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

$P(\omega ,\theta )=F({\omega}_{x},{\omega}_{y}){|}_{{\omega}_{x}=\omega \mathrm{cos}\theta ,{\omega}_{y}=\omega \mathrm{sin}\theta}.$

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

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

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

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

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

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

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

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