1. Introduction

The applicability of super-resolution techniques to the study of dynamical biological processes is typically limited by their long acquisition times. Localization-based super-resolution techniques known as PALM (Photoactivated Localization Microscopy) or STORM (Stochastic Optical Reconstruction Microscopy) rely on a common principle: sources that lie within the same diffraction-limited volume are separated by a sequential activation process, which introduces a temporal separation between source points [4]. Within each frame, a small and random fraction of probes are activated by illumination. Merging all the single-molecule positions obtained from successive frames produces the final image [1, 5]. Since only a small fraction of probes are imaged per frame, a certain number of frames are required in order to obtain a reliable reconstructed image. Multiplying this number by the typical acquisition time of frames (typically in the 10–100ms range) allows us to obtain an expression for the minimal time required to obtain an image at a nanometer-scale resolution (denoted T). A typical reported value is T ∼ 30 min for whole-cell imaging at 10 nm resolution [3], which is too long a time for studying many dynamic processes in living cells: e.g. the contraction of actomyosin units [6], reorganization of focal adhesion complexes [7], or protein cluster formation within the plasma membrane [8,9].

Stochastic microscopy is also prone to localization errors. These errors may either originate from emissions outside of the region of interest (ROI) or from localization error due to overlapping point spread functions (PSF). In this study, we do not try to estimate the risk of such overlapping spread functions - such estimation has already been the subject of numerous theoretical investigations [10–13]. We rather consider the level of the background noise as an experimental input. To filter noisy observations, it is generally assumed that spurious detections are mostly localized in regions with a low count-density: e.g. the density-based spatial clustering of applications with noise algorithm filters noise by eliminating observations whose nearest neighbors are further than a threshold distance [14]. Here, we consider that an elementary region of space belongs to the ROI if it has collected more than observations during the duration of the experiment T. If the number of observations is insufficient, the reconstructed structural image displays voids within the ROI – i.e. parts of the ROI are assigned to the background noise. We refer to these voids as stochastic aberration. This leads to the following main questions of the present paper:

It is generally thought that in widefield stochastic techniques, the acquisition time T is essentially controlled by the density of activated fluorophores per frame (denoted ρ), but not on the total volume of the ROI (S). The latter property represents a substantial advantage over raster-scan-based techniques, e.g. STimulated Emission Depletion (STED) microscopy, where the image is exempt of stochastic aberration but at the cost of an acquisition time that linearly increases with the size of the field of view T ∝ S [15]. For widefield stochastic techniques, arguments based on the Nyquist–Shannon sampling theorem or on Fourier–transform measures typically lead to T_N ∝ 1/ρσ [16,17]. Indeed, the Nyquist criterion states that the mean distance between sampling points must be at least two-fold smaller than the desired resolution [16]. Therefore, to attain a resolution of length δs, at least 2^D evenly spaced observations are needed per resolution volume δs^D. Equivalently, this condition reads ρTσ = 1, in order to guarantee that, on average, there is one observation per elementary volume σ = (δs/2)^D. However, the condition ρTσ = 1 does not imply that every elementary volume has collected at least one observation: some elementary volumes may have collected several observations while most others have none.

In this paper, we argue that the imaging completion time T should be expected to depend on the size of the field of view due to the random (hence uneven) spatial distribution of events. Based on our stochastic model, we derive the relation:

Table 1. List of notations used in the paper.

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We refer to the result of Eq. (1) as the coupon-collector scaling due to its similarity with the solution of the coupon-collector problem [20–22].

Our derivation of Eq. (1) applies to experimental situations in which a high reconstruction fidelity is needed. While we first derive Eq. (1) in the limit of a 100% coverage of the ROI, we show that Eq. (1) holds for a near complete image reconstruction even, i.e. when a non-vanishing fraction of missing pixels is tolerable (see Sec. 3.2). Obtaining such reliable reconstruction can be of critical interest in order to infer structural information, e.g. when evaluating the integrity of a DNA segment [18, 37] or the tensegrity of the actin network within a cell [19]. Indeed, a broken actin filament cannot support tension, similarly to a nano-wire which cannot conduct current when cut in two. Yet, in other experimental contexts in which a high fraction of missed pixels in the reconstructed image is tolerable, the image completion time depends only marginally on the size of the sample (see Sec. 3.7).

