Background and Mathematical Analysis of Diffusion MRI Methods.

The addition of a pair of magnetic field gradient pulses had initially provided the measurement of spin motion with nuclear magnetic resonance (NMR) techniques. In the adaptation of DW-NMR techniques to magnetic resonance imaging (MRI), the taxonomy of mathematical models is divided in two categories: model matching and spectral methods. In this review, the methods are summarized starting from early diffusion weighted (DW) NMR models followed up with their adaptation to DW MRI. Finally, a newly introduced Fourier analysis based unifying theory, so-called Complete Fourier Direct MRI, is included to explain the mechanisms of existing methods.