Gibbs Ringing and Filters

• In addition to reducing resolution, there are other distinct consequences of the truncation that can be viewed in images.
• Specifically, consider point at the edge of a phantom. This is a point discontinuity in the spin density, very similarly mathematically to a delta function.
  
  
  
  
  
  
  
• Near the step observe the sinc-like behavior of the point-spread function
• Magnitude of Gibbs ringing is fixed at 9% . Assume edge occurs at $x_0$
  

  $$\lim_{N \rightarrow \infty}\left\vert \hat{\rho}(x_0^\pm )-... ...\simeq 0.09 \left\vert \rho (x_0^+)-\rho (x_0^-)\right\vert$$ (13.12)

  
  

• Can effect the frequency, and therefore the distance over which you observe the Gibbs ringing, by remembering what the point spread is for the rect function (symmetric for illustration)
  

  $$h_{w}^{sym}(x) = W{\rm sinc}{(\pi Wx)}$$ (13.13)

  
  

• As you make the $k$-space window larger the Gibbs ringing doesn't go away, but it looks better
  (64 by 64) (128 by 128)
  
  
  (256 by 256) (512 by 512)
• Alternatively, see that Gibbs frequency is given by $W$ so it undergoes a complete cycle every 2 pixels, regardless of the resolution of the experiment.
  
  
  
  
  
  
  
• If can stand to blur pixel out into its two nearest neighbors should be able to greatly reduce the appearance of Gibbs Ringing.
• Filter that does this is referred to as the Hanning filter
  

  $$H_{Hanning}(k) = \frac{1+\cos{\left(\frac{2\pi k}{W}\right)}}{2} = \cos^{2}\left(\frac{\pi k}{W} \right)$$ (13.14)

  
  
  
  
  
  
  
  
  
  
• Take the Fourier transform of this function gives you three $\delta$ functions in $x$-space, frequency is chosen to put delta functions in adjacent pixels
  

  $$h_{Hanning}(x) = \frac{1}{2}\delta (x) +\frac{1}{4}(\delta (x-\Delta x)+\delta (x+ \Delta x))$$ (13.15)

  
  

• Convolve this with $\rho(x)$ and will smear point in one voxel into its nearest neighbors.