In this section we describe the implemented phase correction procedure, and illustrate the generation of the data used for the experiments, such as the acquisition setup, the generation of a synthetic phase, and the SNR convention.

The phase correction takes into account a complex DWI

where x and y represent the pixel coordinates, r and i indicate the real and imaginary parts, and j is the imaginary unit. If /DWI_{xy} is a good estimation of the phase, then the phase-corrected image is obtained via complex rotation

where /DWI_{xy} and |DWI|_{xy} are the original noisy phase and magnitude. The real part of the phase-corrected complex DWI, <(DWI^y), contains the signal (tissue contrast) plus Gaussian distributed noise, whereas the imaginary part, 3(DWI?), only contains noise. Henceforth, any classical diffusion modeling and reconstruction taking into account additive Gaussian noise can be performed on <(DWI^y), where the noise floor is absent.

The effectiveness of phase correction clearly depends on the quality of the phase estimation. In this work we implement a total variation method, known to better preserve discontinuities in the images [14]. Particularly, for each complex DWI image u_{0} 2 rDWI_{xy}, iDWI_{xy} defined on coordinates x 2 X, y 2 Y, we find the image u such that it is the minimizer of

where X is the regularization parameter expressing the attachment to data. The estimates of rDWIxy and iDWI^, obtained with Eq. (3) are then used to compute /DWI_{xy} and perform the complex rotation in Eq. (2).