A major challenge in single particle reconstruction from cryo-electron microscopy is to establish a reliable ab initio three-dimensional model using two-dimensional projection images with unknown orientations. Common-lines-based methods estimate the orientations without additional geometric information. However, such methods fail when the detection rate of common-lines is too low due to the high level of noise in the images. An approximation to the least squares global self-consistency error was obtained in [A. Singer and Y. Shkolnisky, SIAM J. Imaging Sci., 4 (2011), pp. 543-572] using convex relaxation by semidefinite programming. In this paper we introduce a more robust global self-consistency error and show that the corresponding optimization problem can be solved via semidefinite relaxation. In order to prevent artificial clustering of the estimated viewing directions, we further introduce a spectral norm term that is added as a constraint or as a regularization term to the relaxed minimization problem. The resulting problems are solved using either the alternating direction method of multipliers or an iteratively reweighted least squares procedure. Numerical experiments with both simulated and real images demonstrate that the proposed methods significantly reduce the orientation estimation error when the detection rate of common-lines is low.

中文翻译：

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使用最小非平方偏差确定冷冻电镜图像的方向。

从低温电子显微镜重建单粒子的一个主要挑战是使用未知方向的二维投影图像建立可靠的从头开始的三维模型。基于公共线的方法无需额外的几何信息即可估计方向。然而，由于图像中的高噪声水平，当公共线的检测率太低时，这种方法会失败。在 [A. Singer 和 Y. Shkolnisky, SIAM J. Imaging Sci., 4 (2011), pp. 543-572] 通过半定规划使用凸松弛。在本文中，我们引入了一个更鲁棒的全局自洽误差，并表明相应的优化问题可以通过半定松弛来解决。为了防止估计观看方向的人为聚类，我们进一步引入了一个谱范数项，该项作为约束或正则化项添加到松弛最小化问题中。由此产生的问题可以使用乘法器的交替方向方法或迭代重新加权最小二乘程序来解决。模拟和真实图像的数值实验表明，当共线检测率较低时，所提出的方法显着降低了方向估计误差。由此产生的问题可以使用乘法器的交替方向方法或迭代重新加权最小二乘程序来解决。模拟和真实图像的数值实验表明，当共线检测率较低时，所提出的方法显着降低了方向估计误差。由此产生的问题可以使用乘法器的交替方向方法或迭代重新加权最小二乘程序来解决。模拟和真实图像的数值实验表明，当共线检测率较低时，所提出的方法显着降低了方向估计误差。