Hankel transform

Not to be confused with the determinant of the Hankel matrix of a sequence.

In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind $J ν (kr)$ . The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r-axis. The necessary coefficient $F ν$ of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.

Definition

The Hankel transform of order ν of a function f(r) is given by:

F_{\nu }(k)=\int _{0}^{\infty }f(r)J_{\nu }(kr)\,r\operatorname {d}\!r

where $J_{\nu }$ is the Bessel function of the first kind of order $\nu$ with $\nu \geq -{\frac {1}{2}}$ . The inverse Hankel transform of $F ν (k)$ is defined as:

f(r)=\int _{0}^{\infty }F_{\nu }(k)J_{\nu }(kr)k\operatorname {d}\!k

which can be readily verified using the orthogonality relationship described below.

Domain of definition

Inverting a Hankel transform of a function f(r) is valid at every point at which f(r) is continuous provided that the function is defined in (0, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and

\int _{0}^{\infty }|f(r)|\,r^{{{\frac {1}{2}}}}\operatorname {d}\!r<\infty .

However, like the Fourier Transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example $f(r)=(1+r)^{{-3/2}}$ .

Alternative definition

An alternative definition says that the Hankel transform of g(r) is:^[1]

h_{\nu }(k)=\int _{0}^{\infty }g(r)J_{\nu }(kr)\,{\sqrt {kr}}\operatorname {d}\!r

The two definitions are related:

{\text{If }}g(r)=f(r){\sqrt r}{\text{ then }}h_{\nu }(k)=F_{\nu }(k){\sqrt k}.

This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse:

g(r)=\int _{0}^{\infty }h_{\nu }(k)J_{\nu }(kr)\,{\sqrt {kr}}\operatorname {d}\!k

The obvious domain now has the condition

\int _{0}^{\infty }|g(r)|\operatorname {d}\!r<\infty

but this can be extended. According to the reference given above, we can take the integral as the limit as the upper limit goes to infinity (an improper integral rather than a Lebesgue integral) and in this way the Hankel transform and its inverse work for all functions in L²(0, ∞).

Orthogonality

The Bessel functions form an orthogonal basis with respect to the weighting factor r:

\int _{0}^{\infty }J_{\nu }(kr)J_{\nu }(k'r)r\operatorname {d}\!r={\frac {\delta (k-k')}{k}},\qquad k,k'>0.

The Plancherel theorem and Parseval's theorem

If f(r) and g(r) are such that their Hankel transforms $F ν (k)$ and $G ν (k)$ are well defined, then the Plancherel theorem states

\int _{0}^{\infty }f(r)g(r)r\operatorname {d}\!r=\int _{0}^{\infty }F_{\nu }(k)G_{\nu }(k)k\operatorname {d}\!k.

Parseval's theorem, which states:

\int _{0}^{\infty }|f(r)|^{2}r\operatorname {d}\!r=\int _{0}^{\infty }|F_{\nu }(k)|^{2}k\operatorname {d}\!k,

is a special case of the Plancherel theorem. These theorems can be proven using the orthogonality property.

Relation to other function transforms

Relation to the Fourier transform (circularly symmetric case)

The Hankel transform of order zero is essentially the $2$ -dimensional Fourier transform of a circularly symmetric function.

Consider a $2$ -dimensional function $f (r)$ of the radius vector $r$ . Its Fourier transform is:

F({\mathbf {k}})=\iint f({\mathbf {r}})e^{{i{\mathbf {k}}\cdot {\mathbf {r}}}}\operatorname {d}\!{\mathbf {r}}.

