Additional Filtering

• If additional filters are applied, then can also write out what resolution will be in these cases
  

  $$\Delta x_{MRI,filter} \equiv \mbox{MR spatial resolution with filter}$$

  $$= \frac{1}{h_{ws,filter}(0)}\int_{-L/2}^{L/2}dx\; h_{ws,filter}(x)$$

  $$= \frac{1}{h_{ws,filter}(0)}\int_{-L/2}^{L/2}dx\; \Delta k \sum_{p=-n}^{n-1} H_{filter}(p\Delta k)e^{i2\pi p\Delta kx}$$

  $$= \frac{1}{\Delta k}\left[\frac{H_{filter}(0)} {\sum_{p'=-n}^{n-1}H_{filter}(p'\Delta k)}\right]$$ (13.9)

  
  

• If $H_{filter}(p'\Delta k) = H_{filter}(0)$ for all $p$ then
  

  $$\Delta x_{MRI,filter} = \frac{1}{2n\Delta k} = \frac{FOV}{2n} = \Delta x$$ (13.10)

  
  
  BUT assume $H_{filter}(0) > H_{filter}(x)$ for all $x$ so $\Delta x_{MRI,filter}$ is always larger than $\Delta x_{ws}$.
  

  $$\Delta x_{MRI,filter} \geq \frac{1}{2n\Delta k} \equiv \frac{1}{W} = \Delta x$$ (13.11)

  
  

• other definitions of resolution exist
  • Full Width at Half Maximum of the point spread function
  • Full Width at Tenth Maximum of the point spread function
  • Look for first zero crossing of function (won't work for all).
  • Some $x$ giving fixed area less than the total area under filter