Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 014627, USA.
IEEE Trans Image Process. 2012 Feb;21(2):742-53. doi: 10.1109/TIP.2011.2165552. Epub 2011 Aug 22.
We introduce a novel algorithm to recover sparse and low-rank matrices from noisy and undersampled measurements. We pose the reconstruction as an optimization problem, where we minimize a linear combination of data consistency error, nonconvex spectral penalty, and nonconvex sparsity penalty. We majorize the nondifferentiable spectral and sparsity penalties in the criterion by quadratic expressions to realize an iterative three-step alternating minimization scheme. Since each of these steps can be evaluated either analytically or using fast schemes, we obtain a computationally efficient algorithm. We demonstrate the utility of the algorithm in the context of dynamic magnetic resonance imaging (MRI) reconstruction from sub-Nyquist sampled measurements. The results show a significant improvement in signal-to-noise ratio and image quality compared with classical dynamic imaging algorithms. We expect the proposed scheme to be useful in a range of applications including video restoration and multidimensional MRI.
我们提出了一种从噪声和欠采样测量中恢复稀疏低秩矩阵的新算法。我们将重建问题表述为一个优化问题，在该问题中，我们最小化数据一致性误差、非凸谱惩罚和非凸稀疏惩罚的线性组合。我们通过二次表达式将不可微的谱和稀疏惩罚项在准则中进行了-majorize 化，以实现一个迭代三步交替最小化方案。由于这些步骤中的每一步都可以通过解析或使用快速方案来评估，因此我们得到了一个计算效率高的算法。我们在亚奈奎斯特采样测量的动态磁共振成像 (MRI) 重建的背景下演示了该算法的实用性。结果表明，与经典的动态成像算法相比，该算法在信噪比和图像质量方面有了显著的提高。我们预计，所提出的方案将在包括视频恢复和多维 MRI 在内的一系列应用中非常有用。
