Abstract

Recently Ferrari et al. [J. Opt. Soc. Am. A 16, 2581 (1999)] presented an algorithm for the numerical evaluation of the Hankel transform of nth order. We demonstrate that this formulation can be interpreted as an application of the projection slice theorem.

© 2001 Optical Society of America

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References

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  1. J. A. Ferrari, D. Perciante, A. Dubra, “Fast Hankel transform of nth order,” J. Opt. Soc. Am. A 16, 2581–2582 (1999).
    [CrossRef]
  2. E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
    [CrossRef]
  3. E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
    [CrossRef]
  4. D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
    [CrossRef]
  5. A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
    [CrossRef]
  6. A. V. Oppenheim, G. V. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
    [CrossRef]
  7. B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
    [CrossRef]
  8. I. N. Sneddon, The Use of Integral Transform (McGraw-Hill, New York, 1972).
  9. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, UK, 1966).
  10. A. M. Cormack, “Representation of a function by its line integrals with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
    [CrossRef]
  11. A. M. Cormack, “Representation of a function by its line integrals with some radiological Applications II,” J. Appl. Phys. 35, 2908–2913 (1964).
    [CrossRef]
  12. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1968).
  13. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University, Cambridge, UK, 1962).
  14. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1978).
  15. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

1999 (1)

1991 (1)

B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

1986 (1)

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

1985 (1)

E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

1983 (1)

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

1980 (1)

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

1978 (1)

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

1964 (1)

A. M. Cormack, “Representation of a function by its line integrals with some radiological Applications II,” J. Appl. Phys. 35, 2908–2913 (1964).
[CrossRef]

1963 (1)

A. M. Cormack, “Representation of a function by its line integrals with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1968).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1978).

Cormack, A. M.

A. M. Cormack, “Representation of a function by its line integrals with some radiological Applications II,” J. Appl. Phys. 35, 2908–2913 (1964).
[CrossRef]

A. M. Cormack, “Representation of a function by its line integrals with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

Dubra, A.

Ferrari, J. A.

Frish, G. V.

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Hansen, E. W.

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

Lighthill, M. J.

M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University, Cambridge, UK, 1962).

Martinez, D. R.

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Mook, D. R.

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Perciante, D.

Sneddon, I. N.

I. N. Sneddon, The Use of Integral Transform (McGraw-Hill, New York, 1972).

Suter, B. W.

B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, UK, 1966).

IEEE Trans. Acoust. Speech Signal Process. (3)

E. W. Hansen, “Fast Hankel transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 666–671 (1985).
[CrossRef]

E. W. Hansen, “Correction to ‘Fast Hankel transform algorithms’,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 623–624 (1986).
[CrossRef]

D. R. Mook, “An algorithm for the numerical calculation of Hankel and Abel transforms,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 979–985 (1983).
[CrossRef]

IEEE Trans. Signal Process. (1)

B. W. Suter, “Fast nth order Hankel transform algorithm,” IEEE Trans. Signal Process. 39, 532–536 (1991).
[CrossRef]

J. Acoust. Soc. Am. (1)

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “Computation of the Hankel transform using projections,” J. Acoust. Soc. Am. 68, 523–529 (1980).
[CrossRef]

J. Appl. Phys. (2)

A. M. Cormack, “Representation of a function by its line integrals with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

A. M. Cormack, “Representation of a function by its line integrals with some radiological Applications II,” J. Appl. Phys. 35, 2908–2913 (1964).
[CrossRef]

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

A. V. Oppenheim, G. V. Frish, D. R. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978).
[CrossRef]

Other (6)

I. N. Sneddon, The Use of Integral Transform (McGraw-Hill, New York, 1972).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, UK, 1966).

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1968).

M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University, Cambridge, UK, 1962).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1978).

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

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Equations (62)