With no loss of generality, we can pick a polar coordinate system $(r, θ)$ such that the $k$ vector lies on the $θ = 0$ axis(in K-space). The Fourier transform is now written in these polar coordinates as:

F({\mathbf {k}})=\int _{{r=0}}^{\infty }\int _{{\theta =0}}^{{2\pi }}f(r,\theta )e^{{ikr\cos(\theta )}}\,r\operatorname {d}\!\theta \operatorname {d}\!r

where $θ$ is the angle between the $k$ and $r$ vectors. If the function $f$ happens to be circularly symmetric, it will have no dependence on the angular variable $θ$ and may be written $f (r)$ . The integration over $θ$ may be carried out, and the Fourier transform is now written:

F({\mathbf {k}})=F(k)=2\pi \int _{0}^{\infty }f(r)J_{0}(kr)r\operatorname {d}\!r

which is just $2 π$ times the zero-order Hankel transform of $f (r)$ . For the reverse transform,

f({\mathbf {r}})={\frac {1}{(2\pi )^{2}}}\iint F({\mathbf {k}})e^{{-i{\mathbf {k}}\cdot {\mathbf {r}}}}\operatorname {d}\!{\mathbf {k}}={\frac {1}{2\pi }}\int _{0}^{\infty }F(k)J_{0}(kr)k\operatorname {d}\!k

so $f (r)$ is $1 / 2 π$ times the zero-order Hankel transform of $F (k)$ .

Relation to the Fourier transform (radially symmetric case in $n$ -dimensions)

For an $n$ -dimensional Fourier transform,

F({\mathbf k})=\int f({\mathbf r})e^{{-i{\mathbf k}\cdot {\mathbf r}}}d^{n}{\mathbf r}

if the function $f$ is radially symmetric, then:^[2]

k^{{n/2-1}}F(k)=(2\pi )^{{n/2}}\int _{0}^{\infty }r^{{n/2-1}}f(r)J_{{n/2-1}}(kr)r\,dr

Relation to the Fourier transform (general case)

To generalize: If $f$ can be expanded in a multipole series,

f(r,\theta )=\sum _{{m=-\infty }}^{\infty }f_{m}(r)e^{{im\theta }},

and if $θ k$ is the angle between the direction of $k$ and the $θ = 0$ axis,

{\begin{aligned}F({\mathbf k})&=\int _{0}^{\infty }r\operatorname {d}\!r\,\int _{0}^{{2\pi }}\operatorname {d}\!\theta \,f(r,\theta )e^{{ikr\cos(\theta -\theta _{k})}}\\&=\sum _{m}\int _{0}^{\infty }r\operatorname {d}\!r\,\int _{0}^{{2\pi }}\operatorname {d}\!\theta \,f_{m}(r)e^{{im\theta }}e^{{ikr\cos(\theta -\theta _{k})}}\\&=\sum _{m}e^{{im\theta _{k}}}\int _{0}^{\infty }r\operatorname {d}\!r\,f_{m}(r)\int _{0}^{{2\pi }}\operatorname {d}\!\varphi \,e^{{im\varphi }}e^{{ikr\cos \varphi }}&&\varphi =\theta -\theta _{k}\\&=\sum _{m}e^{{im\theta _{k}}}\int _{0}^{\infty }r\operatorname {d}\!r\,f_{m}(r)2\pi i^{m}J_{m}(kr)\\&=2\pi \sum _{m}i^{m}e^{{im\theta _{k}}}\int _{0}^{\infty }f_{m}(r)J_{m}(kr)r\operatorname {d}\!r.\\&=2\pi \sum _{m}i^{m}e^{{im\theta _{k}}}F_{m}(k)\end{aligned}}

where $F m (k)$ is the $m$ -th order Hankel transform of $f m (r)$ .

Functions inside a limited radius

Additionally, if $f m$ is sufficiently smooth near the origin and zero outside a radius $R$ , it may be expanded into a Chebyshev series,

f_{m}(r)=r^{m}\sum _{{t\geq 0}}f_{{mt}}\left(1-\left({\tfrac {r}{R}}\right)^{2}\right)^{t},\qquad 0\leq r\leq R.

