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

Abstract

Localization microscopy approaches with enhanced depth-of-field (EDoF) are commonly optimized using the Cramér-Rao bound (CRB) as a criterion. It is widely believed that the CRB can be attained in practice by using the maximum-likelihood estimator (MLE). This is however an approximation, of which we define in this paper the precise domain of validity. Exploring a wide range of settings and noise levels, we show that the MLE is efficient when the signal-to-noise ratio (SNR) is such that the localization standard deviation of a single molecule is less than 20 nm. Thus, our results provide an explicit and quantitative validity boundary for the use of the MLE in EDoF localization microscopy setups optimized with the CRB.

On the equivalence of binary phase masks optimized for localization or detection in extended depth-of-field localization microscopy

“…We also observe that 𝐽 det ( 𝝆 opt ) tends asymptotically to 𝐽 det (0) as the photon balance 𝑁 0 /𝑏 increases, that is, when the dominant source of noise is the shot noise due to the fluorescence signal (this situation is called Scenario A in Ref. [14]). It means that in this case, the use of a phase mask does not improve nor deteriorate fluorophore detection: the contrast between the fluorophore and the background is sufficient for a correct detection over the DoF range 𝜓 max = 1𝜆 without the help of a mask.…”

“…To extend the depth over which fluorescent emitters can be detected, we propose to optimize annular binary phase masks using the criterion based on the Bhattacharyya distance defined in Eq. (14). A reasonable criterion for mask optimization is to maximize the smallest value of the Bhattacharyya distance within the targeted EDoF range [−𝜓 max , 𝜓 max ].…”

“…To facilitate the comparison between the masks that are optimal for detection and those optimal for localization obtained in Ref. [14], we will assume that the emitter is located at the center of a pixel so that 𝜽 0 = (0, 0). the value of the criterion obtained without a mask.…”

“…Small values of 𝑁 0 /𝑏 correspond to situations where the dominant source of noise is due to background fluctuations (we shall call this situation Scenario B, as in Ref. [14]). In this scenario, the PSF of the imaging system is drowned in the background noise when no mask is used.…”

“…This metric is then used in Section 5 to optimize phase masks for detection over a prescribed DoF range. In Section 6, these optimal masks are compared to the masks that were optimized for localization-only in a previous study [14]. Section 7 is devoted to concluding remarks and perspectives.…”