Abstract

In some measurement techniques the profile, fx, of a function should be obtained from the data on measured slope fx by integration. The slope is measured in a given set of points, and from these data we should obtain the profile with the highest possible accuracy. Most frequently, the integration is carried out by numerical integration methods [Press et al., Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987)] that assume different kinds of polynomial approximation of data between sampling points. We propose the integration of the function in the Fourier domain, by which the most-accurate interpolation is automatically carried out. Analysis of the integration methods in the Fourier domain permits us to easily study and compare the methods’ behavior.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. C. Irick, R. Krishna Kaza, and W. R. McKinney, Rev. Sci. Instrum. 66, 2108 (1995).
    [CrossRef]
  2. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).
  3. C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function fx when only dfx/dx is known at discrete measurement points,” Proc. SPIE4782 (to be published).
  4. L. Yaroslavsky and M. Eden, Fundamentals of Digital Optics (Birkhauser, Boston, Mass., 1996).
    [CrossRef]
  5. J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice-Hall, Englewood Cliffs, N.J., 1999).
  6. L. Yaroslavsky, Proc. SPIE 4667, 120 (2002).
    [CrossRef]

2002 (1)

L. Yaroslavsky, Proc. SPIE 4667, 120 (2002).
[CrossRef]

1995 (1)

S. C. Irick, R. Krishna Kaza, and W. R. McKinney, Rev. Sci. Instrum. 66, 2108 (1995).
[CrossRef]

Eden, M.

L. Yaroslavsky and M. Eden, Fundamentals of Digital Optics (Birkhauser, Boston, Mass., 1996).
[CrossRef]

Elster, C.

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function fx when only dfx/dx is known at discrete measurement points,” Proc. SPIE4782 (to be published).

Fink, K. D.

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

Irick, S. C.

S. C. Irick, R. Krishna Kaza, and W. R. McKinney, Rev. Sci. Instrum. 66, 2108 (1995).
[CrossRef]

Kaza, R. Krishna

S. C. Irick, R. Krishna Kaza, and W. R. McKinney, Rev. Sci. Instrum. 66, 2108 (1995).
[CrossRef]

Mathews, J. H.

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice-Hall, Englewood Cliffs, N.J., 1999).

McKinney, W. R.

S. C. Irick, R. Krishna Kaza, and W. R. McKinney, Rev. Sci. Instrum. 66, 2108 (1995).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

Weingärtner, I.

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function fx when only dfx/dx is known at discrete measurement points,” Proc. SPIE4782 (to be published).

Yaroslavsky, L.

L. Yaroslavsky, Proc. SPIE 4667, 120 (2002).
[CrossRef]

L. Yaroslavsky and M. Eden, Fundamentals of Digital Optics (Birkhauser, Boston, Mass., 1996).
[CrossRef]

Proc. SPIE (1)

L. Yaroslavsky, Proc. SPIE 4667, 120 (2002).
[CrossRef]

Rev. Sci. Instrum. (1)

S. C. Irick, R. Krishna Kaza, and W. R. McKinney, Rev. Sci. Instrum. 66, 2108 (1995).
[CrossRef]

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function fx when only dfx/dx is known at discrete measurement points,” Proc. SPIE4782 (to be published).

L. Yaroslavsky and M. Eden, Fundamentals of Digital Optics (Birkhauser, Boston, Mass., 1996).
[CrossRef]

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB (Prentice-Hall, Englewood Cliffs, N.J., 1999).

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

Comparison of the frequency responses of the Fourier, trapezoidal, Simpson’s and 3/8 Simpson methods of integration.

Fig. 2
Fig. 2

(a) Slope of the test function. Bold curve, original function; thin curve, with added noise. (b) Original and reconstructed profiles obtained with the four methods. In (a) and (b) the curves essentially overlap (c), (d) [enlarged portions of (b): (c) profiles obtained with the Fourier method and the Simpson rule, (d) profiles obtained with the Fourier method and the 3/8 Simpson rule; the oscillating curves correspond to the Simpson and 3/8 Simpson methods.

Equations (18)

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

f1=0,  fk=fk-1+Δx2fk-1+fk.
fk=fk-2+Δx3fk-2+4fk-1+fk.
fk=fk-3+3Δx8fk-3+3fk-2+3fk-1+fk.
Fr=1Nk=0N-1fk expi2πkrN.
fk=1Nr=0N-1ηrFr exp-i2πkrN,
ηr=0,r=0-Ni2πr,r=1,2,,N/2-1-12π,r=N/2ηN-r*,r=N/2+1,,N-1;
fkT-fk-1T=Δx2fk-1+fk,
fkS-fk-2S=Δx3fk-2+4fk-1+fk,
fk3S-fk-33S=3Δx8fk-3+3fk-2+3fk-1+fk.
FrT1-expi2πr/N=Δx2Fr1+expi2πr/N,
FrS1-expi4πr/N=Δx3Fr1+4 expi2πr/N+expi4πr/N,
Fr3S1-expi6πr/N=3Δx8Fr×1+3 expi2πr/N+3 expi4πr/N+expi6πr/N,
Fr=1Nk=0N-1fk expi2πkrN.
ηrT=-cosπr/N2i sinπr/N,  r=1,,N-1,
ηrS=-cos2πr/N+23i sin2πr/N,  r= 1,,N-1,
ηr3S=-cos3πr/N+3 cosπr/Ni sin3πr/N,  r= 1,,N-1.
Nrout=ηr2Nrinp.
σout2=r=0N-1ηr2σinp2.

Metrics