Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem.
Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem.
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DOI：
10.1137/130935434
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发表时间：
2015-01-22
影响因子：
2.1
通讯作者：
Singer A
中科院分区：
文献类型：
--
作者：
Katsevich E;Katsevich A;Singer A
在低温电子显微镜（cryo-EM）中，显微镜可以生成一个随机方向的分子拷贝样本的俯视图。低温电镜单粒子重建（SPR）的问题是利用在未知方向拍摄的噪声二维投影图像集来重建分子的三维（3D）结构。在某些情况下，被检查的分子表现出结构变异性，这对SPR提出了根本性的挑战。非均质性问题是绘制分子构象态空间的任务。以前有人提出，三维分子协方差矩阵的首特征向量可以用来解决非均质性问题。估计协方差矩阵是具有挑战性的，因为只观察到分子的投影，而不是分子本身。本文提出了一个由样本的噪声投影估计协方差的一般问题。该问题与矩阵补全问题和高维主成分分析问题有着密切的联系。我们提出了一个估计量并证明了它的相合性。当存在有限多个异质性类别时，估计协方差矩阵的谱显示了类别的数量。估计量可以作为某线性系统的解得到。在冷冻电镜的情况下，要反转的线性算子，我们称之为投影协方差变换，是涉及结构变化的层析问题协方差估计的重要对象。反转它需要应用一个类似于层析成像中的斜坡滤波器的滤波器。我们设计了一个基，其中这个线性算子是稀疏的，因此尽管它的大小很大，但可以跟踪反转。我们通过合成数据集的数值实验证明了我们的算法对高水平噪声的鲁棒性。
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.
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