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Radon Transform--Cylinder


RadonCylinder

Let the two-dimensional cylinder function be defined by

 f(x,y)={1   for r<R; 0   for r>R.
(1)

Then the Radon transform is given by

 R(p,tau)=int_(-infty)^inftyint_(-infty)^inftyf(x,y)delta[y-(tau+px)]dydx,
(2)

where

 delta(x)=1/(2pi)int_(-infty)^inftye^(-ikx)dk
(3)

is the delta function. Rewriting in polar coordinates then gives

 R(p,tau)=1/(2pi)int_(-infty)^inftyint_0^(2pi)int_0^Re^(-ik(rsintheta-prcostheta))rdrdthetadk 
 =1/(2pi)int_(-infty)^inftye^(iktau)int_0^(2pi)int_0^Re^(-ikr(sintheta-pcostheta))rdrdthetadk.
(4)

Now use the harmonic addition theorem to write

 sintheta-pcostheta=sqrt(1+p^2)cos(theta+phi)=sqrt(1+p^2)costheta^',
(5)

with phi a phase shift. Then

R(p,tau)=1/(2pi)int_(-infty)^inftye^(iktau)int_0^R(int_0^(2pi)e^(-iksqrt(1+p^2)rcostheta^')dtheta^')rdrdk
(6)
=1/(2pi)int_(-infty)^inftye^(iktau)int_0^R2piJ_0(ksqrt(1+p^2)r)rdrdk
(7)
=int_(-infty)^inftye^(iktau)int_0^RJ_0(ksqrt(1+p^2)r)rdrdk.
(8)

Then use

 int_0^zt^(n+1)J_n(t)dt=z^(n+1)J_(n+1)(z),
(9)

which, with n=0, becomes

 int_0^ztJ_0(t)dt=zJ_1(z).
(10)

Define

t=ksqrt(1+p^2)r
(11)
dt=ksqrt(1+p^2)dr
(12)
rdr=(tdt)/(k^2(1+p^2)),
(13)

so the inner integral is

int_0^(Rsqrt(1+p^2))J_0(t)(tdt)/(k^2(1+p^2))=1/(k^2(1+p^2))kRsqrt(1+p^2)J_1(kRsqrt(1+p^2))
(14)
=(J_1(kRsqrt(1+p^2)))/(ksqrt(1+p^2))R,
(15)

and the Radon transform becomes

R(p,tau)=R/(sqrt(1+p^2))int_(-infty)^infty(e^(iktau)J_1(kRsqrt(1+p^2)))/kdk
(16)
=(2R)/(sqrt(1+p^2))int_0^infty(cos(ktau)J_1(kRsqrt(1+p^2)))/kdk
(17)
={2/(1+p^2)sqrt(R^2(1+p^2)-tau^2) for tau^2<R^2(1+p^2); 0 for tau^2>=R^2(1+p^2).
(18)

Converting to R^' using p=cotalpha,

R^'(r,alpha)=2/(sqrt(1+cot^2alpha))sqrt((1+cot^2alpha)R^2-r^2csc^2alpha)
(19)
=2/(cscalpha)sqrt(csc^2alphaR^2-r^2csc^2alpha)
(20)
=2sqrt(R^2-r^2),
(21)

which could have been derived more simply by

 R^'(r,alpha)=int_(-sqrt(R^2-r^2))^(sqrt(R^2-r^2))dy.
(22)

See also

Radon Transform, Radon Transform--Delta Function, Radon Transform--Gaussian, Radon Transform--Square

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Cite this as:

Weisstein, Eric W. "Radon Transform--Cylinder." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RadonTransformCylinder.html

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