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AN EFFICIENT AND STABLE ALGORITHM FOR NUMERICAL EVALUATION OF HANKEL TRANSFORMS  

Singh, Om P. (Department of Applied Mathematics, Institute of Technology, Banaras Hindu University)
Singh, Vineet K. (Birla Institute of Technology and Science-Pilani, Goa Campus, PDPM Indian Institute of Information Technology, Design and Manufacturing)
Pandey, Rajesh K. (Birla Institute of Technology and Science-Pilani, Goa Campus, PDPM Indian Institute of Information Technology, Design and Manufacturing)
Publication Information
Journal of applied mathematics & informatics / v.28, no.5_6, 2010 , pp. 1055-1071 More about this Journal
Abstract
Recently, a number of algorithms have been proposed for numerical evaluation of Hankel transforms as these transforms arise naturally in many areas of science and technology. All these algorithms depend on separating the integrand $rf(r)J_{\upsilon}(pr)$ into two components; the slowly varying component rf(r) and the rapidly oscillating component $J_{\upsilon}(pr)$. Then the slowly varying component rf(r) is expanded either into a Fourier Bessel series or various wavelet series using different orthonormal bases like Haar wavelets, rationalized Haar wavelets, linear Legendre multiwavelets, Legendre wavelets and truncating the series at an optimal level; or approximating rf(r) by a quadratic over the subinterval using the Filon quadrature philosophy. The purpose of this communication is to take a different approach and replace rapidly oscillating component $J_{\upsilon}(pr)$ in the integrand by its Bernstein series approximation, thus avoiding the complexity of evaluating integrals involving Bessel functions. This leads to a very simple efficient and stable algorithm for numerical evaluation of Hankel transform.
Keywords
Hankel Transform; Bernstein Polynomial; Random Noise;
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1 I. N. Sneddon, The use of Integral Transforms, McGraw-Hill , 1972.
2 B.W. Suter, Fast nth order Hankel transform algorithm,IEEE Trans. Signal Process 39 (1991), 532-536.   DOI   ScienceOn
3 B.W. Suter and R.A. Hedges, Understanding fast Hankel transform,J. Opt. Soc. Am. A 18 (2001),717-720.   DOI   ScienceOn
4 Li. Yu, M. Huang, M. Chen, W. Huang and Z. Zhu Quasi-discrete Hankel transform, Opt.Lett. 23 (1998), 409-411.   DOI   ScienceOn
5 A.E. Siegman, Quasi Fast Hankel Transform, J. Optics. Lett 1 (1977), 13-15.   DOI
6 O.P. Singh, On pseudo-differential operator ${(-x^{-1}D)}^v$, J. Math. Anal. Appl. 191 (1995), 450-459.   DOI   ScienceOn
7 0.P. Singh and J.N. Pandey, The Fourier-Bessel series representation of the pseudo differential operator ${(-x^{-1}D)}^v$, Proc. Amer. Math. Soc. 115 (1992), 969-976.
8 V.K. Singh,O.P. Singh and R.K. Pandey, Numerical evaluation of Hankel transform by using linear Legendre multi-wavelets,Compu. Phys. Commun. 179 (2008),424-429.   DOI   ScienceOn
9 A.V. Oppenheim, G.V. Frish and D.R. Martinez, Component of Hankel transform using projections ,J. Acoust. Soc. Am. 68 (1980), 523-529.   DOI   ScienceOn
10 V.K. Singh,O.P. Singh and R.K. Pandey, Efficient algorithms to compute Hankel transforms using Wavelets,Compu. Phys. Commun. 179 (2008),812-818.   DOI   ScienceOn
11 D. Patella, Gravity interpretation using the Hankel transform, Geophysical Prospecting 28 (1980), 744-749.   DOI   ScienceOn
12 V. Magni, G. Cerullo and D. Silvestri, High-accuracy fast Hankel transform for optical beam Propagation,J. Opt. Soc. Am. A 12 (1992), 2031-2033.
13 E.B. Postnikov, About calculation of the Hankel transform using preliminary wavelet transform,J. Appl. Math. 6 (2003),319-325.
14 J.J. Reis, R.T. Lynch and J. Butman, Adaptive Harr transform video bandwith reduction stem for RPV's, in Proceeding of the annual meeting on society of Photo Optic Instrumentation Enginering (SPIE),San Diego, CA (1976), 24-45.
