Dear Oliver, dear SPM-ers,

In "Detecting activations in PET and fMRI : levels of inference and power", Friston
et al., Neuroimage 4 : 223-235 1995 it is stated that :

"Usual estimates of smoothness fail when the reasonable lattice assumption is
violated. In our work we side-step this issue by simply interpolating the data to
reduce voxel size or smoothing the data to increase smoothness."

Of course you cannot get additional information by interpolating the data, but I
thought it was a good way to avoid the loss of spatial resolution which occurs when
smoothing with a kernel of width 2-3 times the acquisition voxel size. With
interpolated data it would be possible to reach the criterion "voxel size < FWHM",
i.e. a sufficient smoothness for the lattice assumption to hold, with very little
smoothing, i.e. nearly no loss in spatial resolution.

Could you or anyone else comment this ?

Thanks a lot,

Vincent



> Dear Dirk,
>
> There is probably little advantage and definite dis-advantages to
> sub-sampling data in this fashion.
>
> Firstly, some interpolation algorithms (e.g. bilinear) are only
> intended to improve the apprearance of displayed images and so do not
> necessarily preserve all of the image information. To prevent this loss
> a band-limited interpolation scheme is necessary (also sometimes known
> as "Fourier", "full sinc" or "ideal" interpolation).
>
> Seondly, the interpolation will increase the amount of data that have
> to be stored and processed (by a factor of 4 in the example you
> mention).
>
> Thirdly, data that have undergone interpolation will exhibit a very
> specific form of spatial autocorrelation. No energy should (generally)
> or will (ideally) be present at the higher spatial frequency points
> introduced by interpolation. In k-space terms ideal interpolation is
> equivalent to (and may actually be implemented by) acquiring only the
> centre of a larger k-space grid (e.g. a 64x64 matix inside a 128x128
> grid).
>
> If no enery is present in the data the same will be true for the
> residuals after a design matrix has been fitted. However, the Gaussian
> random field theory component of SPM presumes a Gaussian residual
> spatial power spectral density. (This can be expressed in terms of
> spatial auto-correlation, which by the Wiener-Khintchine theorem is the
> Fourier transform of the power spectral density). This condition is
> normally (at least approximately) enforced by smoothing the data with a
> Gaussian kernel of 2-3 times the ORIGINAL voxel size. This smoothing
> is, by the convolution theorem, equivalent to multiplying the PSD by a
> Gaussian which has fallen to a low value by the edge of the ACQUIRED
> portion of k-space. Just by padding k-space does not change this and so
> no gain in resolution in an SPM is possible simply by subsampling the
> data.
>
> These points apply equally for any sub-sampling interpolation performed
> off-line including within spatial normalisation by SPM, which, by
> default (I think), sub-samples the output to 2x2x2 mm.
>
> Oliver



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