so that

{\begin{aligned}F({\mathbf k})&=2\pi \sum _{m}i^{m}e^{{im\theta _{k}}}\sum _{t}f_{{mt}}\int _{0}^{R}r^{m}\left(1-\left({\tfrac {r}{R}}\right)^{2}\right)^{t}J_{m}(kr)r\operatorname {d}\!r&&(*)\\&=2\pi \sum _{m}i^{m}e^{{im\theta _{k}}}R^{{m+2}}\sum _{t}f_{{mt}}\int _{0}^{1}x^{m}(1-x^{2})^{t}J_{m}(kxR)x\operatorname {d}\!x&&x={\tfrac {r}{R}}\\&=2\pi \sum _{m}i^{m}e^{{im\theta _{k}}}R^{{m+2}}\sum _{t}f_{{mt}}{\frac {t!2^{t}}{(kR)^{{1+t}}}}J_{{m+t+1}}(kR).\end{aligned}}

The above can be viewed as a more general case that is not as constrained as the previous case in the previous section. The numerically important aspect is that the expansion coefficients $f mt$ are accessible with Discrete Fourier transform techniques. Insertion into the previous formula yields

This is one flavor of fast Hankel transform techniques.

Relation to the Fourier and Abel transforms

The Hankel transform is one member of the FHA cycle of integral operators. In two dimensions, if we define $A$ as the Abel transform operator, $F$ as the Fourier transform operator and $H$ as the zeroth order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that:

FA=H.

In other words, applying the Abel transform to a $1$ -dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.

Some Hankel transform pairs

^[3]

$f(r)\,$	$F_{0}(k)\,$
$1\,$	${\tfrac {\delta (k)}{k}}$
${\tfrac {1}{r}}$	${\tfrac {1}{k}}$
$r\,$	$-{\tfrac {1}{k^{3}}}$
$r^{3}\,$	${\tfrac {9}{k^{5}}}$
$r^{{m}}\,$	${\frac {2^{{m+1}}\Gamma \left({\tfrac {m}{2}}+1\right)}{k^{{m+2}}\Gamma \left(-{\tfrac {m}{2}}\right)}},\qquad -2<\Re (m)<-{\tfrac {1}{2}}$
${\frac {1}{{\sqrt {r^{2}+z^{2}}}}}\,$	${\frac {e^{{-k\|z\|}}}{k}}$
$\frac{1}{z^2+r^2}$	$K_{0}(kz)\qquad z\in {\mathbf {C}}$
${\tfrac {e^{{iar}}}{r}}$	${\frac {i}{{\sqrt {a^{2}-k^{2}}}}}\qquad a>0,k<a.$
${\tfrac {e^{{iar}}}{r}}$	${\frac {1}{{\sqrt {k^{2}-a^{2}}}}}\qquad a>0,k>a.$
$e^{{-{\frac {1}{2}}a^{2}r^{2}}}$	${\frac {1}{a^{2}}}e^{{-{\tfrac {k^{2}}{2a^{2}}}}}$
${\frac {1}{r}}J_{0}(lr)e^{{-sr}}\,$	${\frac {2}{\pi {\sqrt {(k+l)^{2}+s^{2}}}}}K{\bigg (}{\sqrt {{\frac {4kl}{(k+l)^{2}+s^{2}}}}}{\bigg )}$
$-r^{2}f(r)\,$	${\frac {\operatorname {d}^{2}\!F_{0}}{\operatorname {d}\!k^{2}}}+{\frac {1}{k}}{\frac {\operatorname {d}\!F_{0}}{\operatorname {d}\!k}}$

$f(r)\,$	$F_{{\nu }}(k)\,$
$r^{s}\,$	${\frac {2^{{s+1}}}{k^{{s+2}}}}{\frac {\Gamma \left({\tfrac {1}{2}}(2+\nu +s)\right)}{\Gamma ({\tfrac {1}{2}}(\nu -s))}}$
$r^{{\nu -2s}}\Gamma \left(s,r^{2}h\right)\,$	${\tfrac {1}{2}}\left({\tfrac k2}\right)^{{2s-\nu -2}}\gamma \left(1-s+\nu ,{\tfrac {k^{2}}{4h}}\right)\,$
$e^{{-r^{2}}}r^{\nu }U\left(a,b,r^{2}\right)\,$	${\frac {\Gamma (2+\nu -b)}{2\Gamma (2+\nu -b+a)}}\left({\tfrac k2}\right)^{\nu }e^{{-{\frac {k^{2}}{4}}}}\,_{1}F_{1}\left(a,2+a-b+\nu ,{\tfrac {k^{2}}4}\right)$
$r^{n}J_{\mu }(lr)e^{{-sr}}\,$	Expressable in terms of elliptic integrals.^[4]
$-r^{2}f(r)\,$	${\frac {\operatorname {d}^{2}\!F_{\nu }}{\operatorname {d}\!k^{2}}}+{\frac {1}{k}}{\frac {\operatorname {d}\!F_{\nu }}{\operatorname {d}\!k}}-{\frac {\nu ^{2}}{k^{2}}}F_{\nu }$

