Truncating the MRI Signal

• in presence of constant gradient $k \propto t$, but we can't sample forever.

• signal as a function of $k$ is going to be truncated

• What does the final sampled and truncated MRI signal look like? Multiply by sampling function and rect function for truncation.
  

  $${\rm s}_{m}(k) = s(k) \cdot u(k) \cdot {\rm rect}\left( \frac{k+\frac{1}{2}\Delta k}{W}\right)$$

  $$= \Delta k \sum_{p = -n}^{n-1} s(p\Delta k)\delta(k - p \Delta k)$$ (11.9)

  
  

• Note: For lots of reasons in MRI generally work with powers of 2, or at least even numbers when sampling. With an even number of points, don't have a center point to match up with $k=0$, choose to set up equations such that point $n+1$ of $2n$ points collected corresponds to $k=0$. Strictly speaking if we also shift the rect function, will introduce a bit of linear phase, but usually disregard this in discussions.

• Let's disregard effect of sampling for a minute (it doesn't go away), what is the effect of truncating the data
  

  $${\rm s}_{m}(k) = s(k) \cdot {\rm rect}\left( \frac{k+\frac{1}{2}\Delta k}{W}\right)$$ (11.10)

  
  
  where $W = 2n \Delta k$ is the width of the sampling window.

• Convolution theorem tells us that
  

  $$\hat{\rho}_{truncated}(x) = \rho(x) * W sinc (\pi W x) e^{-i \pi x \Delta k}$$ (11.11)

  
  
  What does this imply for an image?

  
  
  
  
  
  
  
  
  
  
  
  
  
  

• Put it all back together to see what the final reconstructed image looks like
  

  $$\hat{\rho}(x) = \rho(x) * U(x) * \left (W\,{\rm sinc}{(\pi W x)\,e^{-i\pi x \Delta k}}\right)$$ (11.12)

  
  

• If look at as an integral over the measured signal
  

  $$\hat{\rho}(x) \equiv \int_{-\infty}^{\infty} dk\, {s}_{m}(k) e^{i 2 \pi k x}$$

  $$= \Delta k \int_{-\infty}^{\infty} dk \sum_{p = -n}^{n-1} s(p\Delta k)\delta(k - p \Delta k)e^{i 2 \pi k x}$$

  $$= \Delta k \sum_{p=-n}^{n-1} s(p \Delta k) e^{i 2 \pi p \Delta k x}$$ (11.13)

  
  
  see that it is again a periodic function.

• Truncation doesn't take away the Nyquist sampling criteria.