Circularly Symmetric Functions: Hankel Transforms of Zeroth Order, or, Fourier-Bessel Transforms

1
Circularly Symmetric Functions Hankel
Transforms of Zeroth Order, or, Fourier-Bessel
Transforms
What if our function is expressed in polar
coordinates? and is separable in those
coordinates?
Why is this of interest?
Useful for modeling circular sources, lesions,
lenses, etc.
2
Transforming coordinates,
u
y
?
r
?
v
?
x
rd?
rd?
d?
dr
3
Hankel Transform, continued
transforming variables...
where
(Using Trig Identity)
So,
But, this can be simplified by using
4
Bessel functions
Jn(x) is a Bessel function of the first kind of
order n.
Useful identities,
and
where J1(x) is a Bessel function of the first
kind of order 1.
5
Hankel Transform, continued (2)
(from previous slide)
But,
Subbing in Jo(2?r?) yields
which is not a function of ?.
Thus, the function is circularly symmetric in
both domains.
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The Inverse Hankel Transform
Circularly symmetric in space
Circularly symmetric in spatial frequency
Notice no difference in sign between forward and
inverse transforms
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Example Hankel transform of a circle
Consider the Fourier Transform of a circle

• (circular symmetry)

assume
y
Consider g r(r) circ(r)
r 1
x
Let
Then,
and
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Hankel Transform of Circle, continued
Subbing in r2?r? yields
Note
So
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Jinc Function
We define the jinc function as
Similar to sinc function, only sinc has zeros are
at equal intervals, jinc zeros vary
10
Hankel Transform of Circ
circ(r)
F.T fft(circ(r)) jinc(r)
Log10(abs(fft(circ(r)))
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Hankel Transform Scaling Property
The only difference between the Fourier and
Hankel transform scaling property is the scalar
1/a2 The scalar takes into account that the
function is expanding or contracting in 2
dimensions.