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Radon Transform--Delta Function

For a delta function at (x_0,y_0),

R(p,tau)=int_(-infty)^inftyint_(-infty)^inftydelta(x-x_0)delta(y-y_0)delta[y-(tau+px)]dydx
(1)
=1/(2pi)int_(-infty)^inftyint_(-infty)^inftyint_(-infty)^inftye^(-ik[y-(tau+px)])delta(x-x_0)delta(y-y_0)dkdydx
(2)
=1/(2pi)int_(-infty)^inftye^(iktau)[int_(-infty)^inftye^(-iky)delta(y-y_0)dyint_(-infty)^inftye^(ikpx)delta(x-x_0)dx]dk
(3)
=1/(2pi)int_(-infty)^inftye^(iktau)e^(-iky_0)e^(ikpx_0)dk
(4)
=1/(2pi)int_(-infty)^inftye^(ik(tau+px_0-y_0))dk=delta(tau+px_0-y_0).
(5)

See also

Radon Transform, Radon Transform--Cylinder, Radon Transform--Gaussian, Radon Transform--Square

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Cite this as:

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

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