In cryo-electron microscopy, the three-dimensional (3D) electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy two-dimensional images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For n images of size N-by-N pixels, we propose an algorithm for calculating this covariance estimator with computational complexity O ( n N 4 + κ N 6 log N ) , where the condition number κ is empirically in the range 10-200. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in six dimensions. This is then solved by the conjugate gradient method with an appropriate circulant preconditioner. The result is the first computationally efficient algorithm for consistent estimation of the 3D covariance from noisy projections. It also compares favorably in runtime with respect to previously proposed nonconsistent estimators. Motivated by the recent success of eigenvalue shrinkage procedures for high-dimensional covariance matrix estimation, we incorporate a shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We evaluate our methods on simulated datasets and achieve classification results comparable to state-of-the-art methods in shorter running time. We also present results on clustering volumes in an experimental dataset, illustrating the power of the proposed algorithm for practical determination of structural variability.

中文翻译：

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噪声层析投影的结构变异。

在冷冻电子显微镜中，分子集合的三维 (3D) 电势沿着任意观察方向投影，以产生嘈杂的二维图像。代表这些势的体积图通常表现出大量的结构变异性，这由它们的 3D 协方差矩阵描述。通常，该协方差矩阵近似低秩，可用于对体积进行聚类或估计构象空间的内在几何形状。我们将此协方差矩阵的估计公式化为线性反问题，从而产生一致的最小二乘估计量。对于大小为 N×N 像素的 n 个图像，我们提出了一种计算协方差估计量的算法，其计算复杂度为 O ( n N 4 + κ N 6 log N ) ，其中条件数 κ 根据经验在 10-200 范围内。其效率依赖于对正规方程等效于六维反卷积问题的观察。然后通过共轭梯度法和适当的循环预处理器来解决这个问题。结果是第一个计算高效的算法，用于从噪声投影中一致地估计 3D 协方差。与之前提出的非一致性估计器相比，它在运行时也具有优势。受最近用于高维协方差矩阵估计的特征值收缩程序取得成功的推动，我们采用了一种收缩程序，可以提高较低信噪比下的精度。我们在模拟数据集上评估我们的方法，并在更短的运行时间内实现与最先进的方法相当的分类结果。我们还展示了实验数据集中的聚类体积结果，说明了所提出的算法在实际确定结构变异性方面的能力。