next up previous
Next: Calculating the smoothness of Up: Smoothness estimation Previous: Smoothness a la SPM

A more robust smoothness estimator

The estimator described in section 2.2.1 becomes increasingly inaccurate for sub voxel width filter sizes. As the filter width decreases equation 12 becomes increasingly inaccurate until the estimation in equation  14 is no longer valid. An alternative estimate [2] offers a more robust result when evaluating images with small spatial correlation. Consider the following definitions

$\displaystyle S^2$ $\displaystyle = \frac{1}{N} \sum_i^N \frac{1}{M} \sum_t^M {S_{i,t}}^2$ (16)
$\displaystyle (\nabla S)^2$ $\displaystyle = \frac{1}{N} \sum_i^N \frac{1}{M} \sum_t^M \left( S_{i,t} - S_{i-\vec{1},t} \right)^2$ (17)
$\displaystyle SS_-$ $\displaystyle = \frac{1}{N} \sum_i^N \frac{1}{M} \sum_t^M \left( S_{i,t} S_{i - \vec{1}, t} \right)$ (18)

where $ \vec{1}$ is a unit vector along the dimension currently under consideration. $ S^2$ is the individual voxel variance, $ (\nabla S)^2$ the variance of the difference between each voxel and its edgewise neighbours, and $ SS_-$ is the correlation of two neighbouring voxels.

The approach outlined in 2.2.1 calculates the quantity

$\displaystyle s = \sqrt \frac{1}{2 (\nabla S)^2}.$ (19)

Forman et al derive the alternate formula

$\displaystyle s = \sqrt {\frac{- 1}{4 \cdot ln \left( 1 - \frac{(\nabla S)^2}{2 S^2} \right)}}.$ (20)

Mark Jenkinson [4] has provided an extensive derivation for this formula as well as pointing out another, computationally cleaner, variation on the same theme

$\displaystyle s = \sqrt {\frac{- 1}{4 \cdot ln \left( \frac{SS_-}{S^2} \right) } }.$ (21)

Equations 20 and 21 remain accurate for relatively small spatial smoothnesses and it is equation 21 that you will find in the FSL code base.


next up previous
Next: Calculating the smoothness of Up: Smoothness estimation Previous: Smoothness a la SPM
David Flitney 2001-11-29