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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 24, Iss. 6 — Jun. 1, 2007
  • pp: 1609–1616

Zernike–Bessel representation and its application to Hankel transforms

Charles Cerjan  »View Author Affiliations

JOSA A, Vol. 24, Issue 6, pp. 1609-1616 (2007)

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The duality between the well-known Zernike polynomial basis set and the Fourier–Bessel expansion of suitable functions on the radial unit interval is exploited to calculate Hankel transforms. In particular, the Hankel transform of simple truncated radial functions is observed to be exact, whereas more complicated functions may be evaluated with high numerical accuracy. The formulation also provides some general insight into the limitations of the Fourier–Bessel representation, especially for infinite-range Hankel transform pairs.

© 2007 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.1960) Diffraction and gratings : Diffraction theory
(070.2590) Fourier optics and signal processing : ABCD transforms
(100.3020) Image processing : Image reconstruction-restoration
(100.6890) Image processing : Three-dimensional image processing

ToC Category:
Image Processing

Original Manuscript: October 27, 2006
Revised Manuscript: January 3, 2007
Manuscript Accepted: January 5, 2007
Published: May 9, 2007

Virtual Issues
Vol. 2, Iss. 7 Virtual Journal for Biomedical Optics

Charles Cerjan, "Zernike-Bessel representation and its application to Hankel transforms," J. Opt. Soc. Am. A 24, 1609-1616 (2007)

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  1. G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999), pp. 210-216.
  2. J. Markham and J.-A. Conchello, "Numerical evaluation of Hankel transforms for oscillating functions," J. Opt. Soc. Am. A 20, 621-630 (2003). [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), Section 9.2 and Appendix VII.
  4. F. Lado, "Equation of state of the hard-disk fluid from approximate integral equations," J. Chem. Phys. 49, 3092-3096 (1968). [CrossRef]
  5. H. Fisk Johnson, "An improved method for computing the discrete Hankel transform," Comput. Phys. Commun. 43, 181-202 (1987). [CrossRef]
  6. D. Lemoine, "The discrete Bessel transform algorithm," J. Chem. Phys. 101, 3936-3984 (1994). [CrossRef]
  7. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, "Quasi-discrete Hankel transform," Opt. Lett. 23, 409-411 (1998). [CrossRef]
  8. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004). [CrossRef]
  9. A. Prata, Jr., and W. V. T. Rusch, "Algorithm for computation of Zernike polynomials expansion coefficients," Appl. Opt. 28, 749-754 (1989). [PubMed]
  10. A. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (U. S. Government Printing Office, 1972), Chap. 22.
  11. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas andTheorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 128.
  12. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966), Chap. XVIII.
  13. The computer package, MATHEMATICA (Wolfram Research, Inc.), conveniently includes an algorithm for the roots of the derivatives of the Bessel functions.
  14. F. S. Gibson and F. Lanni, "Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy," J. Opt. Soc. Am. A 8, 1601-1613 (1991).
  15. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966), p. 216.
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, 1980), pp. 1037, 8.965.

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