Radon Transform—Cylinder

Author

Eric W. Weisstein
February 8, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegralTransforms/RadonTransformCylinder.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RadonTransformCylinder.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Integral

Integrate[Exp[-I k r(Sin[θ]-p Cos[θ])]r,{θ,0,2π}]

Transform θ so that sin(θ)-p cos(θ) becomes (Sqrt[(p^2)+1]) cos(θ')≡s cos(θ')

Bug: Integrate cannot handle symbolic parameter expressed as (Sqrt[(p^2)+1])

Integrate[Exp[-I k r Sqrt[1+p^2] Cos[θ]] r,{θ,0,2π},Assumptions->k>0&&s>1&&r>0]

Integrate[Exp[-I k r s Cos[θ]] r,{θ,0,2π},Assumptions->k>0&&s>1&&r>0]

Integrate[%,{r,0,R}]

v5.01

Integrate[(((E^(I k τ)) R BesselJ[1,k R s])/(k s)),{k,-∞,∞},Assumptions->s>1&&R>0&&τ∈Reals]

v5.2

Integrate[Exp[-I k r Sqrt[1+p^2] Cos[θ]] r,{θ,0,2π},Assumptions->k>0&&s>1&&r>0]//Timing

Integrate[(((E^(I k τ)) R BesselJ[1,k R s])/(k s)),{k,-∞,∞},Assumptions->s>1&&R>0&&τ∈Reals]//Timing

v6

Integrate[Exp[-I k r Sqrt[1+p^2] Cos[θ]] r,{θ,0,2π},Assumptions->k>0&&s>1&&r>0]//Timing

Integrate[(((E^(I k τ)) R BesselJ[1,k R s])/(k s)),{k,-∞,∞},Assumptions->s>1&&R>0&&τ∈Reals]//Timing

Plot

Plot3D[If[t^2>(1+p^2),0,2/(1+p^2) Sqrt[1+p^2-t^2]],{p,-20,20},{t,-20,20},PlotPoints->50,Boxed->False,PlotRange->{0.,2.},AxesLabel->TraditionalForm/@{p,τ,R[p,τ]}]