O'Reilly logo

Image Reconstruction by Gengsheng Lawrence Zeng

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Finally, using the definition of the 2D Fourier transform yields

P(ω,θ)=F( ω x , ω y ) | ω x =ωcosθ, ω y =ωsinθ .

In the polar coordinate system, the central slice theorem can be expressed as

P(ω,θ)= F polar (ω,θ).

2.6.4Derivation of the FBP algorithm

We start with the 2D inverse Fourier transform in polar coordinates

f(x,y)= 0 2π 0 F polar (ω,θ) e 2πiω(xcosθ+ysinθ) ωdωdθ.

Because F polar (ω,θ)= F polar (ω,θ+π), we have

f(x,y)= 0 π F polar (ω,θ)|ω| e 2πiω(xcosθ+ysinθ) dωdθ.

By using the central slice theorem, we can replace F by P:

f(x,y)= 0 π P(ω,θ)|ω| e 2πiω(xcosθ+ysinθ) dωdθ.

We recognize that ω| is the ramp filter. Let Q(ω, θ)=|ω|P(ω

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required