The complex DWIs have in all cases been created by generating a synthetic phase image, Ф^,, associated with a magnitude image, M^. The phase images are created in order to mimic the outcome of subject movements. We assume a bi-dimensional sinusoidal wave oriented along the direction v = (v_{x}, v_{y}) with frequencies f_{x}, f_{y} and initial shifts ф_{х}, ф_{у}

where w_{x}, w_{y} are scale parameters: in this case they correspond to the width of the image along the corresponding direction (w_{x} = card.X), w_{y} = card(Y)). Eventually, constant phase patches are added. Assuming to have the ground-truth images of magnitude M_{xy} and phase Ф_{ху}, the latter resulting from Eq. (4), then

where p^{r}xy,p^{[1]}_{xy} 2 N.0, a^{2}). The noise is added with a value of a calculated according to the DW-MRI convention a = (^card[p(X x Y)]^{_1}P_{xy} p(x, y/M^f^ /SNR_{0},

where SNR_{0} is defined on the magnitude image without diffusion weighting М^{ъ}=°, and p 2{0,1} is a mask defined on the pairs (x, y), e.g., a mask of the tissue-related signal like the brain mask. The Rician magnitude |DWI|_{xy} and the phase /DWI_{xy }are calculated from the real and imaginary parts in Eq. (5).

The data used for the experiments is a HCP brain dataset corrected for eddy currents where we selected DWIs of interest for b 2 [0,1000, 3000] s/mm^{2}. Other experiments use Phantomas [ 15] to obtain the ground-truth magnitude images, M_{xy}. This software requires input with a geometrical description of tissue structures and fiber bundles. We used the well known geometry produced for the HARDI reconstruction challenge 2013.^{[1]} We generated DWIs for a 3-shells scheme with b 2 {1000,2000,3000}s/mm^{2}, 51 samples per shell, with samples uniformly distributed within and among shells [16].

The phase-corrected real DWIs can contain negative values: the noise is zero- mean Gaussian and the noise floor is absent. Therefore, the DTI reconstruction is performed by non-linearly en forcing signal positivity, and MAP is performed with Laplacian regularization imposing positivity on the recovered Ensemble Average Propagator [17].