The paper is organized as follows. We first present the experimental protocols for the PALM, STORM, and Total Internal Reflection Microscopy (TIRM) experiments. We then define two image rendering schemes, called patch method (PM) and box-filling method (BFM), respectively (see Fig. 1). We then prove Eq. (1) in several experimental situations. In particular, we investigate the effect of correlations between successive frames, motivated by the analogy between the gold nanoparticle experiments and PALM techniques which rely on organic dyes whose blinking statistics exhibit time-correlations [23]. Correlated fluorophore blinking events in PALM are analogous to the multiple returns of the Brownian particles to the illuminated region in the TIRM setup, while bleaching events in PALM correspond to the escape of the Brownian particles far from the illuminated region in the TIRM setup.

 

Fig. 1 Structural reconstruction by stochastic super-localization microscopy. Probes (colored dots) are bound to a structure of interest (green line). (a) Circular patches representation: the diffraction pattern (left), from which super-resolution algorithms yield the coordinates of the center of the pattern. In the patch method representation (middle), each point coordinate is represented by a disk with a radius σ (blue circle: uncertainty of localization algorithm). Patches progressively accumulate as the acquisition time is increased, eventually covering the whole structure of interest (rightmost circle in (a): disks of different colors for separate time frames). (b–c) Box-filling representation: (b) The field of view is divided into N = 9 pixels among which F = 5 pixels contain probes. (c) Evolution of the map of the cumulative number of observations $M_j^{(t)}$ for each pixel j and for each frame t. At t = 50, all pixels have been observed at least r = 10 times.

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We conclude our article by presenting a procedure to estimate the risk of stochastic aberrations in real-time during the acquisition. Our method could be combined with a real-time monitoring algorithm (e.g. [36]) to provide a stopping criterion for data acquisitions.

2. Methods

2.1. Experimental setups

We analyzed the sequence of localization events from three different experimental setups: (i) STORM images of cells with tagged microtubule structures, (ii) PALM images of a silane sample with quasi-uniform sampling in fluorophores, (iii) TIRM of Brownian particles on top of a homogeneous surface. We briefly present the experimental protocols for these three setups.

2.2. Two image rendering methods

Super-resolution techniques rely on the localization of the center of diffraction-limited spots, which provides a set of points. However, a spatial extension needs to be attributed to each point to obtain an image that is readable to the human eye. In this manuscript, we are interested in the following two image rendering methods.

2.3. Statistics of events

We assume that localization events are distributed according to a homogeneous Poisson process, such that the probability density dP that an event occurs in an infinitesimal space of volume ds within an elementary element of the ROI reads dP = ρds [28]. The local density of observations ρ can be expressed as ρ = fd in which d represents the density of fluorophores and f is the fraction of fluorophores detected at each frame (f is typically determined by the dosage of the activation pulse [2]).

We further assume that the local density of events ρ is time-independent, thereby neglecting fluorophore bleaching [23]. Our time-independent assumption corresponds to two situations, in which either (i) the total number of fluorophores remains large compared to the number of bleached fluorophores, or (ii) if the activation laser intensity is increased as a function of time in order to balance the effect of bleaching.

2.4. Estimation of the structure size

The size of the ROI is generally unknown a priori. The maximum likelihood estimator of the number of pixels F (BFM) reads $\widehat{F}(t)={\sum }_{j=1}^F\text{min}\left(M_j^{(t)},1\right)$, where $M_j^{(t)}$ is the cumulative number of measures of the pixel j. For example $M_j^{(t)}=0$ if the pixel j has never collected any event after t frames, and $M_j^{(t)}\ge 1$ if the pixel has been observed at least once after t frames (see Fig. 1). Similarly, within the PM framework, the maximum likelihood estimator of the structure volume consists of the covered volume at the time t. These two estimators are biased as they tend to underestimate the ROI length/area/volume in 1D/2D/3D, respectively.