$K n (z)$ is a modified Bessel function of the second kind. $K (z)$ is the complete elliptic integral of the first kind.

The expression

{\frac {\operatorname {d}^{2}\!F_{0}}{\operatorname {d}\!k^{2}}}+{\frac {1}{k}}{\frac {\operatorname {d}\!F_{0}}{\operatorname {d}\!k}}

coincides with the expression for the Laplace operator in polar coordinates $(k, θ)$ applied to a spherically symmetric function $F 0 (k)$ .

The Hankel transform of Zernike polynomials are essentially Bessel Functions (Noll 1976):

R_{n}^{m}(r)=(-1)^{{{\frac {n-m}{2}}}}\int _{0}^{\infty }J_{{n+1}}(k)J_{m}(kr)\operatorname {d}\!k

for even $n - m \geq 0$ .

Generalized Hankel transform for fan-beam geometry

The image reconstruction in polar coordinates with parallel beam CT projections can be done with Hankel transform. This theory has been generalized to fan-beam geometry. Generalized Bessel function is utilized. ^[5]

References

insert

↑ Louis de Branges (1968). Hilbert spaces of entire functions. London: Prentice-Hall. p. 189. ISBN 978-0133889000.
↑ Faris, William G. (2008-12-06). "Radial functions and the Fourier transform: Notes for Math 583A, Fall 2008" (PDF). University of Arizona, Department of Mathematics. Retrieved 2015-04-25.
↑ Papoulis, Athanasios (1981). Systems and Transforms with Applications to Optics. Florida USA: Krieger Publishing Company. pp. 140–175. ISBN 0898743583.
↑ Kausel, E.; Irfan Baig, M. M. (2012). "Laplace transform of products of Bessel functions: A visitation of earlier formulas". Quarterly of Applied Mathematics. 70: 77. doi:10.1090/s0033-569x-2011-01239-2. open access.
↑ Zhao S.R., H. Halling, Image Reconstruction for Fan Beam Tomography Using a New Integral Transform Pair, published on: International Symposium on Computerized Tomography in Novosibirsk, Russia, August 10–14, 1993. Abstracts ed. M.M. Lavrentev, P125