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f˜n(ρ)=2π0rf (r)Jn(2πρτ)dr,
fˇ(x)=Tnf(r)=2|x| f(r) Tn(x/r)[1-(x/r)2]1/2dr,
Φn(x)=2(-i)n0f((x2+y2)1/2)cos(nθ[x, y])dy,
f˜n(ρ)=F1Φn(x)=-Φn(x)exp(i2πρx)dx,
Φn(x)=(-i)nTn f(r).
Tn f(r)=2|x| f(r) Tn(x/r)[1-(x/r)2]1/2dr.
Tn f(r)=20f((y2+x2)1/2)Tn(x/(y2+x2)1/2)dy.
Tnf(r)=20f((y2+x2)1/2)×cos{n cos-1[x/(y2+x2)1/2]}dy.
(-i)nTn f(r)
=2(-i)n0f((x2+y2)1/2)cos(nθ [x, y])dy,
=Φ(x).
F2 f(r)cos(nθ)=002πf(r)cos(nθ)×exp[i2πρr cos(θ-ϕ)]rdθdr.
F2 f(r)cos(nθ)=0rf(r)02πcos(nβ+nϕ)×exp(i2πρr cos β)rdβdr.
F2 f(r)cos(nθ)=cos(nϕ)0rf(r)02πcos(nβ)
×exp(i2πρr cos β)dβdr
-sin(nϕ)0rf(r)02πsin(nβ)
×exp(i2πρr cos β)dβdr.
F2 f (r)cos(nθ)=cos(nϕ)0rf(r)02πcos(nβ)×exp(i2πρr cos β)dβdr.
Jn(x)=(-i)n2π02πexp[i(nβ+x cos β)]dβ,
Jn(x)=(-i)n2π02πcos(nβ)exp(ix cos β)dβ+i (-i)n2π02πsin(nβ)exp(ix cos β)dβ.
Jn(x)=(-i)n2π02πcos(nβ)exp(ix cos β)dβ.
02πcos(nβ)exp(i2πρr cos β)dβ=in2πJn(2πρr).
F2 f(r)cos(nθ)=cos(nϕ)in2π0rf(r)Jn(2πρr)dr,
F2 f(r)cos(nθ)=cos(nϕ)inHn f(r).
R f(r)cos(nθ)
=02π0f(r)cos(nθ)δ(ρ-r cos(θ-ϕ))rdrdθ,
Rf(r)cos(nθ)
=0rf(r)02πcos(nβ+nϕ)δ(ρ-r cos β)dβdr.
 
δ ( f(x))=nδ(x-xn)|f(xn)|
02πcos(nβ+nϕ)δ(ρ-r cos β)dβ
=cos[n cos-1(ρ/r)+nϕ]+cos[-n cos-1(ρ/r)+nϕ]r[1-(ρ/r)2]1/2
=2 cos[n cos-1(ρ/r)]cos(nϕ)r[1-(ρ/r)2]1/2,rρ.
0 2πcos(nβ+nϕ)δ(ρ-r cos β)dβ=2Tn(ρ/r)cos(nϕ)r[1-(ρ/r)2]1/2,
rρ.
Rf(r)cos(nθ)=cos(nϕ)2|ρ|f(r)Tn(ρ/r)r[1-(ρ/r)2]1/2dr.
Rf(r)cos(nθ)=cos(nϕ)Tn f(r).
F1Rf(r)cos(nθ)=cos(nϕ)F1Tn f(r).
F2 f(r)cos(nθ)=cos(nϕ)F1R f(r).
Hn f(r)=(-in)F1Tn f(r)=F1Φn(x).
f˜(x)=F1A f(r),
Tn f(r)=2|x|R f(r) Tn(x/r)[1-(x/r)2]1/2dr,
(-i)nTn f(r)=2(-i)n0(R2-x2)1/2f((x2+y2)1/2)×cos(nθ [x, y])dy=Φn(x).
02πexp(ix cos ϕ)exp(-ikϕ)dϕ
=-ππexp(ix cos ϕ)cos(kϕ)dϕ.
02πexp(ix cos ϕ)exp(-ikϕ)dϕ
=02πexp(ix cos ϕ)[cos(-kϕ)+i sin(-kϕ)]dϕ.
02πexp(ix cos ϕ)exp(-ikϕ)dϕ
=-ππexp(ix cos ϕ)[cos(kϕ)-i sin(kϕ)]dϕ.
-ππexp(ix cos ϕ)sin(kϕ)dϕ=0,
-ππexp(ix cos ϕ)sin(kϕ)dϕ
=12i-ππexp(ix cos ϕ)[exp(ikϕ)-exp(-ikϕ)]dϕ
=12i-ππexp(ix cos ϕ+kϕ)dϕ--ππexp(ix cos ϕ-kϕ)dϕ.
-ππexp(ix cos ϕ-kϕ)dϕ
=-ππexp i[x cos(-ϕ)-kϕ)]dϕ
=-π-πexp i[x cos(ϕ)+kϕ)]dϕ
(replaceϕby-ϕ)
=-ππexp i[x cos(ϕ)+kϕ)]dϕ
(reversethelimitsofintegration).
-ππexp(ix cos ϕ)sin(kϕ)dϕ
=12i-ππexp i[x cos(ϕ)+(kϕ)]dϕ
--ππexp i[x cos(ϕ)+(kϕ)]dϕ=0.

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