2.5. Mathematical definition of the image completion time

We call image completion time the minimal number of frames required to obtain a complete image of the ROI. More precisely, the image completion time T is the random variable (called stopping time) that corresponds to the minimal number of frames such that, in every pixel, the each cumulative number of observations is larger or equal to the threshold r: ${\text{min}}_j\left(M_j^{(T)}\right)=r$; where j refers to any pixel (BFM) or point (PM) within the ROI. The centile of T is defined as:

2.6. Construction of the ROI and parameter estimate

In the context of imaging cellular structures, we are confronted by the problem that the ROI is unknown prior to imaging. Here, we show how we construct the ROI in the context of microtubule labelling by Alexa-tubulin as shown in Fig. 2. The sample contains the total number of collected frames (T_t = 2.5 · 10⁵) and the number of events is large (10⁷). We divide the total field of view into N = 512 × 512 pixels, which corresponds to a spatial precision of σ = 160 nm. This value of σ is significantly higher than the localization precision (typically of the order of 20 nm) as we seek to obtain a large number of image completion times.

 

Fig. 2 STORM imaging of the microtubule meshwork within a fibroblast cell, labelled by (left – a,c,e) Alexa-tubulin and (right – b,d,f) tdEos-tubulin fluorophores. (a–b) Accumulated number of observations per pixel of size σ = (160 nm)². (c–d) Zoomed in region of crossed microtubules, in scatter (called patch-method) and density (called box-filling) representations (σ = (100 nm)², error bar: 500 nm) (e) Analysis of the dispersion in the event probability density within the structure of interest. We fix a lower bound on the probability densities ρ = 2.4 ·10⁻⁶ nm⁻².frame⁻¹ for Alexa-tubulin and ρ = 1.2·10⁻⁶ nm⁻².frame⁻¹ for tdEos-tubulin. (f) Our theoretical expression predicts the centile t_0.05 of the image acquisition time, expressed in number of frames. The solid lines represent the analytical result assuming an exponential dispersion (parameters are fixed by the fit of Fig. 2. (e) – see Sec. 4.2). The dashed line represent the coupon-collector scaling from Eq. (5), and the data point with error bars represent experimental results with bootstrap estimation of errors. See Dataset 1 (Files 1 and 2) for underlying values [38] and Code 1 [39].

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Since we expect the ROI to consist of regions of space with a higher density of events compared to the background noise, we define the ROI as the set of pixels in which the total number of collected events exceeds a given threshold R. We consider R = 100 for the Alexa-tubulin data with σ = 160 nm, as this corresponds to a minimal density of events within the ROI ρ_min = 1.5·10⁻⁷ nm⁻².frame⁻¹ and to a mean density of events ρ_min = 2.8·10⁻⁷ nm⁻².frame⁻¹. The resulting image in shown in Fig. 2(d). Within the ROI, the natural estimator of the rate of observation per pixel is p̂_1,i = 1 − p̂_0,i where p̂_0,i is the fraction frames in which there is no observation in the pixel i. The latter estimator should be statistically relevant, meaning that, with 95% probability, the real rate of observation per pixel lies in the confidence interval

2.7. Simulations

Both in the BFM and PM frameworks, the volume of the ROI is tessellated into a grid of elementary squares. In the BFM, each event covers a single elementary square; while in the PM, each patch σ covers a square matrix of elementary squares. In both frameworks, we generate a large sample of coverage events and we analyze the resulting distribution of coverage times using the Matlab function prctile.

3. Results and discussion

3.1. The image completion time follows a coupon-collector scaling

We first discuss Eq. (1) in the context of the BFM representation. Assuming that observations occurring in separate pixels are independent, the probability that exactly M pixels have been observed at least once (r = 1) reads: $𝕇\left[{\widehat{F}}^{(t)}=M\right]=\left(\begin{array}{c}F\\ M\end{array}\right)p_0^{(M-F)t}{\left(1-p_0^t\right)}^M$, where p₀ = 1 − p₁, and p₁ is the probability, per pixel and per frame, that there is at least one event. In particular, the probability that the estimator F̂^(t) is equal to its target value F reads 𝕇 [F̂^(t) = F] = (1 − (1 − p₁)^t)^F.

