# Maximum spherical harmonic degree lmax¶

The lmax value refers to the highest harmonic order $$l$$ included in the Spherical Harmonics series. It determines the highest angular frequency band to be included in the series: higher lmax values allow sharper details to be represented, but require more coefficients to be stored and processed.

## 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 antipodally-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 if lmax 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 fewer coefficients for any given value of lmax: 2 for lmax=2, 3 for lmax=4, 4 for lmax=6, 5 for lmax=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 set lmax 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.
• If performing Track Orientation Density Imaging (TODI) using tckmap -tod, then the apodized point spread functions (aPSFs) can be generated at any value of lmax for 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 amp2response command is a special case, and the maximum permissible lmax is vastly greater than the maximum practical value.