In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. To understand and design k-space sampling patterns, a theoretical framework is needed to analyze how well arbitrary sampling patterns reconstruct unsampled k-space using receive coil information. As shown here, reconstruction from samples at arbitrary locations can be understood as approximation of vector-valued functions from the acquired samples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between approximation theory and parallel imaging. Theoretical tools from approximation theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional image-domain g-factor noise analysis to both noise amplification and approximation errors in k-space. This is demonstrated with numerical examples.

中文翻译：

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并行磁共振成像作为再现核希尔伯特空间中的近似值

在磁共振成像 (MRI) 中，数据样本在空间频域（k 空间）中收集，通常通过在笛卡尔网格上进行耗时的逐行扫描来收集。可以通过使用多个接收器（并行成像）同时采集数据以及使用更有效的非笛卡尔采样方案来加速扫描。为了理解和设计 k 空间采样模式，需要一个理论框架来分析任意采样模式如何使用接收线圈信息重建未采样的 k 空间。如此处所示，从任意位置的样本重建可以理解为对获取样本的向量值函数的逼近，并使用再现核希尔伯特空间 (RKHS) 和由接收线圈的空间灵敏度定义的矩阵值内核进行公式化. 这在近似理论和并行成像之间建立了正式的联系。然后可以使用来自近似理论的理论工具来理解 k 空间中的重建，并将样本选择影响的分析扩展到传统图像域 g 因子噪声分析之外的 k 空间中的噪声放大和近似误差。这是通过数值例子来证明的。