The cryo-electron microscopy reconstruction problem is to find the three-dimensional (3D) structure of a macromolecule given noisy samples of its two-dimensional projection images at unknown random directions. Present algorithms for finding an initial 3D structure model are based on the "angular reconstitution" method in which a coordinate system is established from three projections, and the orientation of the particle giving rise to each image is deduced from common lines among the images. However, a reliable detection of common lines is difficult due to the low signal-to-noise ratio of the images. In this paper we describe two algorithms for finding the unknown imaging directions of all projections by minimizing global self-consistency errors. In the first algorithm, the minimizer is obtained by computing the three largest eigenvectors of a specially designed symmetric matrix derived from the common lines, while the second algorithm is based on semidefinite programming (SDP). Compared with existing algorithms, the advantages of our algorithms are five-fold: first, they accurately estimate all orientations at very low common-line detection rates; second, they are extremely fast, as they involve only the computation of a few top eigenvectors or a sparse SDP; third, they are nonsequential and use the information in all common lines at once; fourth, they are amenable to a rigorous mathematical analysis using spectral analysis and random matrix theory; and finally, the algorithms are optimal in the sense that they reach the information theoretic Shannon bound up to a constant for an idealized probabilistic model.

中文翻译：

---

通过特征向量和半定规划确定冷冻电镜中公共线的三维结构（）。

冷冻电子显微镜重建问题是在给定未知随机方向的二维投影图像的噪声样本的情况下，找到大分子的三维 (3D) 结构。用于寻找初始 3D 结构模型的现有算法基于“角度重构”方法，其中从三个投影建立坐标系，并从图像之间的公共线推导出产生每个图像的粒子的方向。然而，由于图像的低信噪比，难以可靠地检测公共线。在本文中，我们描述了通过最小化全局自洽误差来寻找所有投影的未知成像方向的两种算法。在第一个算法中，最小值是通过计算从公共线导出的专门设计的对称矩阵的三个最大特征向量获得的，而第二种算法基于半定规划 (SDP)。与现有算法相比，我们的算法有五方面的优势：首先，它们以非常低的共线检测率准确估计所有方向；其次，它们非常快，因为它们只涉及几个顶级特征向量或稀疏 SDP 的计算；第三，它们是非顺序的，并且一次使用所有公共行中的信息；第四，它们可以使用谱分析和随机矩阵理论进行严格的数学分析；最后，算法是最优的，因为它们达到了信息论香农绑定到一个理想化概率模型的常数。