thanks for your comments, John. sorry for having to ask this novice
question as a beginner to signal processing: so is it right that the
results of smoothing in the Fourier space should be closer to what the
real data is than smoothing in the spatial (temporal) domain, because
how data is interpolated between the centres of the voxels are
considered? That is, take a data set without any spatial smoothing and
run the stats, theoretically the results should look more like the one
smoothed in the Fourier space than in the spatial (temporal) domain?
erik
On 1/17/06, John Ashburner <[log in to unmask]> wrote:
>
> > is it possible that spatial smoothing results in different smoothed dataset
> > when conducted in the spatial domain and in the frequency domain (made
> > equivalent by setting cut-off freq = 1/FWHM)?
>
> It is very possible. An image can be considered as a discrete representation
> of a continuous function. An image is often convolved by summing over the
> voxels, weighting each voxel by the height of the e.g. Gaussian at each
> point. This does not consider how the data is interpolated between the
> centres of the voxels. If this is considered, then the operation would be
> treated as a proper integration. Convolving in Fourier space would take this
> interpolation into consideration. Also, if you examine the function
> spm_smoothkern.m, you will see that it generates a smoothing kernel that
> assumes that the images are continuously interpolated.
>
> Another difference relates to the boundary conditions. If you smooth via
> Fourier transforms, then you assume that outside the FOV, you have the image
> repeating ad infinitum in all directions. This boundary condition can be
> incorporated into spatial (temporal) smoothing procedures, but it is normally
> not the case (e.g. in SPM).
>
> Best regards,
> -John
>
>
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