| Line 20: | Line 20: | ||
<math>(x,y):</math>= the coordinates of the system the original object resides in (as seen in Figure 1) | <math>(x,y):</math>= the coordinates of the system the original object resides in (as seen in Figure 1) | ||
| − | <math>(r,z):</math>= the coordinates of the system the projection resides rotated at an angle <math>\theta</math> relative to | + | <math>(r,z):</math>= the coordinates of the system the projection resides in rotated at an angle <math>\theta</math> relative to the object's coordinate system (as seen in Figure 2) |
<math>\rho:</math>= the frequency variable corresponding to <math>r</math> | <math>\rho:</math>= the frequency variable corresponding to <math>r</math> | ||
| Line 40: | Line 40: | ||
<math>P_{\theta}({\rho}) = F(\rho\cos(\theta),\rho\sin(\theta)) \ </math> | <math>P_{\theta}({\rho}) = F(\rho\cos(\theta),\rho\sin(\theta)) \ </math> | ||
<br /> | <br /> | ||
| + | |||
| + | This means... | ||
---- | ---- | ||
=Proof= | =Proof= | ||
| + | |||
| + | <font size="3">Method 1</font> | ||
| + | |||
| + | <font size="3">Method 2</font> | ||
| + | Let | ||
| + | |||
---- | ---- | ||
Revision as of 08:32, 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
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 in rotated at an angle $ \theta $ relative to the object's coordinate system (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)) \ $
This means...
Proof
Method 1
Method 2
Let
References:
[1] C. A. Bouman. ECE 637. Class Lecture. Digital Image Processing I. Faculty of Electrical Engineering, Purdue University. Spring 2013.
