MRI-Denoising Algorithm

The proposed iterative denoising algorithm requires an initial estimate. We obtain an initial estimate entirely based on the knowledge of the noise model, without any use of Markov prior. Thus, the initialization is a ML estimate of the image. The MRI-denoising algorithm finally produces the MAP image estimate as follows:

1. 
   Infer the prior PDF $P ({\bf Z})$ (as described in Section 5.3) by minimizing the KL divergence, using the EM algorithm, between the observed corrupted-signal Markov PDF and its estimate derived from the prior-PDF model. The prior PDF is represented by a Parzen-window sum of isotropic Gaussian kernels with means $\{ {\bf z}_u \}_{u \in \mathcal{U}}$ and standard deviation $\sigma$.
   
2. 
   Obtain an initial denoised ML image ${\bf\hat x}^0 = \{ \hat x_t^0 \}_{t \in \mathcal{T}}$:
   

   $$\forall t \in \mathcal{T}, \hat x_t^0 = \mathop{\mbox{argmax }}_{x_t} P ( \tilde x_t \vert x_t ).$$ (134)

   
   
   We compute the mode of each likelihood PDF numerically using the iterative mode-seeking mean-shift procedure [60,57].
   
3. 
   Given the denoised-image estimate ${\bf\hat x}^m$ at iteration $m$, obtain the next estimate ${\bf\hat x}^{m+1}$ as
   

   $$\forall t \in \mathcal{T}, \hat x_t^{m+1} = \hat x_t^{m}$$

   $$+ \frac { \sum_{u \in \mathcal{U}} G (\hat {\bf y}_t^m - {\bf y}_... ...cal{U}} G (\hat {\bf y}_t^m - {\bf y}_u, \sigma) G (\hat x_t^m - x_u, \sigma) }$$

   $$- \frac {\hat x_t^m} {\sigma^2} + \frac {\tilde x_t} {\sigma^2} \... ...\tilde x_t \hat x_t^m / \sigma^2) } { I_0 (\tilde x_t \hat x_t^m / \sigma^2) },$$ (135)

   
   
   where all the symbols have the same meaning as in Section 5.4.
   
4. 
   If $\parallel {\bf\hat x}^{m+1} - {\bf\hat x}^m \parallel_2 < \epsilon$, where $\epsilon$ is small threshold, then stop, otherwise go to Step 3.