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Introduction

The Fourier Slice Theorem elucidates how the projections measured by a medical imaging device can be used to reconstruct the object being scanned. From those projections a Continuous Time Fourier Transform (CTFT) is taken. Then according to the theorem, an inverse Continuous Space Fourier Transform (CSFT) can be used to form the original object,$ f(x,y) $. There are two proofs that will be demonstrated.

Fourier Slice Theorem

Given:

$ (x,y): $= the coordinates of the system the original object resides in (as seen in Figure 1)

$ (r,z): $= the coordinates of the system the projection resides rotated at an angle $ \theta $ relative to $ (x,y) $(as seen in Figure 2)

$ \rho: $= the frequency variable corresponding to $ r $

$ u: $= the frequency variable corresponding to $ x $

$ v: $= the frequency variable corresponding to $ y $

The Fourier Slice Theorem (FST) states that if
$ \begin{align} P_{\theta}({\rho}) &= CTFT \{p_\theta(r)\} \\ F(u,v) &= CSFT\{f(x,y)\} \end{align} $

Then
$ P_{\theta}({\rho}) = F(\rho\cos(\theta),\rho\sin(\theta)) \ $

Proof

References:
[1] C. A. Bouman. ECE 637. Class Lecture. Digital Image Processing I. Faculty of Electrical Engineering, Purdue University. Spring 2013.

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