Smoothness a la SPM

The current implementations of SPM (SPM99) use the smoothness estimation as described in Kiebel et al [5].

For a Gaussian random field the smoothness is defined as $W = \vert\boldsymbol{\Lambda}\vert ^{-{1}/{2D}}$ where $D$ is the dimensionality of the field and $\boldsymbol{\Lambda}$ the covariance matrix of it's first partial derivatives. An unbiased estimator for the covariance of the partial derivatives is given by

$$\hat{\lambda}_{jk} = \frac{\nu - 2}{(\nu - 1)N} \sum_{i}^{N} \fra... ...c{\partial S_{it}}{\partial x_j} \frac{\partial S_{it}} {\partial x_k} \right),$$ (10)

where $\nu$ is the number of degrees of freedom, $N$ the number of voxel positions ($i$) and $M$ the number of time points ($t$) in a FMRI time series. In fact we don't have to evaluate this equation for the whole covariance matrix. The off-diagonal elements will be zero so we only need to calculate the diagonal elements of $\boldsymbol{\Lambda}$

$$\hat{\lambda}_{jj} = \frac{\nu - 2}{(\nu - 1)N} \sum_{i}^{N} \frac{1}{M} \sum_{t}^{M} \left( \frac{\partial S_{it}}{\partial x_j} \right)^2.$$ (11)

The partial derivative is evaluated via the gradient operator

$$\nabla S_{i} = \frac{S_{i+} - S_{i-}}{2 \delta d},$$ (12)

for non-edge voxels, where $S_{i+}$ and $S_{i-}$ are $S_i$s neighbouring voxels in the dimension along which the derivative is currently being evaluated.

Now we can calculate the FWHM of the theoretical Gaussian responsible for the observed smoothness

$$FWHM_i = \sqrt{8 \cdot ln (2) \cdot W_{ii}},$$ (13)

where $W_{ii}$ is the variance, $\sigma^2$, of the Gaussian point-spread-function computed from the covariance measure for the $i$th dimension

$$W_{ii} = \frac{1}{2 \lambda_{ii}}.$$ (14)

A combined value for the FWHM is given by the geometric mean of the various FWHM$_i$

$$FWHM = \left[\prod_{i=0}^{D}FWHM_i\right]^{1/D}.$$ (15)