Optical microscopy is an invaluable tool to visualize biological processes at the cellular scale. In the recent past, there has been significant interest in studying these processes at the single molecule level. An important question that arises in single molecule experiments concerns the estimation of the distance of separation between two closely spaced molecules. Presently, there exists different experimental approaches to estimate the distance between two single molecules. However, it is not clear as to which of these approaches provides the best accuracy for estimating the distance. Here, we address this problem rigorously by using tools of statistical estimation theory. We derive formulations of the Fisher information matrix for the underlying estimation problem of determining the distance of separation from the acquired data for the different approaches. Through the Cramer-Rao inequality, we derive a lower bound to the accuracy with which the distance of separation can be estimated. We show through Monte-Carlo simulations that the bound can be attained by the maximum likelihood estimator. Our analysis shows that the distance estimation problem is in fact related to the localization accuracy problem, the latter being a distinct problem that deals with how accurately the location of an object can be determined. We have carried out a detailed investigation of the relationship between the Fisher information matrices of the two problems for the different experimental approaches considered here. The paper also addresses the issue of a singular Fisher information matrix, which presents a significant complication when calculating the Cramer-Rao lower bound. Here, we show how experimental design can overcome the singularity. Throughout the paper, we illustrate our results by considering a specific image profile that describe the image of a single molecule.

中文翻译：

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单分子显微镜中距离估计方法的随机分析：量化光子限制成像系统的分辨率限制

光学显微镜是在细胞尺度上可视化生物过程的宝贵工具。最近，人们对在单分子水平上研究这些过程产生了浓厚的兴趣。单分子实验中出现的一个重要问题涉及估计两个紧密间隔的分子之间的分离距离。目前，存在不同的实验方法来估计两个单分子之间的距离。但是，尚不清楚这些方法中的哪一种为估计距离提供了最佳精度。在这里，我们通过使用统计估计理论的工具严格解决这个问题。我们推导出 Fisher 信息矩阵的公式，用于确定从不同方法获取的数据的分离距离的潜在估计问题。通过 Cramer-Rao 不等式，我们推导出可以估计分离距离的准确度的下限。我们通过 Monte-Carlo 模拟表明可以通过最大似然估计器获得界限。我们的分析表明，距离估计问题实际上与定位精度问题有关，后者是一个独特的问题，涉及如何准确地确定物体的位置。我们对这里考虑的不同实验方法的两个问题的 Fisher 信息矩阵之间的关系进行了详细调查。该论文还解决了奇异 Fisher 信息矩阵的问题，该问题在计算 Cramer-Rao 下界时带来了显着的复杂性。在这里，我们展示了实验设计如何克服奇点。在整篇论文中，我们通过考虑描述单个分子图像的特定图像配置文件来说明我们的结果。