Gaskill, Jack D. (1978). Linear Systems, Fourier Transforms, and Optics. New York: John Wiley & Sons. ISBN 0-471-29288-5.
Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 0-8493-2876-4.
Smythe, William R. (1968). Static and Dynamic Electricity (3rd ed.). New York: McGraw-Hill. pp. 179–223.
Offord, A. C. (1935). "On Hankel transforms". Proceedings of the London Mathematical Society. 39 (2): 49–67. doi:10.1112/plms/s2-39.1.49.
Eason, G.; Noble, B.; Sneddon, I. N. (1955). "On certain integrals of Lipschitz-Hankel type involving products of Bessel Functions". Philosophical Transactions of the Royal Society A. 247 (935): 529–551. doi:10.1098/rsta.1955.0005. JSTOR 91565.
Kilpatrick, J. E.; Katsura, Shigetoshi; Inoue, Yuji (1967). "Calculation of integrals of products of Bessel functions". Mathematics of Computation. 21 (99): 407–412. doi:10.1090/S0025-5718-67-99149-1.
MacKinnon, Robert F. (1972). "The asymptotic expansions of Hankel transforms and related integrals". Mathematics of Computation. 26 (118): 515–527. doi:10.1090/S0025-5718-1972-0308695-9. JSTOR 2003243.
Linz, Peter; Kropp, T. E. (1973). "A note on the computation of integrals involving products of trigonometric and Bessel functions". Mathematics of Computation. 27 (124): 871–872. doi:10.2307/2005522. JSTOR 2005522.
Noll, Robert J (1976). "Zernike polynomials and atmospheric turbulence". Journal of the Optical Society of America. 66 (3): 207–211. Bibcode:1976JOSA...66..207N. doi:10.1364/JOSA.66.000207.
Siegman, A. E. (1977). "Quasi-fast Hankel transform". Opt. Lett. 1 (1): 13–15. Bibcode:1977OptL....1...13S. doi:10.1364/OL.1.000013.
Magni, Vittorio; Cerullo, Giulio; De Silverstri, Sandro (1992). "High-accuracy fast Hankel transform for optical beam propagation". J. Opt. Soc. Am. A. 9 (11): 2031–2033. Bibcode:1992JOSAA...9.2031M. doi:10.1364/JOSAA.9.002031.
Agnesi, A.; Reali, Giancarlo C.; Patrini, G.; Tomaselli, A. (1993). "Numerical evaluation of the Hankel transform: remarks". Journal of the Optical Society of America A. 10 (9): 1872. Bibcode:1993JOSAA..10.1872A. doi:10.1364/JOSAA.10.001872.
Barakat, Richard (1996). "Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy". Applied Mathematics Letters. 9 (5): 21–26. doi:10.1016/0893-9659(96)00067-5. MR 1415467.
Ferrari, José A.; Perciante, Daniel; Dubra, Alfredo (1999). "Fast Hankel transform of nth order". J. Opt. Soc. Am. A. 16 (10): 2581–2582. Bibcode:1999JOSAA..16.2581F. doi:10.1364/JOSAA.16.002581.
Secada, José D. (1999). "Numerical evaluation of the Hankel transform". Comp. Phys. Comm. 116 (2–3): 278–294. Bibcode:1999CoPhC.116..278S. doi:10.1016/S0010-4655(98)00108-8.
Wieder, Thomas (1999). "Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL". ACM Trans. Math. Softw. 25 (2): 240–250. doi:10.1145/317275.317284.
Knockaert, Luc (2000). "Fast Hankel transform by fast sine and cosine transforms: the Mellin connection" (PDF). IEEE Trans. Signal Process. 48 (6): 1695–1701. doi:10.1109/78.845927.
Zhang, D. W.; Yuan, X.-C.; Ngo, N. Q.; Shum, P. (2002). "Fast Hankel transform and its application for studying the propagation of cylindrical electromagnetic fields". Opt. Exp. 10 (12): 521–525. doi:10.1364/oe.10.000521.
Markham, Joanne; Conchello, Jose-Angel (2003). "Numerical evaluation of Hankel transforms for oscillating functions". J. Opt. Soc. Am. A. 20 (4): 621–630. Bibcode:2003JOSAA..20..621M. doi:10.1364/JOSAA.20.000621.
Perciante, César D.; Ferrari, José A. (2004). "Fast Hankel transform of nth order with improved performance". J. Opt. Soc. Am. A. 21 (9): 1811. Bibcode:2004JOSAA..21.1811P. doi:10.1364/JOSAA.21.001811.
Gizar-Sicairos, Manuel; Guitierrez-Vega, Julio C. (2004). "Computation of quasi-discrete Hankel transform of integer order for propagating optical wave fields". J. Opt. Soc. Am. A. 21 (1): 53–58. Bibcode:2004JOSAA..21...53G. doi:10.1364/JOSAA.21.000053.
Cerjan, Charles (2007). "The Zernike-Bessel representation and its application to Hankel transforms". J. Opt. Soc. Am. A. 24 (6): 1609–1616. Bibcode:2007JOSAA..24.1609C. doi:10.1364/JOSAA.24.001609.

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