MRI TV-Rician Denoising - Biomedical Engineering Systems and Technologies - page 243

two-fold: to improve the DT image quality through accurate Rician denosing so
allowing shorter scanning time. The data we used consist of a DW-MR brain volume
provided by Fundacion CIEN-Fundacion Reina Sofıa which was acquired with a 3 Tesla
General Electric scanner equipped with an 8-channel coil. The DW images have been
obtained with a single-shot spin-eco EPI sequence (FOV=24cm, TR=9100, TE=88.9,
slice thickness=3mm, spacing=0.3, matrix size=128x128, NEX=2 ). The DW-MRI data
consists on a volume obtained with b=0/mm 2 and 15 volumes with b=1000s/mm 2 cor-
responding with the gradient directions specified in [14]. These DW-MR images, which
represent diffusion measurements along multiples directions, are denoised with the pro-
posed method previously to the Diffusion Tensorial Image reconstruction, which was
done with the 3d Slicer tools 2 . In Figure 6(a) we show a slice of the original DWI data
corresponding to the (1, 0, 0) gradient direction where the affecting noise is clearly vis-
ible. The complete DW-MRI data volume is denoised using the proposed method. The
Rician noise standard deviation ( σ ) has been estimated for each slice of each gradient
direction following [3], while we used a value of λ = σ/ 2 for the denoising. The slice
resulting from the denoising process is shown in Figure 6(b). It can be observed how in
the denoised images the noise has been removed but the details and the edges have been
fully preserved, as we should expect when the exact TV model is solved. The effect of
this denoising process over the reconstructed tensor and their derived scalar measure-
ments (obtained with the 3d Slicer tools) is presented in Figures 7 and 8. Figure 7 shows
a Fractional Anisotropy image where the structures and details are clearly enhanced if
the DW-MRI volume is denoised previously. When finer details are considered the de-
noising step is yet more crucial. For instance in Figure 8 the main eigenvector of the
tensor is represented, where the noise on the original DWI data cause inhomogeneities
(see Figure 8(a)) in the eigenvectors field which are product of the noise (Figure 8(b)).

6 Conclusions

In this notes we address the problem of the numerical computation of the solution of
the variational formulation of the Rician denoising model proposed in [4]. We deduce
a semi-implicit formulation for the gradient flow which leads to the resolution of ROF
like-problems at each step of the time discretization. This is accomplished efficiently
using a gradient descent for the dual variable associated to the primal ROF model.
While our study is preliminary it indicates how to obtain fast numerical solutions for
Rician denoising. This is specially interesting when Diffusion Weighted Images (DWI)
are considered for Diffusion Tensor Images reconstruction whereas they have poor res-
olution and low SNR which makes Rician denoising necessary.
Challenging mathematical issues arise about the existence, uniqueness and conver-
gence, when

0 , of weak bounded variation solutions of the quasilinear elliptic
equations considered in this paper (i.e. (8) and (12)) and the gradient flow analysis of
their parabolic counterpart ((10) and (13)) when t

→

. A rigorous justification of
the above arguments is desired. Nevertheless this approach is the mostly used regu-
larization technique to approximate and compute the minimizer of the total variation
energy and its variants (see [15]).

→ + ∞

2

Free available in http://www.slicer.org/