用于无直流磁共振成像的离散卷积核。
A discrete convolution kernel for No-DC MRI.
作者信息
Zeng Gengsheng L, Li Ya
机构信息
Department of Engineering, Weber State University, Ogden, UT 84408, USA.
Department of Radiology, University of Utah, Salt Lake City, UT 84108, USA.
出版信息
Inverse Probl. 2015 Aug;31(8). doi: 10.1088/0266-5611/31/8/085006. Epub 2015 Jul 9.
An analytical inversion formula for the exponential Radon transform with an imaginary attenuation coefficient was developed in 2007 (2007 23 1963-71). The inversion formula in that paper suggested that it is possible to obtain an exact MRI (magnetic resonance imaging) image without acquiring low-frequency data. However, this un-measured low-frequency region (ULFR) in the k-space (which is the two-dimensional Fourier transform space in MRI terminology) must be very small. This current paper derives a FBP (filtered back-projection) algorithm based on You's formula by suggesting a practical discrete convolution kernel. A point spread function is derived for this FBP algorithm. It is demonstrated that the derived FBP algorithm can have a larger ULFR than that in the 2007 paper. The significance of this paper is that we present a closed-form reconstruction algorithm for a special case of under-sampled MRI data. Usually, under-sampled MRI data requires iterative (instead of analytical) algorithms with L-norm or total variation norm to reconstruct the image.
2007年，人们推导出了带有虚衰减系数的指数拉东变换的解析反演公式（2007年，第23卷，第1963 - 71页）。该论文中的反演公式表明，有可能在不采集低频数据的情况下获得精确的磁共振成像（MRI）图像。然而，在k空间（在MRI术语中是二维傅里叶变换空间）中的这个未测量低频区域（ULFR）必须非常小。本文通过提出一个实用的离散卷积核，基于尤氏公式推导了一种滤波反投影（FBP）算法。为该FBP算法推导了点扩散函数。结果表明，所推导的FBP算法能拥有比2007年论文中更大的ULFR。本文的意义在于，我们针对欠采样MRI数据的一种特殊情况给出了一种封闭形式的重建算法。通常，欠采样MRI数据需要使用具有L范数或总变分范数的迭代（而非解析）算法来重建图像。
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