Penczek Pawel A, Renka Robert, Schomberg Hermann
Department of Biochemistry and Molecular Biology, The University of Texas-Houston Medical School, 6431 Fannin, MSB6.218, Houston, Texas 77030, USA.
J Opt Soc Am A Opt Image Sci Vis. 2004 Apr;21(4):499-509. doi: 10.1364/josaa.21.000499.
We describe a fast and accurate direct Fourier method for reconstructing a function f of three variables from a number of its parallel beam projections. The main application of our method is in single particle analysis, where the goal is to reconstruct the mass density of a biological macromolecule. Typically, the number of projections is extremely large, and each projection is extremely noisy. The projection directions are random and initially unknown. However, it is possible to determine both the directions and f by an iterative procedure; during each stage of the iteration, one has to solve a reconstruction problem of the type considered here. Our reconstruction algorithm is distinguished from other direct Fourier methods by the use of gridding techniques that provide an efficient means to compute a uniformly sampled version of a function g from a nonuniformly sampled version of Fg, the Fourier transform of g, or vice versa. We apply the two-dimensional reverse gridding method to each available projection of f, the function to be reconstructed, in order to obtain Ff on a special spherical grid. Then we use the three-dimensional gridding method to reconstruct f from this sampled version of Ff. This stage requires a proper weighting of the samples of Ff to compensate for their nonuniform distribution. We use a fast method for computing appropriate weights that exploits the special properties of the spherical sampling grid for Ff and involves the computation of a Voronoi diagram on the unit sphere. We demonstrate the excellent speed and accuracy of our method by using simulated data.
我们描述了一种快速且准确的直接傅里叶方法，用于从多个平行光束投影中重建三变量函数(f)。我们方法的主要应用在于单颗粒分析，其目标是重建生物大分子的质量密度。通常，投影数量极大，且每个投影都有极高的噪声。投影方向是随机的且最初未知。然而，通过迭代过程可以确定方向和(f)；在迭代的每个阶段，都必须解决此处所考虑类型的重建问题。我们的重建算法与其他直接傅里叶方法的区别在于使用了网格化技术，该技术提供了一种有效的手段，可从函数(g)的非均匀采样版本(Fg)（(g)的傅里叶变换）计算出(g)的均匀采样版本，反之亦然。我们将二维反向网格化方法应用于待重建函数(f)的每个可用投影，以便在特殊的球面网格上获得(Ff)。然后我们使用三维网格化方法从(Ff)的这个采样版本重建(f)。此阶段需要对(Ff)的样本进行适当加权，以补偿其非均匀分布。我们使用一种快速方法来计算适当的权重，该方法利用了(Ff)的球面采样网格的特殊性质，并涉及在单位球面上计算Voronoi图。我们通过使用模拟数据证明了我们方法的卓越速度和准确性。
