Increasing the smoothness of already smoothed Nifti images

Hello everyone,

I am working an fMRI data set which has been smoothed with a 3mm fwhm Gaussian kernel. I would like to increase the smoothness of the data by applying additional Gaussian smoothing on top of this already smoothed data. Ideally I would like to have an equivalent effect like if applying a single Gaussian smoothing of 5mm fwhm.
I know from Gaussian blur on 2D images that I would get the equivalent effect of a 5mm kernel smoothing if I apply additional smoothing of 4 mm kernel on top of images that were already smoothed with 3mm kernel, as sqrt(3^2 + 4^2) = 5. I wonder whether this also apply to 3D brain volume?
I have tested it on the standard mask and T1 images and found that the result of 3mm + 4mm smoothing is very similar, but not identical to the result of a single 5mm smoothing. Do you know whether there is a better way to calculate what kernel size I need in order to get a smoothing effect of 5 mm kernel?

Thanks so much,
Yuan-hao

Going by gut feeling here, and haven’t done any math, but I think the same logic applies to 3D as 2D. The values you’re averaging while smoothing are the same distance away, but now you’re fetching them from a sphere about each voxel center, rather than a circle. The distances are still Euclidean, so the Pythagorean theorem still applies.

Regarding:

You’re generally not going to get identical results from performing two operations as one theoretically equivalent operation because there are interpolation and floating point rounding artifacts that will occur in each case, so I wouldn’t worry too much about it. If you do the same on a 2D image, you should still get some deviation, but less because there are fewer inputs to each voxel.

Another thing you can consider is looking at AFNI’s 3dBlurToFWHM which actually calculates the smoothness of the image and smooth until you reach a target.