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Radon Transform--Square

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RadonSquare
R(p,tau)=int_(-infty)^inftyint_(-infty)^inftyf(x,y)delta[y-(tau+px)]dydx,
(1)

where

f(x,y)={1 for x,y in [-a,a]; 0 otherwise
(2)

and

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

is the delta function.

R(p,tau)=1/(2pi)int_(-a)^aint_(-a)^aint_(-infty)^inftye^(-ik[y-(tau+px)])dkdydx
(4)
=1/(2pi)int_(-infty)^inftye^(iktau)[int_(-a)^ae^(-ky)dyint_(-a)^ae^(ikpx)dx]dk
(5)
=1/(2pi)e^(iktau)1/(-ik)[e^(-iky)]_(-a)^a1/(ikp)[e^(ikpx)]_(-a)^adk
(6)
=1/(2pi)int_(-infty)^inftye^(iktau)1/(k^2p)[-2isin(ka)][2isin(kpa)]dk
(7)
=2/(pip)int_(-infty)^infty(sin(ka)sin(kpa)e^(iktau))/(k^2)dk
(8)
=4/(pip)int_0^infty(sin(ka)sin(kpa)cos(ktau))/(k^2)dk
(9)
=2/(pip)int_0^infty(sin[k(tau+a)]-sin[k(tau-a)])/(k^2)sin(kpa)dk
(10)
=2/(pip){int_0^infty(sin[k(tau+a)]sin(kpa))/(k^2)dk-int_0^infty(sin[k(tau-a)]sin(kpa))/(k^2)dk}.
(11)

From Gradshteyn and Ryzhik (2000, equation 3.741.3),

int_0^infty(sin(ax)sin(bx))/(x^2)dx=1/2pisgn(ab)min(|a|,|b|),
(12)

so

R(p,tau)=1/p{sgn[(tau+a)pa]min(|tau+a|,|pa|)-sgn[(tau-a)pa]min(|tau-a|,|pa|)}.
(13)

This can also be written explicitly in the form

R(p,tau)=(|a+ap-tau|+|a+ap+tau|-|a-ap-tau|-|a-ap+tau|)/(2p).
(14)

See also

Radon Transform, Radon Transform--Cylinder, Radon Transform--Delta Function, Radon Transform--Gaussian

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Cite this as:

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

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