# Gaussian kernels: convert FWHM to sigma

When smoothing images and functions using Gaussian kernels, often we have to convert a given value for the full width at the half maximum (FWHM) to the standard deviation of the filter (sigma, $\sigma$). This happens because the implementation generally is in terms of sigma, while the FWHM is the more popular parameter in certain areas. The conversion is trivial, but it may well worth write it up here.

The probability density function (pdf) for the Gaussian distribution with mean $\mu$ and standard deviation $\sigma$ is:

$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

If the filter is centered at the origin, the mean is 0 and the FWHM is the distance between the $-x_w$ and $+x_w$ that produces the half of the peak. For the normal distribution, the mean is the same as the mode (peak) and we have then to find the $x_w$ that will produce $f(x_w) = f(\mu)/2$:

$\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{x_{w}^{2}}{2\sigma^2}} = \frac{1}{2} \frac{1}{\sigma\sqrt{2\pi}}$

For $\sigma \neq 0$ and solving for $x_w$:

$x_w = \pm \sqrt{2\sigma^2\ln 2}$

The FWHM is $+x_w - (-x_w)=2 x_w$:

$\text{FWHM}=2\sqrt{2\sigma^2\ln 2}=\sigma\sqrt{8\ln 2}$

Which gives 2.35482004503 as the conversion factor, i.e., $\text{FWHM} \approx 2.355\cdot \sigma$.

Some software packages, such as SPM and FreeSurfer, interact with the user in terms of FWHM, whereas others, such as FSL, prefer $\sigma$. The relation above allows converting quickly between one and other representation.

# Converting OASIS brains to NIFTI

The OASIS dataset consists of a number of T1-weighted mri brain scans, which has been kindly offered online at http://www.oasis-brains.org. The dataset can be downloaded for free after accepting an academic agreement available at the website. The data was released in analyze 7.5 file format. This format, despite having been used by the neuroimaging community for many years, suffers from not including orientation information. The nifti format addresses this concern, but when converting, some care has to be taken to ensure that there are no left-right flips. Fortunately, for this dataset, during acquisition a capsule of vitamin E was placed next to the head of each subject, on the left side, which is very helpful to identify the correct side (Marcus et al., 2007).

There are still problems, though. The nifti format has not been fully implemented in all common software packages and worse, some packages interpret differently the information contained in the header. Images that look fine in fsl‘s FSLview or FreeSurfer‘s Freeview may look stretched or shrunken in spm for instance. And images that look fine in both, may still be oriented incorrectly in Mango. Although a bit cumbersome, the procedure below ensures that the oasis images can be converted from analyze to nifti in a way that it can correctly read and shown by Mango, fsl, FreeSurfer and spm. The procedure uses exclusively fsl tools:

1. Convert to nifti:
fslchfiletype NIFTI_GZ OAS1_0001_MR1_mpr-1_anon.hdr
2. Make sure there is no undesired orientation information:
fslorient -deleteorient OAS1_0001_MR1_mpr-1_anon.nii.gz
3. Set the sform_code as 2, which is for “aligned anatomy”. Although this is still in native, not aligned space, it ensures that software will read them appropriately:
fslorient -setsformcode 2 OAS1_0001_MR1_mpr-1_anon.nii.gz
4. Set the sform as the following matrix:
fslorient -setsform  0 0 -1.25 0  1 0 0 0  0 1 0 0  0 0 0 1  OAS1_0001_MR1_mpr-1_anon.nii.gz
5. Swap the order of the data. Again, this isn’t really necessary, except to ensure that different applications will all read correctly:
fslswapdim OAS1_0001_MR1_mpr-1_anon.nii.gz RL PA IS OAS1_0001_MR1_mpr-1_anon.nii.gz
6. fsl tries to preserve orientation and, when the voxels are reordered, it modifies the header accordingly, resulting in no net transformation when seen with fsl tools. To resolve this, it’s necessary to change the header again, now the qform:
fslorient -setqform -1.25 0 0 0  0 1 0 0  0 0 1 0  0 0 0 1  OAS1_0001_MR1_mpr-1_anon.nii.gz

These steps can be placed inside a simple loop within the shell for either Linux and Mac, like below (click here to download):

#!/bin/bash

# Directory where the OASIS data is located
ROOTDIR=/Volumes/HD2/oasis-dataset-416subj/original

# Directory to save results
NIFTIDIR=${ROOTDIR}/../nifti # Go to the directory with the data cd${ROOTDIR}

# For each subject
for s in * ; do

# Some feedback in the screen
echo ${s} # Create directory to save the results, if not existing mkdir -p${NIFTIDIR}/${s} # Directory of the original, raw data cd${ROOTDIR}/${s}/RAW # For each acquisition for a in *.hdr ; do # Do each of the 6 steps described in the blog${FSLDIR}/bin/fslchfiletype NIFTI_GZ ${a}${NIFTIDIR}/${s}/${a%.hdr}
${FSLDIR}/bin/fslorient -deleteorient${NIFTIDIR}/${s}/${a%.hdr}
${FSLDIR}/bin/fslorient -setsformcode 2${NIFTIDIR}/${s}/${a%.hdr}
${FSLDIR}/bin/fslorient -setsform 0 0 -1.25 0 1 0 0 0 0 1 0 0 0 0 0 1${NIFTIDIR}/${s}/${a%.hdr}
${FSLDIR}/bin/fslswapdim${NIFTIDIR}/${s}/${a%.hdr} RL PA IS ${NIFTIDIR}/${s}/${a%.hdr}${FSLDIR}/bin/fslorient -setqform -1.25 0 0 0  0 1 0 0  0 0 1 0  0 0 0 1  ${NIFTIDIR}/${s}/\${a%.hdr}
done
done

echo "Done!"


The reference for the oasis dataset is: