The reference for ability to estimate anisotropic response functions of a higher maximal spherical harmonic degree than is supported by the number of unique DWI directions is listed in the help page of the `amp2response`

command (which all `dwi2response`

algorithms rely upon for this specific step). The abstract describing the method is accessible here.

On the topic of repeating diffusion directions, I’ll drop the phrase “condition number” into the mix for anyone interested. This is one of the many things reported by the `dirstat`

command for assessing direction schemes.

Let’s say you have 60 volumes, and so expect an *lmax*=8 fit to be achievable in each voxel. One of the first steps in spherical deconvolution is a conversion of the DWI data from discrete signal intensities as a function of direction of diffusion sensitisation into the continuous spherical harmonic basis. This is just a transformation via matrix multiplication. If your 60 directions are well-distributed, this transformation is well-posed, and the condition number of the transformation will be close to 1.0, indicating that fluctuations in the input image intensities would propagate “nicely” to corresponding changes in the spherical harmonic coefficients. If you actually only have 20 *unique* directions, you *don’t actually have enough information* to fit *lmax*=8 spherical harmonic coefficients, *regardless* of how many repeats of each direction you may have acquired. In this case, the condition number will approach infinity, reflecting how the resulting spherical harmonic coefficients are *completely unpredictable*, and not robust to even tiny fluctuations in the input image data^{1}.

The reason this limitation doesn’t arise specifically in response function estimation is because the method linked utilises data from across multiple image voxels, and the fibre orientations in those voxels are all different.

Rob

^{1} Interestingly, doing motion correction actually mitigates this slightly, as it means that the diffusion sensitisation directions *relative to the image sample* are no longer precisely equivalent, even if the applied diffusion sensitisation gradients were identical. But it’s nevertheless still preferable to get as much unique image information as possible rather than repeats.