## 3 Experiments
### Evaluation Metrics  
To quantitatively compare the different methods, for the synthetic dataset, we used the peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and root mean square error (RMSE) as the evaluation metrics. Moreover, to evaluate the accuracy for the denoised fiber orientations, we also calculated the angular error map (AEM) and the corresponding average angular error (AAE), defined by:  
\[\begin{split} AEM_{i}=\frac{180}{\pi}\operatorname{acos}(\left| \mathit{dif}_{i}^{gt}\cdot\mathit{dif}_{i}^{gt}\right|)& i=1,2, \ldots,Num_{voxel}\\ AAE=\frac{1}{Num_{voxel}}\sum_{i=1}^{Num_{voxel}}AEM_{i} \end{split} \tag{13}\]  
where \(\mathit{dif}_{i}^{gt}\) and \(\mathit{dif}_{i}^{est}\) represent the ground-truth fiber orientation and estimated fiber orientation, respectively, \(|\cdot|\) means dot product, \(\operatorname{acos}(\cdot)\) indicates arccosine, and \(Num_{voxel}\) is the number of voxels of the region of interest (ROI).  
For the real dataset, since there is no ground-truth for the noise-free cardiac DW images, we estimated the residual map of the DW images. If the residual map is more randomly distributed without less structure information, the corresponding denoising result is better. In addition, we also calculated the non-reference-image quality metrics, including signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR), defined by:  
\[SNR=\frac{1}{num}\sum_{i=1}^{num}\frac{\left|\text{mean}(I^{f})- \text{mean}(I_{i}^{b})\right|}{\text{std}(I_{i}^{b})} \tag{14}\] \[CNR=\frac{1}{num}\sum_{i=1}^{num}\frac{\text{max}(I^{f})-\text{ min}(I^{f})}{\text{mean}(I^{f})\text{std}(I_{i}^{b})}\]  
where \(I^{f}\) represents the image intensity of the myocardium region; \(I_{i}^{b}\) indicates the image intensity of the \(i^{th}\) background region; \(num\) indicates the number of the selected background regions; and \(\text{max}(\cdot)\), \(\text{min}(\cdot)\), \(\text{mean}(\cdot)\), and \(\text{std}(\cdot)\) are the maximum, minimum, mean, and standard deviation of the image intensity in the region, respectively. SNR can reflect the level of noise removal, while CNR evaluates whether it can distinguish the interested region and background from the denoised image. The bigger CNR and SNR indicate the better denoised image quality. In this work, for the ex vivo pig hearts, \(num\) was selected as 1, which means that all regions except the heart were considered as background; for the in vivo human hearts, since the in vivo DW images contain other organs, it was impossible to consider all the regions other than the heart as the background. Accordingly, we selected four small regions with less structure information as the background (\(num=4\)).  
In addition to the DW images, we also computed the diffusion tensor images with the least-squares method that embedded in DIPY [36], from which the fractional anisotropy (FA), mean diffusivity (MD), and fiber orientations were derived to evaluate the denoising results in terms of diffusion metrics. FA and MD are defined by:  
\[FA=\sqrt{\frac{1}{2}}\frac{\sqrt{\left(\lambda_{1}-\lambda_{2} \right)^{2}+\left(\lambda_{1}-\lambda_{3}\right)^{2}+\left(\lambda_{2}-\lambda _{3}\right)^{2}}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}}}, \tag{15}\] \[MD=\frac{\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)}{3},\]  
where \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are eigenvalues of the diffusion tensor images. From the fiber orientations, the helix angle (HA) and transverse angle (TA) images were further calculated. The detailed definition can be seen in the work of [PERSON] et al. [37]. If there are no extreme values in the FA and MD maps and the residual maps of FA and MD do not contain structure information, or if the helix angles change smoothly from positive at the endocardium to negative at the epicardium and most of the transverse angles are close to zero, the denoising result is acceptable.