Summary

In practice the smoothness is estimated using the following steps:

1. Normalise the 4D residuals, $R_t(\ensuremath{\mathbf{x}}) \rightarrow S_t(\ensuremath{\mathbf{x}})$, using equation 37.
2. Calculate the derivative approximation $S_t'(\ensuremath{\mathbf{x}})$ at all 4D samples using equation 40.
3. Calculate the 4D average statistic $\lambda_{jj}$ using equation 39.
4. Accumulate the results of $\lambda_{jk}$ into a matrix (assuming non-diagonal elements are zero):
   

   $$\Lambda = \left( \begin{array}{ccc} \frac{1}{2{\sigma_x}^2... ... & 0 \\ 0 & 0 & \frac{1}{2{\sigma_z}^2} \end{array} \right)$$ (41)

   
   

5. Optional: Calculate the filter width matrix $W = (2\Lambda)^{-1}$ and the FWHM values as $\mathrm{FWHM}_j = \sqrt{8 \ln(2) W_{jj}}$.