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Isotropic and anisotropic filtering norm-minimization: A generalization of the TV and TGV minimizations using NESTA
Signal Processing: Image Communication ( IF 3.4 ) Pub Date : 2020-04-18 , DOI: 10.1016/j.image.2020.115856
Jonathan A. Lima , Felipe B. da Silva , Ricardo von Borries , Cristiano J. Miosso , Mylène C.Q. Farias

Compressive sensing (CS) allows for the reconstruction of sparse signals based on measurements acquired at sub-Nyquist sampling rates. Amongst the common CS methods, total variation (TV) minimization is a common approach used to reconstruct images that are approximately piece-wise constant. The discrete gradient operation in TV can be described as a filtering operation using the horizontal and vertical finite differences filters. In this paper, we generalize the TV minimization procedure for any set of digital filters. The proposed method allows one to reconstruct signals that are sparse when filtered with some set of filters, other than the finite difference operators. Our implementation is based on a fast and accurate first-order optimization algorithm which is called NESTA, in a reference to the Nesterov’s algorithm. We incorporate isotropic and anisotropic filtering combinations to the original TV minimization method implemented in the original NESTA algorithm. We also propose 3 forms of the second order total generalized variation (TGV) when using first and second order filters. In order to evaluate the method, we perform a systematic set of experiments using synthetic and real magnetic resonance images, with several sets of filters and cost functions and under different undersampling factors and noise levels. A statistical analysis of the results shows that the best configurations of our method provide a significantly better image quality when compared to the TV and TGV for MRI reconstruction.



中文翻译:

各向同性和各向异性过滤范数最小化:使用NESTA对TV和TGV最小化的一般化

压缩感测(CS)可以基于以次奈奎斯特采样率获取的测量值来重建稀疏信号。在常见的CS方法中,总变化(TV)最小化是用于重建近似分段恒定的图像的常见方法。电视中的离散梯度操作可以描述为使用水平和垂直有限差分滤波器的滤波操作。在本文中,我们概括了任何数字滤波器集的电视最小化程序。所提出的方法允许人们重建用有限差分算子以外的一组滤波器滤波时稀疏的信号。我们的实现是基于一种称为NESTA的快速准确的一阶优化算法,并参考了Nesterov算法。我们将各向同性和各向异性滤波组合结合到原始NESTA算法中实现的原始电视最小化方法中。当使用一阶和二阶滤波器时,我们还提出了三种形式的二阶总广义变化量(TGV)。为了评估该方法,我们使用合成的和真实的磁共振图像进行了系统的实验,并使用了几组滤波器和成本函数,并且在不同的欠采样因子和噪声水平下进行了实验。结果的统计分析表明,与用于MRI重建的TV和TGV相比,我们方法的最佳配置可提供更好的图像质量。当使用一阶和二阶滤波器时,我们还提出了三种形式的二阶总广义变化量(TGV)。为了评估该方法,我们使用合成的和真实的磁共振图像执行了系统的一组实验,其中包含几组滤波器和成本函数,并且在不同的欠采样因子和噪声水平下。结果的统计分析表明,与用于MRI重建的TV和TGV相比,我们方法的最佳配置可提供更好的图像质量。当使用一阶和二阶滤波器时,我们还提出了三种形式的二阶总广义变化量(TGV)。为了评估该方法,我们使用合成的和真实的磁共振图像进行了系统的实验,并使用了几组滤波器和成本函数,并且在不同的欠采样因子和噪声水平下进行了实验。结果的统计分析表明,与用于MRI重建的TV和TGV相比,我们方法的最佳配置可提供更好的图像质量。

更新日期:2020-04-18
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