 / 2 ( 1 2 a i x + b i ) + ( 1 2 a i r 2 + c i ) ln [ x + ( x 2 r 2 ) 1 / 2 ] } ,
I ( x ) = 2 x R E ( r ) ( r 2 x 2 ) 1 / 2 r d r = 2 [ E ( R ) ( R 2 x 2 ) 1 / 2 x R ( r 2 x 2 ) 1 / 2 E ( r ) d r ] , I ( x ) = 2 x [ E ( R ) ( R 2 x 2 ) 1 / 2 + x R E ( r ) ( r 2 x 2 ) 1 / 2 d r ] .
I ( x ) 2 x { E ( R ) ( x 2 r 2 ) 1 / 2 + m ln [ R + ( R 2 x 2 ) 1 / 2 x ] } , I ( x ) 2 x { E ( R ) ( x 2 r 2 ) 1 / 2 + M ln [ R + ( R 2 x 2 ) 1 / 2 x ] } .
σ = { i = 1 i = N [ f i ( r ) exact f i ( r ) calcualted ] 2 / N } 1 / 2 ,
E ( r ) = exp ( r 2 / σ 2 ) ( R = 1.2 , σ = 0.58 ) ,
I ( x ) = σ ( π ) 1 / 2 erf [ ( R 2 x 2 ) 1 / 2 / σ ] exp ( x 2 / σ 2 ) , E ( r ) = r 2 exp ( r 2 / σ 2 ) ( R = 1.2 , σ = 0.58 ) ,
I ( x ) = σ ( π ) 1 / 2 exp ( x 2 / σ 2 ) erf [ ( R 2 x 2 ) 1 / 2 / σ ] ( x 2 + σ 2 / 2 ) σ 2 exp ( R 2 / σ 2 ) ( R 2 x 2 ) 1 / 2 , E ( r ) = ( σ 2 + r 2 ) 1 ( R = 20 , σ = 2 ) ,
I ( x ) = 2 ( x 2 + σ 2 ) 1 / 2 arc tan [ ( R 2 x 2 ) 1 / 2 / ( x 2 + σ 2 ) 1 / 2 ] E ( r ) = 1 3 r 2 + 2 r 3 ( R = 1 ) , T ( x ) = ( 1 x 2 ) 1 / 2 ( 1 5 2 x 2 ) + 3 2 x 4 ln { [ 1 + ( 1 x 2 ) 1 / 2 ] / x } .

Metrics