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 re-
constructed tensor and their derived scalar measure-
ments (obtained with the 3d Slicer tools) is presented
in Figures 7 and 8. Figure 7 shows a color-coded
orientation map created from DTI data. In this im-
age, the principal colors (red, green, and blue) rep-
resent ﬁbers running along the spatial orientations
(x,y, z). Results in 7 shows that the structures are
better deﬁned if the DW-MRI volume is denoised
previously. As evidenced by Figure 8 this effect is
yet more visible in the measurements like the Frac-
tional Anisotropy where the structures and details are
clearly enhanced. When we use a lower value for λ
(Figures 7(c) and 8(c)) we obtain smoother tensorial
images but some details can be better distinguished
when the value of λ is higher (Figures 7(b) and 8(b)).
6 CONCLUSIONS
In this notes we address the problem of the numeri-
cal computation of the solution of the variational for-
mulation of the Rician denoising model proposed in
(Mart
´
ın et al., 2011). We deduce a semi-implicit for-
mulation for the gradient ﬂow which leads to the res-
olution 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 prelim-
inary it indicates how to obtain fast numerical solu-
tions for Rician denoising. This is specially inter-
esting when Diffusion Wheighted Images (DWI) are
considered for Diffusion Tensor Images reconstruc-
tion whereas they have poor resolution and low SNR
which makes Rician denoising necessary.
Challenging mathematical issues arise about the
existence, uniqueness and convergence, when ε → 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 rigor-
ous 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 (Casas et al., 1998)).
The semi-implicit method we propose is well
founded mathematically when the time-discretized
problems are dealt with and it represents a feasible al-
ternative to gaussian denoising for low SNR MR im-
ages. Further study is undoubtedly necessary in order
to make automatic the choice of the parameters in real
medical images. Other possibilities, such as Inverse
scaling, which makes the parameter estimation less
crucial and provide contrast enhanced images shall
also be explored.
ACKNOWLEDGEMENTS
This work was supported by project TEC2009-14587-
C03-03 of the Spanish Ministry of Science. Also we
thank to Mrs. Eva Alfayate, MR-scanner technician
of the Fundaci
´
on Reina Sof
´
ıa, for her professional and
kindly collaboration.
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