## 2 Methods  
### The Overall Structure of the Node2 Node Network  
In DTI, the DW images are acquired from the multiple diffusion gradient directions which are uniformly sampled from the q-space. In view of q-space, each diffusion direction can be taken as a node and the similarity between the different directions can be considered the adjacent matrix between nodes. Accordingly, the DW images along different directions can be expressed in a graph, with the DW signals (reshaped DW image volume along one direction into a vector) being the node features. Considering that the DW images acquired from the adjacent diffusion gradient directions are similar, and the noise in DW images along different directions has the same distribution, this work proposes a novel self-supervised cardiac DW image denoising network, Node2 Node. Its architecture is illustrated in Figure 1, consisting of a noisy-image pair matching block realized with graph framelet transform, and a denoising block based on ResNet. The detailed process of the noisy DW image pair matching is elaborated as follows. Firstly, according to the diffusion gradient directions \(\vec{q}\), we construct a q-space graph. Considering that in the cardiac DTI, all the DW images are acquired with the same \(b\) value, accordingly, during the q-space graph construction, only the diffusion directions are used. Specifically, each direction is considered a graph node, and the adjacent coefficient between two nodes is defined as:  
\[c_{i,j}=\exp(-\frac{1-(\vec{q}_{i}^{T}\vec{q}_{j})^{2}}{2\sigma_{q}^{2}})\quad i,j=1,2,\ldots,N \tag{1}\]where \(\vec{q}_{i}\) and \(\vec{q}_{j}\) represent the \(i_{th}\) and \(j_{th}\) diffusion directions, \(\sigma_{q}\) indicates a hyperparameter to control the adjacent coefficient \(c_{i,j}\) between \(\vec{q}_{i}\) and \(\vec{q}_{j}\), and \(N\) is the total number of diffusion directions. Based on \(c_{i,j}\), the adjacent matrix \(A\) and degree matrix \(D\) of the q-space graph can be derived,  
\[ \begin{split} A_{i,j}=c_{i,j}&\quad i,j=1,2,\ldots,N\\ D_{i}=\sum_{j=1,j\
eq i}^{n}c_{i,j}.\end{split}  \tag{2}\]  
Graph framelet transform is a filter method that is implemented on the graph spectral domain for dividing the signal into different sub-bands, which allows us to more accurately evaluate the feature similarity between the nodes in a band-to-band manner. In this work, GFT is implemented on the q-space graph \(G=(V,E)\). In other words, in graph \(G\), the vertex \(V\) represents the features of nodes. In this work, they are the DW images or signals acquired from a certain direction, and \(E\) indicates the edge between the nodes which is determined by the values of the adjacent matrix. According to the convolution theory, the convolution (filter) in spatial space corresponds to the point-wise product in spectral space. Assuming that the filter in spatial space is \(h\), and the spatial signal is \(s\), the filtering result can be expressed as:  
\[h*s=\mathcal{FFT}^{-1}(\mathcal{FFT}(h)\odot\mathcal{FFT}(s)) \tag{3}\]  
where \(\mathcal{FFT}\) and \(\mathcal{FFT}^{-1}\) represent the Fourier and inverse Fourier transform, respectively, \(\odot\) indicates the point-wise product. Since GFT is performed on the graph in the spectral domain, we should derive the Fourier and inverse Fourier transform of the graph signal. The Fourier transform basis of a graph \(G\) is defined as the eigenvectors of the Laplacian matrix \(L\) of graph \(G\). Given the adjacent matrix and degree matrix, the Laplacian matrix of a graph is derived by \(L=D-A\). Through the eigen-decomposition of matrix \(L\), the pairs  
Figure 1: The overall structure of the Node2 Node, consisting of an image pair matching block and a denoising block. Image pair matching is realized by maximizing the similarity between the GFT coefficients of graph nodes, and denoising is implemented with a ResNet-like structure. of eigenvalues and eigenvectors of \(L\) are derived, denoted as \(\{\lambda_{k},u_{k}\}_{k=0}^{N-1}\).