Introduction

A new improved CNN based denoising method was proposed in the last year ISMRM¹ in which weighted average of slice images are denoised using blind denoising CNN (BDnCNN)² followed by image separation using the inverse matrix calculation. To further improve the denoising performances, we studied a new denoising scheme in which negative weighting coefficients are used that have the effects of inversion of noise distribution of corresponding slice and varying the noise reduction effects of summed image. Proposed scheme was compared with previous proposed method using positive-weighting coefficients.

Method

Figure 1 illustrates the scheme of the parallelized blind image denoising (hereafter ParBID). The first step is the weighting average of slice images with given weight values. Let the k-th slice image be $$$r_k =\rho_k +\delta_k $$$ where $$$\rho_k $$$ is a noise-free image, and $$$\delta_k $$$ is the noise, then weighted average of slice images $$$i_s$$$ are written as follows:
$$i_s = \sum_{k} a_{s,k} r_k, … (1) $$
where $$$a_{s,k}$$$ mean weighting coefficients (hereafter wc). Equation (1) can be written as:
$${\bf I^T= A R^T}, … (2) $$
where $$${\bf R} =(r_1, r_2,..., r_n)$$$, $$${\bf I}=(i_1, i_2,..., i_m) $$$. The second step is blind denoising of the summed images, $$$d_s={\rm DnCNNB}(i_s)$$$ where BDnCNN refers to Blind DnCNN operation², and $$$d_s$$$ indicates the denoised image. The third step is the separation of weighted averaged images by solving generalized inverse problem:
$${\bf P^T= (\bf A^{T} \bf A)^{-1} \bf A^{T} \bf D^{T}}, ... (3)$$
where $$${\bf P} =(p_1, p_2, ..p_n)$$$ and $$${\rm \bf D} =(d_1, d_2, ..d_m)$$$. Image sequence $$$(p_1, p_2, ..p_n)$$$ is the output of ParBID. When negative value wc are used as shown in Fig.2(g) or (h), noise on the image $$$r_1 $$$ is inverted in calculating $$$ -0.3 r_1+1.3 r_2 $$$. Since the distribution of the noise on the summed image i2 (Fig(g)) is completely different from that of the additive summed image $$$i_1$$$ (Fig(e)), the degree of noise reduction in $$$i_2$$$ by blind DnCNN will be different from that of in $$$i_1$$$.
Let the noises on $$$r_1$$$ and $$$ r_2$$$ be $$$n_1$$$ and $$$n_2$$$ and noise reduction factor by BDnCNN be $$$\alpha_1$$$ and $$$\alpha_2$$$, respectively in 2-slice ParBID ($$$\alpha_1, \alpha_2 < 1$$$), the noise on the image $$$d_1, d_2$$$ are $$$\alpha_1(0.7 n_1+0.3 n_2), \alpha_2(-0.3 n_1+1.3 n_2) $$$ and noises on the separated image on $$$p_1 $$$ is calculated as $$$(0.91 \alpha_1+0.09 \alpha_2 ) n_1+0.39(\alpha_1-\alpha_2) n_2 $$$. Generally, denoising of lower S/N image will bring strong denoising effects; i.e. $$$\alpha_2 <\alpha_1 $$$, therefore $$$(0.91 \alpha_1+0.09 \alpha_2 ) n_1 $$$ is smaller than $$$\alpha_1 n_1 $$$ which is the remained noise on $$$p_1 $$$ in ParBID with positive wc. As indicated from a simple estimate, negative wc improve the noise reduction effect.
Generally, denoising of lower S/N image will bring image degradation, however the image (g) is a difference image between $$$r_1$$$ and $$$r_2$$$ and is a kind of edge enhanced image, edges are well remained and therefore image degradation will be suppressed, resulting in improved PSNR of resolved image (m).

Results & Discussions

In 2-slice ParBID, circulant matrix {{0.6,0.4},{0.4,0.6}} was used in previous method¹, whereas asymmetric matrix $$$A_2$$$= {{0.7,0.3},{-0.3,1.3}} was used in this study. The determinant of $$$A_2$$$ is 1 and its inverse matrix is {{1.3,-0.3},{0.3,0.7}}. In the 3-slice ParBID, circulant matrix {{0.5,0.3,0.2},{0.2,0.5,0.3},{0.3,0.2,0.5}} was used in previous studies. In our study, examination were done using $$$A_3$$$={{0.3,-0.8,1.0},{0.2,0.6,0.2},{1.0,-0.8,0.3}} which coefficients were determined by preliminary study. Figure 3(a),(b),(c) show the relationship between PSNR and the number of images used for ParBID with $$$A_3$$$ for noise level 2.5%, 5.0% and 7.5%, respectively. The size of images and slice spacing were 256×256 pixels, 1.2 mm. Results show that the improvement of PSNR using circulant wc matrix (hereafter circ) were small when SN was low in 3-slice ParBID. In contrast, proposed method using asymmetric (hereafter asym) wc matrix show significant increase. It should be noted that improvement of PSNR was obtained only in $$$p_1$$$ in 2-slice ParBID and $$$p_2$$$ in 3-slice ParBID using asymmetric wc.
Figure 3 also shows that 2- and 3-slice parallelized image denoising were also effective for WNNM.
Examples of 5.0% noise level are shown in Fig.4. Denoised images in Single BDnCNN, 2-slice ParBID, 3-slice ParBID using circ and asym wc are shown. As shown in the region pointed by arrows in Fig.(r), fine linear structures were well preserved in proposed 3-slice ParBID-asym. In addition, 3-slice ParBID has excellent weak contrast preservation. 　Application of ParBID to an experimentally obtained MR image (256×256, slice spacing 1.2 mm) is shown in Fig.5. Subimage (a) is the target noisy image, (b) through (e) are denoised image by single slice BDnCNN, 2-slice ParBID-asym, and 3-slice ParBID-asym and 3-slice WNNM-asym respectively. Denoised images (d) and (i) obtained by 3-slice ParBID clearly retain image contrast, as shown in the region indicated by arrows. These images indicate that details of images are preserved to a much higher degree in 3-slice ParBID.

Conclusion

Asymmetric weighting coefficients that contain negative value can further improve the denoising performance of the parallelized blind image denoising.

Acknowledgements

This study was supported in part by JSPS KAKENHI(19K04423) and the Kayamori Foundation of Informational Science Advancement. We would like to thank Canon Medical Systems.