From Eq. (2), we find that the centile of the image completion time reads t_θ = ln (1 − (1 − θ)^1/F) / ln (1 − p₁). With the limits p₁ ≪ 1 and 1 ≪ F, and for sufficiently high centiles (θ < 0.1), we find that the centile of the imaging time reads:

A key feature of Eq. (4) is the non-linear dependence of the imaging time in terms of the number F of pixels within the ROI. This scaling is related to the classical coupon-collector problem [20–22], which determines the minimal number of boxes that need to be bought in order to be almost certain to gather a complete set of coupons, assuming each box contains a random coupon. In the case where each box contains, at random, either 0 (with probability p₀) or 1 coupon, the mean number of bought boxes t (i.e. frames) required to collect all coupons (i.e. all pixels) reads 𝔼 [T] = F(1+1/2+. . .+1/F)/μ with μ = 1 − p₀. When F ≫ 1, 𝔼 [T] ∼ F ln(F)/μ, and the centile of the stopping time reads t_θ ∼ (F/μ) × ln(F/θ) [20]. Notice that the image completion time in Eq. (4) does not depend on the 1D, 2D or 3D nature of the structure; this is expected since pixels are considered to be independent.

3.2. The coupon-collector scaling holds for a near complete coverage

The following simple argument shows that Eq. (1) holds even in the case of near total coverage, i.e. should the final image misses a reduced fraction (e.g. 5%) of the total number of pixels within the ROI.

Proof If a single pixel i is missing after t frames, the additional number of frames Δt that is required to detect an event within this pixel i is of the order of the total number of pixels, i.e. Δt ∝ F. As Δt ≪ T ∝ F ln(F), we expect that T_{near−complete} ≈ T. This shows that, the near-completion time is approximatively equal to the completion time T provided that the missing fraction of pixels is small.

3.3. The coupon-collector scaling holds when redundant observations per pixel are required

To distinguish relevant observations from spurious ones, we consider that a pixel need to have been observed a minimal number of observations (denoted r) in order to be considered as being part of the ROI. We find that the centile of the image completion time reads

 

Fig. 3 Centile of the image completion time as a function of the redundant number of observations per pixel r, in the cases of (a) a homogeneous sample, for 3 values of the number of pixels within the ROI F (orange: F = 10³; blue: F = 10³; green F = 10³); the coupon collector scaling (dashed lines) fits to the exact solution of image completion time (solid line) for sufficiently small value of r. (b) an exponential dispersion in the fluorophore densities, with a minimal probability of event p_0m = 0.99 and F = 10⁵: (upper triangle, dark blue) peaked distribution λ = 0 (circle, light orange) narrow dispersion λ = 0.01; (down triangle, magenta) wide dispersion λ = 0.1. A coupon-collector expression Eq. (5) in which the density is identified to the spatially averaged mean density 〈ρ_i〉 (dashed lines, magenta) significantly underestimates the image completion time.

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Proof We consider the case where the number of observations of a pixel at each frame is either 0 or 1 (i.e. p_k = 0 for all k > 1). We define the probability $q_j^{(t)}$ that the pixel m has been observed a number j times at the time t: $q_j^{(t)}=𝕇\left(M_m^{(t)}=j\right)$. Successive observations are considered as independent in time, hence $q_j^{(t+1)}=p_0{q}_j^{(t)}+p_1{q}_{j-1}^{(t)}$, 1 ≤ j ≤ r − 1 for all 1 ≤ j < r. As we are interested in the time required to reach the state j = r, we consider the state j = r to be an absorbing state $q_t^{(t+1)}=q_r^{(t)}+p_1{q}_{r-1}^{(t)}$. As soon as j ≤ t, the probability to have reached j ≤ r − 1 observations of the pixel reads:

The probability that all pixels have been observed r times after t frames is ${\left(q_r^{(t)}\right)}^F$. We are interested in the centile time t_θ given by the condition: $P\left({\left\{F{\delta }_{jr}\right\}}_j\right)={\left(q_r^{(t_{\theta })}\right)}^F=1-\theta$. In order to obtain a simple explicit expression for t_θ, we approximate the probability $q_r^{(t)}$ by its long-time behavior from Eq. (7) (which is valid for θ is sufficiently small or for F sufficiently large) to obtain:

3.4. The coupon-collector scaling holds in the presence of spatial inhomogeneities

In this section, we discuss the case of a non-homogeneous rate of activation, which will turn to be crucial to analyze STORM and PALM localization sequences (see Fig. 2). We model the non-homogeneity in the spatial distribution of observations by assuming that, among pixels, the no-event probability p₀ (per frame and per pixel) is distributed according to a probability distribution ψ(q). Under this assumption, the probability that, in a given pixel i, there has been more than r observations reads:

As justified by the analysis live cell experiments (see Sec. 4.2), we focus on the case of a hitting rate p₀ that is exponentially distributed:

We point out that a naive application of Eq. (5) after estimation of p₀ in terms of the spatially averaged mean probability (i.e. averaged over ψ(q)) would lead to a drastic underestimation of the image completion time (see the dashed magenta curve in Fig. 3(b).).

Proof We consider the small pixel limit assumption in which at most one observation per pixel and per frame can occur. The probability that, within a random pixel i, the accumulated number of events is larger than r reads:

 

Fig. 4 Spatial inhomogeneities: estimation of the parameters C_i from Eq. (16) within the phase space (λ, p_0,m): (a) C₁, (b) C₂ and (c) C₃.

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3.5. The coupon-collector scaling holds with the patch image-rendering method

We now consider that the image is obtained from the accumulation of circular patches, whose radius σ corresponds to the spatial resolution. The patch centers are distributed according to a homogeneous Poisson distribution within the ROI of volume S.

We use previous analyses concerning the centile n_θ of the number of circular patches required to cover a circle [31] or a square [32]. In the small patch limit σ/S ≪ 1, we expect that t_θ ∼ n_θ/μ where μ is the number of events per frame. Therefore, following [31] and [32], we find that Eq. (1) corresponds to the time required to obtain a r-fold coverage of a D-dimensional ROI of total volume S by circular patches of volume σ. Our simulations agree with the value γ₁ = 0 predicted in [31]; our simulations further indicate that γ₂ ≈ 2 and γ₃ ≈ 3.

At first order, the identity between Eq. (1) and Eq. (5) is in the ratio σ/S ≪ 1, suggesting that the contribution of overlaps between patches tends to vanish in the limit σ ≪ S. At second order and for any value of the spatial dimension d, Eq. (5) corresponds to a value γ_D = −1 in Eq. (1). Such discrepancy at second order in the ratio σ/S ≪ 1 between Eq. (1) and Eq. (5) is discussed in [31].

Using the PM representation, we can directly answer the following question: what is the probability P(∊) that a hole of size ∊ in the reconstructed image after t frame corresponds to a genuine gap in the structure? We identify P(∊) as the empty-space distribution defined in [33], leading to P(∊) = 1 − exp(−ρt ∊^D/Ω), where Ω = π^D/2/Γ [1 + D/2] is the volume of a sphere of radius 1. We expect the latter relation on P(∊) to hold within the BFM framework, providing the probability that a connected set of n = ∊/σ missing pixels corresponds to a genuine hole.

3.6. The coupon-collector scaling holds in the presence of correlations between frames

In the Brownian TIRM experiments, the gold particle may enter, escape, or return within the field of view, leading to correlated observations between successive frames. We encompass these correlated observations through the following box-filling model. We assume that, for each pixel and for each frame, the number of observations is random variable K with a probability law denoted p_k = 𝕇(K = k), k ≥ 0. We then neglect the time between successive correlated events. This model is in qualitative agreement with experimental data as shown in Figs. 5(e) and 5(f) in which we represent the experimental data from [24] and simulated evolutions of the cumulative number of events $M_j^{(t)}$.

