SIAM Journal on Imaging Sciences, Volume 13, Issue 3, Page 1272-1301, January 2020.
Recently, the finite-rate-of-innovation (FRI) based continuous domain regularization is emerging as an alternative to the conventional on-the-grid sparse regularization for compressed sensing (CS) due to its ability to alleviate the basis mismatch between the true support of the shape in the continuous domain and the discrete grid. In this paper, we propose a new off-the-grid regularization for the CS-MRI reconstruction. Following the recent works on two dimensional FRI, we assume that the discontinuities/edges of the image are localized in the zero level set of a band-limited periodic function. This assumption induces the linear dependencies among the Fourier samples of the gradient of the image, which leads to a low rank twofold Hankel matrix. We further observe that the singular value decomposition of a low rank Hankel matrix corresponds to an adaptive tight frame system which can represent the image with sparse canonical coefficients. Based on this observation, we propose a data driven tight frame based off-the-grid regularization model for the CS-MRI reconstruction. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm with a guaranteed global convergence is adopted. Finally, the numerical experiments show that our proposed data driven tight frame based approach outperforms the existing approaches.

中文翻译：

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数据驱动的紧密框架，用于通过离网正则化进行压缩感知MRI重建

SIAM影像科学杂志，第13卷，第3期，第1272-1301页，2020年1月。
最近，基于有限创新率（FRI）的连续域正则化正在出现，可替代传统的基于网格的稀疏正则化压缩感知（CS），因为它具有缓解真实数据之间基本不匹配的能力。在连续域和离散网格中支持形状。在本文中，我们为CS-MRI重建提出了一种新的离网正则化方法。根据二维FRI的最新工作，我们假设图像的不连续性/边缘位于带限周期函数的零级集中。该假设在图像的梯度的傅立叶样本之间引起线性相关性，从而导致低秩双重汉克尔矩阵。我们进一步观察到，低秩汉克尔矩阵的奇异值分解对应于自适应紧框架系统，该系统可以表示具有稀疏规范系数的图像。基于此观察，我们为CS-MRI重建提出了一种基于数据驱动紧密框架的离网正则化模型。为了解决非凸和非光滑模型，采用了具有保证全局收敛性的近端交替最小化算法。最后，数值实验表明，我们提出的基于数据驱动的基于紧框架的方法优于现有方法。为了解决非凸和非光滑模型，采用了具有保证全局收敛性的近端交替最小化算法。最后，数值实验表明，我们提出的基于数据驱动的基于紧框架的方法优于现有方法。为了解决非凸和非光滑模型，采用了具有保证全局收敛性的近端交替最小化算法。最后，数值实验表明，我们提出的基于数据驱动的基于紧框架的方法优于现有方法。