MRI TV-Rician Denoising - Biomedical Engineering Systems and Technologies - page 238

(a) Original phantom (b) Noisy for σ =0 . 025 (c) Noisy for σ =0 . 05 (d) Noisy for σ =0 . 1

(e) Original phantom (f) Denoised for σ =0 . 025 ,

λ =0 . 025

(g) Denoised for σ =0 . 05 ,
λ =0 . 075

(h) Denoised for σ =0 . 1 ,
λ =0 . 125

Fig. 1. The original free noise phantom is shown in images a) and e). In b), c) and d) the con-
taminated phantoms for σ =0 . 025 , 0 . 05 and 0 . 1 respectively. Below, their respective denoised
images e), f) and g) for λ =0 . 025 , 0 . 075 and 0 . 125 .

in fact observed in Figure 2, where using the same values for the algorithms ( τ =10 − 3
and λ =0 . 1 ) the proposed method reach a solution whose energy is smaller than the
obtained by the solution of the first method. This difference caused by the fact that now
we are using the true Total Variation can be also observed in the images of the absolute
difference between the original (free of noise) image and the solutions found by the
two methods. We can see how the image difference corresponding to the solution of
the approximated method (Figure 3 a) presents more structural details than the image
corresponding to the implicit method (Figure 3 b), which confirms that this last method
recovers more structural details, that are eventually lost by the explicit method because
of the approximation.
The other important characteristic of this new formulation is that the diffusion term
is implicitly considered and this provides numerical stability which in turn allows to
increase the value of τ compared to those used in the explicit method, so less iterations
of the algorithm are necessary for time stabilization. In fact if we increase the value of
τ to the value τ =10 − 1 the explicit method becomes unstable and it begins to oscillate
without reaching the minimum of the energy we obtained with τ =10 − 3 .Alsothe
implicit method takes less iterations to reach the same minimum. The performance
of the two algorithms for τ =10 − 1 can be observed in Figure 4 where the energy
computed along the iterates of the implicit method is clearly less than the same energy
calculated along the approximated iterates.
This behaviour is crucial for the selection of the algorithm in so far even if the new
method has more computational cost per iteration (because we solve a ROF problem
at each iteration), we can increase the value of τ in order to reach the solution in less
iterations than the first method, finding a best solution for our problem (in the sense
of figure 4) and spending less time of computation. Finally, in the last figure(figure 5)
we show that this framework is robust in the sense that the same solution is obtained