 

Fig. 5 (a) Sketch of the evolution of the cumulative number of observations $M_j^{(t)}$ in a given voxel j. (a, Inset) Trajectory of the particle around the voxel j (frame) labelled by the observation time. After t ≥ 6, the probe is not detected again. (b) Model: all correlated observations are collapsed into one single instantaneous event. (c–d) Evolution of the cumulative number of observations obtained for 4 pixels (among F = 100), either (c) from the experimental data set of [24], or (d) from Monte-Carlo simulations with time correlations. (e–f) Probability distribution of temporal correlations p_k ∝ exp(−k/k_c) (in log-scale): (e) from experiments (red circles), where the maximum likelihood estimator (MLE) of the exponential model provides the value k_c = 2.9 ± 0.1 (black line); (f) from simulations (blue circles), with k_c = 2.9 and with the same number of events as in the experiments. See Dataset 1 (File 3 - TIRM) for underlying values [38].

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We define the mean and variance of the number of observations per pixel per frame as $\nu ={\sum }_{k=1}^{\infty }k{p}_k$ and ${\sigma }^2={\sum }_{k=1}^{\infty }{k}^2{p}_k-{\nu }^2$, respectively. We assume that the set of probabilities p_k, k ≥ 0 is identical for each of the pixels of the structure to be imaged. In the following proof, we show that, provided that p₁ ≠ 0, the imaging completion time is also given by Eq. (5). However, the 1/(1 − p₀) contribution may significantly differ from the averaged number of events per frame ρσ = μ/S. In particular, at a constant total mean number of events per frame μ, increasing the mean number of correlated events ν increases the imaging time. Similarly to the previous case of spatial heterogeneities (Sec. 3.4), temporal correlations preserve the coupon-collector scaling with r but may significantly affect the value of the image completion time for any value of r.

Proof The probability that a single pixel has collected a number of j observations (1 ≤ j ≤ r − 1) during a sequence of t frames reads:

Following Eq. (2), the centile time t_θ is defined by the relation ${\left(q_r^{(t_{\theta })}\right)}^F=1-\theta$. Provided that p₁ > 0 and under the constraint that ${\sum }_{m=1}^{r}m{j}_m=j$, the set of indexes that maximizes the exponent j_u in Eq. (17) is (j, 0, . . . 0). Moreover, the index j = r − 1 maximizes the exponent j_u = j₁ = j. At the leading order in t ≫ 1, Eq. (17) reads:

3.7. Situations in which the coupon-collector scaling does not hold

First, we expect that the image completion time scales linearly with the image size when a small subset of observations are sufficient to reconstruct the image. Indeed, consider that only M ≪ F different pixels need to be acquired. In this case, the probability 𝕇 [F̂^(t) = M] (defined in Sec. 3.1) is maximal for t ∼ MF/μ in the limit μ/F ≪ 1 and M/F ≪ 1. This case includes situations in which the structure can be inferred from sparse localizations, e.g. by assuming randomly oriented linear shapes [19].

Secondly, when the ratio of the mean density of events within the ROI (ρ_ROI) is comparable to the mean density of observations within the background (ρ_BG), a large number r ≫ F of redundant observations per pixel is required to distinguish the ROI from the background. In this case, the coupon collector scaling is no longer applicable. This is particularly important for experimental conditions in which low exposure times are required, resulting in low signal to noise ratios. Based on the confidence interval Eq. (3) for the observation probability per frame per pixel p_i,1, we find that the minimal number of observations per pixel required to distinguish the ROI from the background diverges as the quadratic inverse in the relative difference of intensities, i.e. as:

We now derive an expression of the image completion time in the limit r ≫ F. Due to the central limit theorem, the number of observations collected in the pixel j converges with t towards a Gaussian distribution: $M_j^{(t)}~𝒩\left(t\mu /F,t{\mathrm{\Sigma }}^2/F\right)$, where μ and Σ² are the mean and variance of the number of observations per frame within the total field of view. The probability distribution of the image completion time T then reads:

4. Comparison to experiments

4.1. Comparison to TIRM experiments

We analyze the TIRM experimental data from [24] within the BFM framework. We determine the statistics of the correlated number of observations K according to the following method: two successive events are assumed to correspond to the return of the same particle if (i) the events occur within the same pixel and (ii) the events are separated by less than Δ = 5 frames. We use a maximum likelihood code to fit the experimental histogram to the exponential distribution p_k = A_kc exp(−k/k_c), where A_kc = 1/(1 − exp(−1/k_c)), and k_c = 2.9 (see Fig. 5(g).); we obtain the values ν = 1/(1 − exp(−1/k_c)) = 3.4 and σ² = 1/(cosh(1/k_c) − 1) = 17.

A straightforward implementation of Eq. (5), which would neglect temporal correlations, leads to a value that is an order of magnitude lower than experimentally observed. As illustrated in Fig. 6, Eqs. (5) and (21) fit to the experimental estimation of the centile time in their respective validity range.

 

Fig. 6 Comparison to the TIRM experimental data from [24]; the centile t_0.05 of the image completion time is presented as a function of the required number of redundant observations per pixel r. (a) With F = 15; showing agreement between (red error bars) the centile estimation from experimental data (solid blue lines) our theoretical prediction from Eq. (5) with modelled time correlations, and (green crosses) simulations with time correlations. (b) With F = 4; showing agreement between (red error bars) the centile estimation from experimental data and (black dots) our theoretical prediction from Eq. (21). Error bars were estimated by bootstrapping [35]. See Dataset 1 (File 3 - TIRM) for underlying values [38].

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4.2. Comparison to PALM and STORM experiments

We analyze the sequence of fluorophore localization events from both PALM and STORM experiments. Non-uniformity in the fluorophore densities leads to heterogeneities in the value of the no-observation probability per frame and per pixel p₀. We fit the distribution of p₀ either

 

Fig. 7 STORM imaging at the entire cell level – the microtubule network is labelled with Alexa-tubulin fluorophores, see Fig. 2(a). (a) Distribution of spatial heterogeneities in the event probability 1 − p₀ (σ = 160 × 160 nm²; F = 3.10⁴): the tail of the experimental distribution (blue circles) is fitted by an exponential distribution (dashed black line). Inset: the field of view in terms of the number of collected observations per pixel. Under the hypothesis of a uniform event density, the distribution should follow a Gaussian distribution around its mean – indicated by the solid red line – with 95% of its statistical weight within the bounds indicated by the dashed line, but this is clearly not the case. (b) The centile t_0.05 (in number of frames), where solid lines represent analytical result with p_0,m = 0.986 and λ = 0.013, and the dashed line represent the coupon-collector scaling from 5, and the error bars represent the experimental results with bootstrap estimation of errors (see Sec. 2.6 for the fitting procedure and construction of the ROI; see Dataset 1 (File 1 - STORM Alexa647) for the underlying values [38].

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Fig. 8 PALM imaging of a silicon wafer with quasi-uniform coating in fluorophores (the ROI corresponds to the whole field of view, with N = F = 2500). (a) Non-uniform fluorophore density leads to spatial heterogeneities in p₀: the experimental distribution (blue boxes) is fitted by a Gaussian distribution (dashed red line). Inset: the field of view in terms of the number of collected observations per pixel. (b) The centile t_0.05 for two distinct samples (blue and red) with identical concentration of fluorophores. The solid lines represent analytical results and the error bars represent experimental results with bootstrap estimation of errors. See Dataset 1 (Files 4 and 5) for the underlying values [38] and Code 1 [39].

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In both cases, our theoretical predictions based on Eq. (11) are in quantitative agreement with the analyzed experimental data. One needs to consider spatial heterogeneities in the labeling densities to obtain the quantitative fits presented in Figs. 2(f), 7(b) and 8(b).

