In this paper, we propose a model for parallel magnetic resonance imaging (pMRI) reconstruction, regularized by a carefully designed tight framelet system, that can lead to reconstructed images with much less artifacts in comparison to those from existing models. Our model is motivated from the observations that each receiver coil in a pMRI system is more sensitive to the specific object nearest to the coil, and all coil images are correlated. To exploit these observations, we first stack all coil images together as a 3-dimensional (3D) data matrix, and then design a 3D directional Haar tight framelet (3DHTF) to represent it. After analyzing sparse information of the coil images provided by the high-pass filters of the 3DHTF, we separate the high-pass filters into effective ones and ineffective ones, and we then devise a 3D directional Haar semi-tight framelet (3DHSTF) from the 3DHTF by replacing its ineffective filters with only one filter. This 3DHSTF is tailor-made for coil images, meanwhile, giving a significant saving in computation comparing to the 3DHTF. With the 3DHSTF, we propose an ${\ell }_1$-3DHSTF model for pMRI reconstruction. Numerical experiments for MRI phantom and in-vivo data sets are provided to demonstrate the superiority of our ${\ell }_1$-3DHSTF model in terms of the efficiency of reducing aliasing artifacts in the reconstructed images.

在本文中，我们提出了一种并行磁共振成像 (pMRI) 重建模型，该模型通过精心设计的紧密框架系统进行规范化，与现有模型相比，该模型可以产生具有更少伪影的重建图像。我们的模型的动机是观察到 pMRI 系统中的每个接收器线圈对离线圈最近的特定对象更敏感，并且所有线圈图像都是相关的。为了利用这些观察结果，我们首先将所有线圈图像堆叠在一起作为一个 3 维 (3D) 数据矩阵，然后设计一个 3D 定向 Haar 紧框架 (3DHTF) 来表示它。在分析了3DHTF的高通滤波器提供的线圈图像的稀疏信息后，我们将高通滤波器分为有效和无效，然后，我们从 3DHTF 设计了一个 3D 定向 Haar 半紧框架 (3DHSTF)，仅用一个滤波器替换其无效的滤波器。此 3DHSTF 是为线圈图像量身定制的，同时与 3DHTF 相比，显着节省了计算量。通过 3DHSTF，我们提出了一个${\ell }_1$-3DHSTF 模型用于 pMRI 重建。提供了 MRI 体模和体内数据集的数值实验，以证明我们的优越性${\ell }_1$-3DHSTF 模型在减少重建图像中的混叠伪影的效率方面。