On the validity domain of maximum likelihood estimators for depth-of-field extension in single-molecule localization...

Area of Science:

• Biophysics
• Optical Microscopy
• Computational Imaging

Background:

• Enhanced depth-of-field (EDoF) techniques in localization microscopy aim to improve 3D imaging capabilities.
• The Cramér-Rao bound (CRB) is frequently used as a theoretical benchmark for optimizing these EDoF methods.
• The maximum-likelihood estimator (MLE) is often assumed to achieve this CRB in practice.

Purpose of the Study:

• To precisely define the operational domain of validity for using the maximum-likelihood estimator (MLE) in enhanced depth-of-field (EDoF) localization microscopy.
• To quantitatively assess the conditions under which the MLE achieves the Cramér-Rao bound (CRB) in EDoF imaging.
• To provide a practical guideline for optimizing EDoF localization microscopy setups.

Main Methods:

• Theoretical analysis of the maximum-likelihood estimator (MLE) performance in EDoF microscopy.
• Numerical simulations across a range of signal-to-noise ratios (SNR) and experimental settings.
• Calculation of localization standard deviations for single molecules under varying conditions.

Main Results:

• The efficiency of the maximum-likelihood estimator (MLE) in EDoF localization microscopy is dependent on the signal-to-noise ratio (SNR).
• The MLE is shown to be efficient when the single-molecule localization standard deviation is less than 20 nm.
• This efficiency boundary is independent of specific EDoF implementation details.

Conclusions:

• The assumption that the maximum-likelihood estimator (MLE) always attains the Cramér-Rao bound (CRB) in EDoF localization microscopy is only valid under specific SNR conditions.
• A quantitative threshold (20 nm localization standard deviation) is established for the reliable use of MLE in CRB-optimized EDoF systems.
• These findings offer crucial insights for the practical implementation and optimization of advanced 3D super-resolution microscopy techniques.