5. Real-time estimation of the risk of stochastic aberration

Experimentally, the number F of pixels within the ROI is generally unknown prior to imaging. Here, we propose a real-time procedure to determine whether the estimated number of pixels is reliable and whether one can safely consider that the image is complete.

We introduced an estimator of number of pixels F̂^(t) in Sec. 2.4. Based on this estimator, we can estimate the probability that the image is complete. For example, within the BFM framework, the probability 𝕇̂ [F̂^(t) = F] = (1 − (1 − μ̂^(t)/F̂^(t))^t)^F̂(t) is an estimator of the probability that the image is complete after t frames.

In Fig. 9(c), we present simulations that indicate that the discrepancy between the estimator and the real probability 𝕇 [F̂^(t) = F] is generally very small (error bars indicate the standard deviation); this shows that, at any given time of the experiments, 𝕇̂ is a good estimate of the probability that the image is complete.

 

Fig. 9 Simulation and real-time imaging methods, with F = 100 and an event probability per frame and per pixel p₁ = 10⁻². (a) Evolution of the mean value of F̂^(t) (solid blue curve) and of P(F̂^(t) = F) (solid magenta curve), where error bars indicate the expected standard deviation to those curves when considering a single random realization – this shows that probability estimated over a single experiments closely follow the true probability of image completion P(F̂^(t) = F). (b) Evolution in terms of the number of frames t of the estimator ${\widehat{t_{0.05}}}^{(5)}$ (solid error bars), which converges to the exact centile t_0.95 = 760 (vertical red line).

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Our analytical expression of the image completion time can then be used to infer the additional number of frames required. We represent the convergence of the estimator ${\widehat{t_{\theta }}}^{(t)}=\left({\widehat{F}}^{(t)}/{\widehat{\mu }}^{(t)}\right)\text{ln}\left({\widehat{F}}^{(t)}/\theta \right)$ to the expected value of the centile time t_θ in Fig. 9(b) and 9(d). After t = 300 frames, we estimate that about 420 additional frames are required, which is consistent with the theoretical value of the centile time t_θ = 760.

Our procedure for analyzing an imaging process in real time is the following: once t frames have been collected,

The above procedure is not specific to any particular criteria for the image completion, e.g. if a large redundancy is required (r ≫ ln(F)), one should use the expressions of Eqs. (20) and (21).

6. Conclusion

Our theoretical model provides a unified framework to describe the temporal resolution of several types of stochastic microscopy techniques. These include STORM and PALM, in which a large number of fluorescent probes are attached to the sample and are stochastically activated, or techniques similar to the Brownian TIRM method in which a smaller number of scattering probes stochastically explore the imaged region. We derive analytical expressions to determine the centile of the imaging time for several types of image completion criteria. When a sufficiently low number of accumulated events per pixel are required, the temporal resolution is shown to be logarithmically coupled to the spatial resolution (pixel size), due to the spatial redundancy of detection events. However, the temporal resolution is linearly coupled to the spatial resolution when a large spatial redundancy of events is needed, as the effects of the localization randomness are averaged out. Our results on the imaging time are readily applicable to estimate the minimal time required to reliably characterize spatial patterns by stochastic imaging, with applications ranging from the detection of protein clusters by PALM [8] to the detection of the electromagnetic field around nano-antennas by Brownian particles [24].

Data accessibility

Experimental dataset and codes are accessible under the following DOIs: https://doi.org/10.6084/m9.figshare.4857137 and https://doi.org/10.6084/m9.figshare.5350057, respectively.

Authors contributions

A. M. M. and G. T. carried out the gold nano-particle experiments and localization analysis, and instigated the theoretical problem. Z. Z., P. K. performed microtubule filaments imaging, and R. C. performed actin filaments imaging. J.-F. R. performed the theoretical calculations, simulations, centile time analysis of the experiments and wrote the manuscript. All authors gave final approval for publication. We have no competing interests.