Athalye Vivek, Lustig Michael, Uecker Martin
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.
Inverse Probl. 2015 Apr 1;31(4):045008. doi: 10.1088/0266-5611/31/4/045008.
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.
在磁共振成像（MRI）中，数据样本是在空间频率域（k空间）中采集的，通常是在笛卡尔网格上逐行进行耗时的扫描。扫描可以通过使用多个接收器同时采集数据（并行成像）以及使用更高效的非笛卡尔采样方案来加速。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利用接收线圈信息重建未采样的k空间。如下所示，从任意位置的样本进行重建可以理解为从采集的样本中逼近向量值函数，并使用具有由接收线圈的空间灵敏度定义的矩阵值核的再生核希尔伯特空间（RKHS）来进行公式化。这在逼近理论和平行成像之间建立了形式上的联系。然后，可以使用逼近理论的理论工具来理解k空间中的重建，并将样本选择效果的分析从传统的图像域g因子噪声分析扩展到k空间中的噪声放大和逼近误差。通过数值示例进行了说明。
