Single-particle electron microscopy is an experimental technique that is used to determine the three-dimensional (3D) structure of biological macromolecules and the complexes that they form. In general, image processing techniques and reconstruction algorithms are applied to micrographs, which are 2D images taken by electron microscopes. Each of these planar images can be thought of as a projection of the macromolecular structure of interest from an a priori unknown direction. A class is defined as a collection of projection images with a high degree of similarity, presumably resulting from taking projections along similar directions. In practice, micrographs are very noisy and those in each class are aligned and averaged in order to reduce the background noise. Errors in the alignment process are inevitable due to noise in the electron micrographs. This error results in blurry averaged images. In this paper, we investigate how blurring parameters are related to the properties of the background noise in the case when the alignment is achieved by matching the mass centers and the principal axes of the experimental images. We observe that the background noise in micrographs can be treated as Gaussian. Using the mean and variance of the background Gaussian noise, we derive equations for the mean and variance of translational and rotational misalignments in the class averaging process. This defines a Gaussian probability density on the Euclidean motion group of the plane. Our formulation is validated by convolving the derived blurring function representing the stochasticity of the image alignments with the underlying noiseless projection and comparing with the original blurry image.

中文翻译：

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单粒子电子显微镜中类别平均的随机运动学模型

单粒子电子显微镜是一种实验技术，用于确定生物大分子的三维 (3D) 结构及其形成的复合物。通常，图像处理技术和重建算法应用于显微照片，这是由电子显微镜拍摄的二维图像。这些平面图像中的每一个都可以被认为是感兴趣的大分子结构从先验未知方向的投影。一个类被定义为具有高度相似性的投影图像的集合，可能是沿相似方向进行投影的结果。在实践中，显微照片非常嘈杂，每个类别中的显微照片都经过对齐和平均以减少背景噪音。由于电子显微照片中的噪声，对准过程中的误差是不可避免的。此错误会导致平均图像模糊。在本文中，我们研究了在通过匹配实验图像的质心和主轴来实现对齐的情况下，模糊参数如何与背景噪声的特性相关。我们观察到显微照片中的背景噪声可以被视为高斯噪声。使用背景高斯噪声的均值和方差，我们推导出类平均过程中平移和旋转错位的均值和方差的方程。这定义了平面欧几里得运动群上的高斯概率密度。