Maximum spherical harmonic degree lmax
¶
What determines 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 lmax
occurs
are as follows:
- Determine the maximum value for
lmax
that 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:
lmax | Required volumes |
---|---|
2 | 6 |
4 | 15 |
6 | 28 |
8 | 45 |
10 | 66 |
12 | 91 |
... | ... |
If
lmax
exceeds 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
lmax
until 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
lmax
selected for FOD estimation will also be reduced iflmax
of 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 = 0
terms). This basis requires far less coefficients for any given value oflmax
: 2 forlmax = 2
, 3 forlmax = 4
, 4 forlmax = 6
, 5 forlmax = 8
and 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.
Reduced lmax
in particular subjects¶
If you find that certain subjects within a cohort have a reduced lmax
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.
Setting 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
lmax
to 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 setlmax
to a lower value than that supported by the data. - In spherical deconvolution, it is possible to set a higher
lmax
than 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 oflmax
for which aPSF data are available (currentlylmax = 16
, since the angular resolution of the original image data is not a limiting factor here. - As described previously, the
amp2response
command is a special case, and the maximum permissiblelmax
is vastly greater than the maximum practical value.