Image Reconstruction

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

$P(\omega ,\theta )=F({\omega }_x,{\omega }_y){|}_{{\omega }_x=\omega \cos \theta ,{\omega }_y=\omega \sin \theta }.$

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

$P(\omega ,\theta )=F_{\text{polar}}(\omega ,\theta ).$

2.6.4Derivation of the FBP algorithm

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

$f(x,y)={\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{F}_{\text{polar}}(\omega ,\theta )e^{2\pi i\omega (x\cos \theta +y\sin \theta )}\omega d\omega d\theta .}}$

Because $F_{\text{polar}}(\omega ,\theta )=F_{\text{polar}}(-\omega ,\theta +\pi ),$ we have

$f(x,y)={\displaystyle \underset{0}{\overset{\pi }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{F}_{\text{polar}}(\omega ,\theta )\left|\omega \right|e^{2\pi i\omega (x\cos \theta +y\sin \theta )}d\omega d\theta .}}$

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

$f(x,y)={\displaystyle \underset{0}{\overset{\pi }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}P(\omega ,\theta )\left|\omega \right|e^{2\pi i\omega (x\cos \theta +y\sin \theta )}d\omega d\theta .}}$

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