# A stochastic analysis of distance estimation approaches in single molecule microscopy: quantifying the resolution limits of photon-limited imaging systems

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DOI: 10.1007/s11045-012-0175-6

- Cite this article as:
- Ram, S., Ward, E.S. & Ober, R.J. Multidim Syst Sign Process (2013) 24: 503. doi:10.1007/s11045-012-0175-6

## Abstract

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.