MRI TV-Rician Denoising - Biomedical Engineering Systems and Technologies - page 237

TV-Rician Denoising Algorithm

- Initialization: given λ , σ , τ 1 , τ 2 , 1 , 2 and f .Set:
• u 0 ≡ 0

β =( τ 1 λ ) /σ 2

•

• γ =(1+ β ) /τ 1

u k u k− 1 ∞ > 1 :

- While −

1. g i,j = 1

u i,j + β r ( u i,j ,f i,j ) f i,j

1+ β 1+ β

2. Set p 0 ≡ 0 . While p n − p n− 1 ∞ > 2 :

= p i,j + τ 2 (

( div p n g k /γ )) i,j ∇ −

p n +1
i,j

•

1+ τ 2 | ( ∇ ( div p n − g k /γ )) i,j |

3. u k +1
i,j

= g i,j − γ ( div p ) i,j

5 Results and Discussion

The theoretical result presented in the previous section have to be numerically con-
firmed in order to asses the well behaviour of the method and also the advantages it
presents when it is compared to the original regularized method which computes the
approximating u solution. In order to assess the performance of our algorithm we
tested it with synthetic and real brain images. The obtained results are presented and
discussed below.

Synthetic Brain Images
The synthetic brain images we used for our study were obtained from the BrainWeb
Simulated Brain Database 1 at the Montreal Neurological Institute [13]. The original
phantoms were contaminated artificially with Rician noise considering the data as a
complex image with zero imaginary part and adding random gaussian perturbations to
both the real and imaginary part, before computing the magnitude image. This process
allows to control the amount and distribution of the Rician noise so providing a gold
standard for our study. For this we used different values of the σ parameter which
represents the variance of the noise ( σ =0 . 025 , σ =0 . 05 and σ =0 . 1 ) and different
values of the λ parameter ( λ =0 . 05 , λ =0 . 1 and λ =0 . 125 ). Notice that, fixed σ
(which can be estimated for real images) the λ parameter is the only one we have to
choose for regularization (as in the gaussian case).
We can observe in Figure 1 the performance of the denoising method based on the
semi-implicit formulation for λ =0 . 05 , λ =0 . 1 and λ =0 . 125 . This implicit method
solves exactly the total variation operator in (6) due to its dual formulation and not its
approximate form as the explicit method which solves the primal formulation, so the
solution obtained should be close to the ideal minimum of (6). This behaviour can be

1

Available at http://www.bic.mni.mcgill.ca/brainweb