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摘要: 从欠采样数据中进行磁共振成像（简称MRI）重建一直是一项具有挑战性和吸引力的任务，因为这是一个病态问题，且伴随着压缩感知理论会具有重要的意义.基于压缩感知的多数先进稀疏表示方法是将图像分割成重叠的图像块，然后每个图像块分开处理.然而，这些方法丢失了信号的重要空间结构，且忽略了对MR图像有强大约束的像素一致性.在文章中我们提出了一种新型的重建方法，这种方法是将最近提出的卷积稀疏编码与梯度域结合起来用于解决上述所提到的问题.不同于基于图像块状的方法，本文提出的算法是直接在整个梯度域图像中获取相邻局部的相关性，利用梯度域图像的全局相关性产生更好的梯度域图像边缘，锐利特征.提出的算法也能够高效的获取暗含在梯度域图像中的局部特征.与对比算法相比，大量的实验结果表明本文算法具有更好的质量重建，且在不同采样方案，不同K-空间加速因子情况下具有更快速的收敛.Abstract: Magnetic resonance imaging (MRI) reconstruction from undersampled data has always been a challenging and fascinating task due to its implicit ill-posed nature and its significance accompanied with the emerging compressed sensing (CS) theory. Most state-of-the-art sparse representation based CS approaches partition the image into overlapped patches, and process each patch separately. These methods, however, lose important spatial structures of the signal of interest, and ignore the consistency of pixels, which is a strong constraint for MR image. In this paper, we propose a new reconstruction method, which builds on recently introduced ideas of convolutional sparse coding in gradient domain (GradCSC) to address above mentioned issue. As opposed to patch-based methods, GradCSC directly operates on the whole gradient image to capture the correlation between local neighborhoods and exploits the gradient image global correlation to produce better edges and sharp features of gradient image. It enables local features implicitly existed in the gradient images to be captured effectively. Extensive experimental results demonstrate that the proposed algorithm achieves higher quality reconstructions than alternative methods and converges quickly at various sampling schemes and k-space acceleration factors.
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Key words:
- Alternating direction method of multipliers /
- convolutional sparse coding(CSC) /
- filter learning /
- gradient image /
- magnetic resonance imaging(MRI)reconstruction
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Fig. 1 One illustration of filters learned. (a) Learned dictionary by DLMRI, (b) Learned dictionary by GradDL, and (c) Learned filter by GradCSC, respectively.
Fig. 2 The reconstruction results of the Lumbar spine image under different undersampling factors with 2D random sampling.
Fig. 3 The reconstruction results of the axial T2-weighted brain image under different undersampling schemes.
Algorithm 1. The GradCSC algorithm 1: Initialization: ${{z}_{k}}^{0}=0$, ${{d}_{k}}^{0}=0$, ${{({{b}^{(i)}})}^{0}}=0$, $i=1, 2$; ${{u}^{0}}=F_{p}^{T}f$
2: For $j=1, 2, \ldots $ repeat until a stop-criterion is satisfied
3: ${{({{w}^{(i)}})}^{j+1}}=\ \frac{{{\nu }_{2}}[{{({{b}^{(i)}})}^{j}}+{{({{\nabla }^{(i)}}u)}^{j}}]+\sum\limits_{k=1}^{K}{d_{_{k}}^{j}*z_{_{k}}^{j}}}{2{{\nu }_{2}}+1}, \ \ i=1, 2$
4: Updating $\{{{d}_{k}}^{j+1}, {{z}_{k}}^{j+1}\}$ from difference images ${{({{w}^{(i)}})}^{j+1}}$ by (13), $i=1, 2$
5: Updating $\{ r_{1}^{^{\ell +1}}, r_{2}^{^{\ell +1}}, r_{3}^{^{\ell +1}} \}$ from the coefficients, and the filters by (14)
6: ${{u}^{j+1}}={{F}^{\rm{-}1}}( \frac{F[{{\nu }_{1}}F_{p}^{T}f+{{\nu }_{2}}{{\nabla }^{T}}({{({{w}^{(i)}})}^{j}}^{+1}-{{({{b}^{(i)}})}^{j}})]}{{{\nu }_{1}}FF_{p}^{T}{{F}_{p}}{{F}^{T}}+{{\nu }_{2}}F{{\nabla }^{T}}{{F}^{T}}F\nabla {{F}^{T}}} )$
7: ${{({{b}^{(i)}})}^{j+1}}={{({{b}^{(i)}})}^{j}}+{{({{\nabla }^{(i)}}u)}^{j+1}}-{{({{w}^{(i)}})}^{j+1}}$, $i=1, 2$
8: End
9: Output ${{u}^{j+1}}$
下载: 导出CSV
Table Ⅰ Reconstruction PSNR Values at Different Undersampling Factors With the Same 2D Random Sampling Trajectories
Undersampling factors 2.5-fold 4-fold 6-fold 8-fold 10-fold 20-fold DLMRI 35.92 35.19 31.26 30.76 30.13 26.39 GradDL 38.26 38.20 34.35 32.89 31.81 27.90 GradCSC 37.39 37.53 35.23 35.45 34.62 30.54
下载: 导出CSV
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