  The present paper introduces a method for substantial reduction of the number
of diffusion encoding gradients required for reliable reconstruction of HARDI
signals. The method exploits the theory of compressed sensing (CS), which
establishes conditions on which a signal of interest can be recovered from its
under-sampled measurements, provided that the signal admits a sparse
representation in the domain of a linear transform. In the case at hand, the
latter is defined to be spherical ridgelet transformation, which excels in
sparsifying HARDI signals. What makes the resulting reconstruction procedure
even more accurate is a combination of the sparsity constraints in the
diffusion domain with additional constraints imposed on the estimated diffusion
field in the spatial domain. Accordingly, the present paper describes a novel
way to combine the diffusion- and spatial-domain constraints to achieve a
maximal reduction in the number of diffusion measurements, while sacrificing
little in terms of reconstruction accuracy. Finally, details are provided on a
particularly efficient numerical scheme which can be used to solve the
aforementioned reconstruction problem by means of standard and readily
available estimation tools. The paper is concluded with experimental results
which support the practical value of the proposed reconstruction methodology.
