ying fu^{1}, xi wu^{1}, yangzhi peng^{2}, and jiliu zhou^{1}

Super-resolution (SR) of diffusion weighted imaging (DWI) data is an ill-posed problem, which can be regularized by exploiting diverse priors learned from image patches. In this work, based on patch-based strategy of SR, we propose a new regularization method to reconstruct DW images, which integrates the sparse representation prior with dictionary learned from external image patches and non-local self-similarity prior learned from internal image patches. Meanwhile, in dictionary learning part, nonparametric Bayesian method is adopted to infer dictionary learning variables such as the size of the dictionary from data automatically. Experimental results demonstrate that the proposed method outperforms current methods in DWI reconstruction.

To solve SR problem of DW images with regularization term, the HR image x can be estimated from LR observation y as follows:

\[\hat{\mathbf{x}}={\rm arg} \min\limits_{\mathbf{x}}\; \{||\mathbf{y}-DH\mathbf{x}||+\lambda R(\mathbf{x})\}\]

where \(||\mathbf{y}-DH\mathbf{x}||\) is is fidelity term, \(R(\mathbf{x})\) is the regularization term and \(\lambda\) is the tune parameters to balance them.

A. External patch-based Regularization

One regularizer of our method enforces that patches have a sparse representation according to a dictionary, and corresponding regularization function is

\[R_(x):=\sum_i{\frac{1}{2}||R_i{x}-D\alpha_i||_2^2+\mu||\alpha_i||_1}\]

We consider a Bayesian nonparametric method called beta process factor analysis (BPFA) 3 driven by an underlying Poisson process, than can infer parameters such as the dictionary size and the sparsity level . And we use variational Bayesian inference to update BPFA variables. And then approximately optimize the objective function is:

\[{\rm arg} \min\limits_{\alpha_i}\;{\sum_i{\frac{1}{2}||R_i{x}-D\alpha_i||_2^2}}\]

subject to \(||\alpha_i||_1<=T\quad \forall i\)

B. Internal patch-based regularization

The other regularizer of our method enforces non-local self-similarity approach:

\[R_2(x):=\sum_i{||\sum_j{x_i-w(x_i,x_j)x_j}||^2}\]

Where \(w(x_i,x_j)\) is the weight assigned to \(x_j \) . This weight depends on the similarity between the neighborhood of \(x_i \)and \(\sum_j{w(x_i,x_j)=1}\).

Lastly, we use the alternating direction method of multipliers (ADMM) for optimization.

**Result**

channels. A DW dual spin-echo, SENSE accelerated msh-EPI was used to acquire the DWI data (b-value: 700 s/mm2; 15 diffusion directions); FOV = 210 × 30 × 21 mm3; matrix size = 300 × 300 with 15 slices and a spatial resolution of 0.7 × 0.7 × 2 mm3. To train, the high-resolution DW images in ten diffusion orientation were used for the dictionary construction step. The patch-size and the truncation level in beta process are set to 8 × 8 and 256, respectively. The coupled learned dictionary is shown as Fig.1. To test, the high-resolution DW images in remaining five diffusion orientations were downsampled by a factor of two using the nearest neighbor interpolation along each axis (i.e., [2 2 2]), which resulted in simulated LR images of 1.4 × 1.4 × 4 mm3 as test datasets. Fig.2 presents one slice of a high-resolution DW reconstruction obtained in a specific diffusion orientation, compared with Bicubic interpolated method (Fig.2b), BPFA method (Fig.2c), and the proposed method (Fig.2d). Finally, each reconstruction result is compared with the original high- resolution image using the PSNR and the SSIM showed by Fig.3 and Fig.4, respectively.

**Conclusion**

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Fig.1.Dictionary trained by BPFA (Left) High-resolution Dictionary, (Right) Low-resolution Dictionary.

Fig.2.Tests of diffusion weighted image reconstruction obtained for different methods. (a) The gold standard. (b) Bicubic interpolated version of the low-resolution DW image. (c) The reconstruction using BPFA method. (d) The reconstruction using our proposed method.

Fig.3.Comparison of PSNR values achieved with a standard bicubic interpolation (red bars) , the BPFA reconstruction (green bars) and the proposed super-resolution reconstruction (blue bars)

Fig.4.Comparison of SSIM values achieved with a standard bicubic interpolation (red bars) , the BPFA reconstruction (green bars) and the proposed super-resolution reconstruction (blue bars)