Updated MK and MW help and definitions in usage

docs/docs/TMI/usage.md

@@ -43,8 +43,9 @@ By default, if none of the below options are used, TMI will not estimate paramet
 - When DKI option is used, b-shells greater than 3000 are automatically excluded from tensor estimation.
 
 ### `-WDKI`
-- Include diffusion kurtosis parameters based on the W definition of the kurtosis tensor in output folder.
+- Output folder will also contain kurtosis parameters that are computed using definitions with the kurtosis tensor, unlike the conventional definitions which compute them using averages of the directional kurtosis. For example, for mean kurtosis mk_wdki is simply the angular average of the dimensionless cumulant tensor $\overline{W({\bf \hat{n}})} =  \tfrac{1}{4 \pi} \! \int_{\mathbb{S}^2}\!\! d\hat{\mathbf{n}}\, W(\hat{\mathbf{n}})$ while the more standard definition (mk_dki) is the angular average of the directional kurtosis \overline{K({\bf \hat{n}})} =  \tfrac{1}{4 \pi} \! \int_{\mathbb{S}^2}\!\! d\hat{\mathbf{n}}\, K(\hat{\mathbf{n}})=  \tfrac{1}{4 \pi} \! \int_{\mathbb{S}^2}\!\! d\hat{\mathbf{n}}\,  \frac{\bar{D}^2 W(\hat{\mathbf{n}})}{ {D(\hat{\mathbf{n}})}^2}.
 - Parameters extracted from the W tensor are labelled as `*_wdki.nii` (e.g. ad_wdki.nii, fa_wdki.nii, mk_wdki.nii).
+- Note that the estimated tensors (D and W) are identical but how we combine the tensor elements to make kurtosis scalar maps differs (DTI maps with _wdki are identical to those with _dki since there is no conflict with DTI-derived metrics).
 
 ### `-fit_constraints <0,0,0>`
 - For use with `-DKI` or `-WDKI`