| Line 19: | Line 19: | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
| − | P_{\theta} | + | P_{\theta}{\rho} &= CTFT \{p_\theta(r)\} \\ |
F(u,v) &= CSFT\{f(x,y)\} | F(u,v) &= CSFT\{f(x,y)\} | ||
\end{align} | \end{align} | ||
| Line 26: | Line 26: | ||
Then <br /> | Then <br /> | ||
| − | <math>P_{\theta}{\rho} = F(\rho\cos(\theta)rho\sin(\theta)) \ | + | <math>P_{\theta}{\rho} = F(\rho\cos(\theta),\rho\sin(\theta)) \ </math> |
| − | </math> | + | <br /> |
| − | + | ||
---- | ---- | ||
=Proof= | =Proof= | ||
Revision as of 06:00, 19 December 2014
Fourier Slice Theorem (FST)
A slecture by ECE student Sahil Sanghani
Partly based on the ECE 637 material of Professor Bouman.
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
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.
