In cryo-electron microscopy (cryo-EM), a microscope generates a top view of a sample of randomly oriented copies of a molecule. The problem of single particle reconstruction (SPR) from cryo-EM is to use the resulting set of noisy two-dimensional projection images taken at unknown directions to reconstruct the three-dimensional (3D) structure of the molecule. In some situations, the molecule under examination exhibits structural variability, which poses a fundamental challenge in SPR. The heterogeneity problem is the task of mapping the space of conformational states of a molecule. It has been previously suggested that the leading eigenvectors of the covariance matrix of the 3D molecules can be used to solve the heterogeneity problem. Estimating the covariance matrix is challenging, since only projections of the molecules are observed, but not the molecules themselves. In this paper, we formulate a general problem of covariance estimation from noisy projections of samples. This problem has intimate connections with matrix completion problems and high-dimensional principal component analysis. We propose an estimator and prove its consistency. When there are finitely many heterogeneity classes, the spectrum of the estimated covariance matrix reveals the number of classes. The estimator can be found as the solution to a certain linear system. In the cryo-EM case, the linear operator to be inverted, which we term the projection covariance transform, is an important object in covariance estimation for tomographic problems involving structural variation. Inverting it involves applying a filter akin to the ramp filter in tomography. We design a basis in which this linear operator is sparse and thus can be tractably inverted despite its large size. We demonstrate via numerical experiments on synthetic datasets the robustness of our algorithm to high levels of noise.

中文翻译：

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低温电磁异质性问题的协方差矩阵估计。

在低温电子显微镜（cryo-EM）中，显微镜会产生分子随机定向复制的样品的顶视图。来自cryo-EM的单粒子重建（SPR）问题是要使用在未知方向上拍摄的所得的嘈杂二维投影图像集来重建分子的三维（3D）结构。在某些情况下，受检分子表现出结构变异性，这对SPR构成了根本性挑战。异质性问题是绘制分子构象状态空间的任务。先前已经提出，可以使用3D分子协方差矩阵的前导特征向量来解决异质性问题。估计协方差矩阵具有挑战性，因为仅观察到分子的投影，但不是分子本身。在本文中，我们从样本的噪声投影中提出了一个协方差估计的一般问题。这个问题与矩阵完成问题和高维主成分分析有着密切的联系。我们提出一个估计量并证明其一致性。当异质性类别有限时，估计协方差矩阵的频谱将揭示类别的数量。可以找到估计器作为某个线性系统的解。在低温电磁场情况下，要反转的线性算子（我们称为投影协方差变换）是涉及结构变化的层析成像问题的协方差估计中的重要对象。反转涉及在层析成像中应用类似于斜坡滤波器的滤波器。我们设计了一个基础，其中线性运算符比较稀疏，因此尽管尺寸很大，也可以很容易地反转。我们通过在合成数据集上进行的数值实验证明了我们的算法对高噪声水平的鲁棒性。