Dear Eric,
I don't think smoothing in Fourier space will necessarily mean that the data
are closer to the original data. Maybe the amount of smoothing applied is
just different.
Best regards,
-John
> 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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