  This paper presents novel single and multi-shell sampling schemes for
diffusion MRI. In diffusion MRI, it is paramount that the number of samples is
as small as possible in order that scan times are practical in a clinical
setting. The proposed schemes use an efficient number of measurements in that
the number of samples is equal to the degrees of freedom in the orthonormal
bases used for reconstruction. Novel reconstruction algorithms based on smaller
subsystems of linear equations, as compared to the standard regularized
least-squares method, are developed for both single and multi-shells sampling
schemes. The smaller matrices used in these novel reconstruction algorithms are
designed to be well-conditioned, leading to improved reconstruction accuracy.
Accurate and robust reconstruction is also achieved through incorporation of
regularization into the novel reconstruction algorithms and using a Rician or
non-central Chi noise model. We quantitatively validate our single and
multi-shell schemes against standard least-squares reconstruction methods to
show that they enable more accurate reconstruction when the number of samples
is equal to the degrees of freedom in the basis. Human brain data is also used
to qualitatively evaluate reconstruction
