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Numerical evaluation of Hankel transforms for oscillating functions.

Joanne Markham1, José-Angel Conchello

  • 1Mallinckrodt Institute of Radiology, Washington University School of Medicine, St. Louis, Missouri 63110, USA. joanne@npg.wustl.edu

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|April 10, 2003
PubMed
Summary

Six numerical methods for Hankel transforms were evaluated. A new projection-slice method offers an excellent balance of accuracy and computational efficiency for oscillating functions.

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Area of Science:

  • Numerical analysis
  • Signal processing
  • Scientific computing

Background:

  • Accurate numerical calculation of Hankel transforms is crucial for various scientific applications.
  • Existing methods may present limitations in terms of accuracy or computational cost for oscillating functions.

Purpose of the Study:

  • To evaluate and compare six distinct numerical methods for calculating zero-order Hankel transforms of oscillating functions.
  • To identify methods that offer an optimal balance between accuracy and computational efficiency.

Main Methods:

  • Implementation of one Filon quadrature method.
  • Implementation of two published projection-slice methods.
  • Development and implementation of a novel projection-slice method for Abel transform computation.

Related Experiment Videos

  • Testing algorithms with an oscillating sweep signal and a 3D point-spread function.

    • Main Results:

    • The Filon quadrature method demonstrated high accuracy but was computationally intensive.
    • The implemented projection-slice methods showed varying degrees of accuracy and efficiency.
    • The newly derived projection-slice method provided a superior compromise between accuracy and computational efficiency.

    Conclusions:

    • The novel projection-slice method is recommended for efficient and accurate numerical calculation of zero-order Hankel transforms for oscillating functions.
    • Projection-slice methods, particularly the new variant, present a practical alternative to traditional methods like Filon quadrature.