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

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