15 J.D. Secada, Numerical evaluation of the Hankel transform,Compu. Phys. Commun. 116 (1999),278-294.   DOI   ScienceOn
16 J. Markham and J.A. Conchello, Numerical evaluation of Hankel transform for oscillating Function,J. Opt. Soc. Am. A 20 (2003), 621-630.   DOI   ScienceOn
17 E.V. Hansen, Correction to Fast Hankel transform algorithms,IEEE Trans. Acoust. Speech Signal Process ASSP- 34 (1986), 623-624.   DOI   ScienceOn
18 D.R. Mook, An algorithm for numerical evaluation of Hankel and Abel transform, IEEE Trans. Acoust. Speech Signal Process ASSP- 31 (1983), 979-985.
19 P.K. Murphy and N.C. Gallagher, fast algorithm for computation of zero-order Hankel transform,J. Opt. Soc. Am. 73 (2003), 1130-1137.
20 A.V. Oppenheim, G.V. Frish and D.R. Martinez, An algorithm for numerical evaluation of Hankel transform,IEEE Proc.66 (1980), 264-265.
21 W.E. Higgins and D.C. Munsons Jr., An algorithm for computing general integer order Hankel Transforms,IEEE Trans. Acoust. Speech Signal Process ASSP- 35 (1987), 86-97.
22 J.A. Ferrari, D. Peciante and A. Durba, Fast Hankel transform of nth order,J. Opt. Soc. Am. A 16 (1999),2581-2582.   DOI
23 W.E. Higgins and D.C. Munsons Jr., A Hankel transform approach to tomographic image Reconstruction, IEEE Trans. Med. Imag. 7 (1988), 59-72.   DOI   ScienceOn
24 L. Knockaret, Fast Hankel transform by fast sine and cosine transform: the Mellin connection,IEEE Trans. Signal Process 48 (2000), 1695-1701.   DOI   ScienceOn
25 V.S. Kulkarni and K.C. Deshmukh , An inverse quasi-static steady-state in a thick circular plate, J. Frank. Inst 345 (2008), 29-38.   DOI   ScienceOn
26 E.V. Hansen, Fast Hankel transform algorithms,IEEE Trans. Acoust. Speech Signal Process ASSP- 33 (1985), 666-671.
27 J. Gaskell, Linear systems, Fourier transforms, and Optics, chapter 11,Wiely, New York (1978).
28 J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York , 1968
29 M. Guizar-Sicairos and J.C. Gutierrez-Vega, Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields ,J. Opt. Soc. Am. A 21 (2004),53.   DOI   ScienceOn
30 E.C. Cavanagh and B.D.Cook , Numerical evaluation of Hankel Transform via Gaussian-Laguerre polynomial expressions, IEEE Trans. Acoust. Speech Signal Process ASSP- 27 (1979), 361-366.
31 A. Erdelyi(Ed.), Tables of Integral Transforms,McGraw-Hill, New York (1954).
32 H.Y. Fan, Hankel transform as a transform between two entangled state representations, Phys. Lett. A 313 (2003), 343-350.   DOI   ScienceOn
33 J.A. Ferrari, Fast Hankel transform of order zero, J. Opt. Soc. Am. A 12 (1995), 1812-1813.   DOI   ScienceOn
34 R. Barakat, E. Parshall and B.H. Sandler, Zero-order Hankel transform algorithms based on Filon quadrature philosophy for diffraction optics and beam propogation,J. Opt. Soc. Am. A 15 (1998), 652-659.
35 R. Barakat and B.H. Sandler, Evaluation of first-order Hankel transforms using Filon quadrature Philosophy, Appl. Math. Lett. 11 (1998), 127-131.
36 S.M. Candel, Dual algorithms for fast calculation of the Fourier Bessel transform, IEEE Trans. Acoust. Speech Signal Process ASSP- 29 (1981), 963-972.
37 A. Agnesi, G.C. Reali, G. Patrini and A. Tomaselli, Numerical evaluation of the Hankel transform: remarks,J. Opt. Soc. Am. A 10 (1993),1872-1874.   DOI   ScienceOn
38 R. Barakat and E. Parshall, Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy, Appl. Math. Lett. 9 (1996), 21-26.