  This paper proposes a multi-shell sampling scheme and corresponding
transforms for the accurate reconstruction of the diffusion signal in diffusion
MRI by expansion in the spherical polar Fourier (SPF) basis. The sampling
scheme uses an optimal number of samples, equal to the degrees of freedom of
the band-limited diffusion signal in the SPF domain, and allows for
computationally efficient reconstruction. We use synthetic data sets to
demonstrate that the proposed scheme allows for greater reconstruction accuracy
of the diffusion signal than the multi-shell sampling schemes obtained using
the generalised electrostatic energy minimisation (gEEM) method used in the
Human Connectome Project. We also demonstrate that the proposed sampling scheme
allows for increased angular discrimination and improved rotational invariance
of reconstruction accuracy than the gEEM schemes.
