Maximum spherical harmonic degree
lmax for my image data?¶
For any command or script operating on data in the spherical harmonic
basis, it should be possible to manually set the maximum harmonic degree
of the output using the
-lmax command-line option. If this is not
provided, then an appropriate value will be determined automatically.
The mechanisms by which this automatic determination of
are as follows:
- Determine the maximum value for
lmaxthat is supported by the number of DWI volumes in the shell being processed (or the total number of non-*b*=0 volumes in a single-shell acquisition). This is the number of coefficients required to store an anitipodally-symmetric spherical harmonic function:
lmaxexceeds 8, reduce to 8. This is primarily based on the findings in this paper.
Check the condition of the transformation between DWIs and spherical harmonics. If the transformation is ill-conditioned (usually indicating that the diffusion sensitisation gradient directions are not evenly distributed over the sphere or half-sphere), reduce
lmaxuntil the transformation is well-conditioned.
As an example: concatenating two repeats of a 30 direction acquisition to produce 60 volumes will not support an ``lmax``=8 fit: the angular resolution of the data set is equivalent to 30 unique directions, and so ``lmax``=6 would be selected (and this would be accompanied by a command-line warning to the user).
In the case of spherical deconvolution, the
lmaxselected for FOD estimation will also be reduced if
lmaxof the provided response function is less than that calculated as above.
The exception to these rules is the new
amp2response command, which
is now called by default in all
dwi2response script algorithms. This
command converts amplitudes on the half-sphere (most likely in the form
of raw DWI image intensities) into a response function intended for use
in spherical deconvolution. This command behaves differently for two
reasons in combination:
- The image data from multiple voxels are combined together in a single fitting procedure, therefore having a much greater number of samples when performing the transformation.
- The data are transformed not to the spherical harmonic basis, but
directly to the zonal spherical harmonic basis (this is the spherical
harmonic basis containing only the
m = 0terms). This basis requires far less coefficients for any given value of
lmax: 2 for
lmax = 2, 3 for
lmax = 4, 4 for
lmax = 6, 5 for
lmax = 8and so on.
The value of
lmax that can be used in this command is therefore
practically unconstrained; though the power in higher harmonic degrees
is much smaller than that in lower degrees. The command is currently
configured to select
lmax = 10 by default, regardless of b-value;
interested readers can find the discussion here.
lmax in particular subjects¶
If you find that certain subjects within a cohort have a reduced
compared to the rest of the cohort when using any command relating to
spherical harmonics, the most likely cause is premature termination of the
diffusion sequence during scanning of those subjects, resulting in a reduced
number of diffusion volumes, and therefore a reduced
lmax according to
the table above.
lmax in different applications¶
The range of permissible values for
lmax depends on the particular
command being used; e.g.:
- For any command that maps image data directly to spherical harmonics, it
is impossible to set
lmaxto a value higher than that supported by the image data. The transformation from DWI data to spherical harmonics simply cannot be done in such a case, as the problem is under-determined. You can of course set
lmaxto a lower value than that supported by the data.
- In spherical deconvolution, it is possible to set a higher
lmaxthan that supported by the data - so-called super-resolved spherical deconvolution. Here, additional information is provided by the non-negativity constraint to make estimation of additional spherical harmonic coefficients possible. However this is not guaranteed: sometimes the algorithm will fail in particular voxels, in cases where there are an insufficient number of directions in which the initial FOD estimate is negative, as the problem remains under-determined.
- If performing Track Orientation Density Imaging (TODI) using
tckmap -tod, then the apodized point spread functions (aPSFs) can be generated at any value of
lmaxfor which aPSF data are available (currently
lmax = 16, since the angular resolution of the original image data is not a limiting factor here.
- As described previously, the
amp2responsecommand is a special case, and the maximum permissible
lmaxis vastly greater than the maximum practical value.