Algorithm 1.

Super-Resolution Reconstruction of Diffusion Weighted Images

Input: Low-resolution 4D diffusion-weighted image T;

Initialize the desired high-resolution image X(0) by upsampling T with nearest neighbor interpolation. Set redundant variables $\begin{array}{ccc}{M}_i^{\left(0\right)}=0,& U_i^{\left(0\right)}=0& i=1,2,3,4\end{array}$;

For each iteration k,

Update Xk by using gradient descent:

$\text{arg}\underset{X}{min}{\sum }_{n=1}^N{\Vert DS{X}_n^{\left(k-1\right)}-T_n\Vert }^2+{\sum }_{i=1}^4\frac{\rho }{2}{\Vert X^{\left(k-1\right)}-M_i^{\left(k-1\right)}+U_i^{\left(k-1\right)}\Vert }^2+{\lambda }_{t\nu }{\sum }_{n=1}^N\int |\nabla X_n^{\left(k-1\right)}|\mathit{dxdydz}$ (7)

Update $M_i^{\left(k\right)}$ by using Singular Value Thresholding (SVT) [18]:

$\begin{array}{c}{M}_i^{\left(k\right)}=\mathit{fol}{d}_i\left[SV{T}_{{\lambda }_{\mathit{rank}}{\alpha }_i/\rho }\left(X_{\left(i\right)}^{\left(k\right)}+U_{i\left(i\right)}^{\left(k-1\right)}\right)\right]\\ \text{with}\mathit{fol}{d}_i\left(M_{i\left(i\right)}\right)=M_i\end{array}$ (8)

$\text{Update}{U}_i^{\left(k\right)}\text{by:}{U}_i^{\left(k\right)}=U_i^{\left(k-1\right)}+\left(X^{\left(k\right)}-M_i^{\left(k\right)}\right)$ (9)

Until difference between iterations ||Xk − Xk−1||/||T|| ≤ ε;

End

Output: Reconstructed high-resolution 4D diffusion-weighted image X.