The first approach of this type in the literature was Tournier’s Spherical Deconvolution [25, 41, 42], used to infer the orientations of multiple populations of white matter fibres—part of a class of methods known as High Angular Resolution Diffusion Imaging (HARDI). Here the signal is modelled by assuming that the measured signal is a convolution of the single fibre response function and an unknown fibre orientation distribution (FOD) function. This is equivalent to assuming that f (x, y) = f (x — y), i.e. a function of displacement only, and explicitly integrating out the radial direction. The approach deconvolves the unknown FOD, assuming or measuring some form for the single fibre response function. Spherical deconvolution approaches have been highly successful in multi-fibre tractography, largely because the technique provides very sharp and informative FOD estimates using fast, linear formulations for the deconvolution process itself. An interesting extension of this approach is by Kaden et al. [27] who allow parameters of the kernel to be fitted alongside the FOD.

Interestingly, it has been shown that other HARDI methods such as PAS-MRI [21] and Q-ball imaging [1, 43] can be shown to be forms of deconvolution, illustrating the power of the convolution formulation of the signal.