The applicability of widefield stochastic microscopy, such as PALM or STORM, is limited by their long acquisition times. Images are produced from the accumulation of a large number of frames that each contain a scarce number of super-resolved localizations. We show that the random and uneven distribution of localizations leads to a specific type of trade-off between the spatial and temporal resolutions. We derive analytical predictions for the minimal time required to obtain a reliable image at a given spatial resolution. We find that the image completion time scales logarithmically with the ratio of the image size to the spatial resolution volume, with second order corrections due to spurious localization within the background noise. We validate our predictions against experimental localization sequences of labeled microtubule filaments obtained by STORM. Our theoretical framework makes it possible to compare the efficiency of emitters, define optimal labeling strategies, and allow implementation of a stopping criterion for data acquisitions that can be performed using real-time monitoring algorithms.
© 2017 Optical Society of America
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 . 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 , which is too long a time for studying many dynamic processes in living cells: e.g. the contraction of actomyosin units , reorganization of focal adhesion complexes , 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 . 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:
- What is the minimal acquisition time required in order to reliably discriminate the ROI from the rest of the field of view?
- How can we reliably discriminate whether a hole in the reconstructed super-resolved image is a genuine gap in the structure rather than an aberration due to a lack of observations?
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 . For widefield stochastic techniques, arguments based on the Nyquist–Shannon sampling theorem or on Fourier–transform measures typically lead to TN ∝ 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 . Therefore, to attain a resolution of length δs, at least 2D evenly spaced observations are needed per resolution volume δsD. 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 for notations). The prefactor ln(S/σ) in Eq. (1) can be significantly larger than 1, e.g. a cell of extension S = 103 μm2 contains 107 squares of area σ = 10−4 μm2 (i.e. a typical PSF area), which leads to ln(S/σ) = 16.
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 . 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 . 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.
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. ) to provide a stopping criterion for data acquisitions.
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.
- STORM microtubule imaging NIH 3T3 mouse fibroblast cells were grown in DMEM containing 10% fetal bovine serum (Life Technologies, Carlsbad, CA). The cells were seeded on glass coverslips coated with 10μg/ml fibronectin (Sigma Aldrich, St. Louis, MO). After spreading for 4h, cells were pre-fixed using 0.3% Glutaraldehyde mixed with 0.2% TritonX-100 in PHEM buffer (60 mM PIPES, 27 mM HEPES, 10 mM EGTA, 8 mM MgSO4, pH 7.0) for 3 min at 37C, and then post-fixed using 4% PFA (Sigma) in PHEM buffer for 15min at 37C. Free aldehydes were then quenched by 5 mg/ml Sodium borohydride (Sigma Aldrich) for 5 min. Samples were washed in PBS and incubated with blocking solution (2% Bovine Serum Albumin in PBS, Sigma Aldrich) for 30 min. Microtubules were probed using mouse anti-α-tubulin (DM1A) monoclonal antibody (abcam, Cambridge, UK) and AlexaFluor 647-conjugated secondary antibody (Life Technologies, Carlsbad, CA). Before imaging, samples were mounted with a reducing imaging buffer containing 40 μg/mL catalase, 100 mM cysteamine, 0.8mg/mL glucose oxidase and 5% glucose (w/v) in PBS (Phosphate Buffered Saline). A Vaseline-lanolin-paraffin mixture was used to seal the samples for STORM imaging. Specimens were imaged using the TIRF mode on a Nikon N-STORM microscope (Nikon Inc., Japan). The camera on the system is a back-illuminated EMCCD camera (Andor Ixon3, Belfast, UK). 100X NA 1.49 Apo TIRF objective lens, and a Cy5 (excitation, 620/60; emission, 700/75) filter set (Chroma Inc) were used. Specimens were illuminated by a 641 nm laser of 100 mW (Coherent inc.) for excitation of Alexa Fluor 647, and a 405 nm laser of 100 mW (Coherent inc.) for photoswitching. A number of 2 · 104 single-molecule frames were acquired with 50 ms exposure time, EM gain of 200, and a read-out speed of 10 MHz. Single-molecule identification and localization were performed using PeakSelector (courtesy of Harald Hess, Howard Hughes Medical Institute) which is a custom-programmed software developed in IDL (Exelis Vis, Boulder, CO). The centroid coordinate of each molecule was determined by 2D-Gaussian non-linear least square fitting. Drift was corrected using an image cross-correlation algorithm .
- PALM quasi-homogeneous sample Biotinylated silane (Methoxy Silane PEG biotin) was mixed with Methoxy Silane to produce a final concentration of 103 molecules/μm2 of Biotin on the surface. Clean cover slips were coated with this silane mixture using vaporization under vacuum. Silane functionalized cover slips were then washed with PBS and incubated with Dylite650 Neutravidin for 1 hour followed by a subsequent wash before PALM imaging. PALM imaging was performed using a Zeiss Elyra microscope. A 100X objective (Alpha Plan Apochromat 100X oil NA 1.46) with 1.6 magnification was used to obtain a final pixel size of 100 nm by 100 nm. The camera on the system is a Andor iXon DU897 512x512 electron multiplier CCD camera. A total of 2 · 104 images were collected with continuous streaming at 50 ms per frame for each sample. PALM images were reconstructed using a custom-made maximum likelihood software .
- TIRM experiments In a recent work , we presented a new stochastic imaging technique to map an electromagnetic field with a nano-scale resolution, using light-scattering Brownian particles as local probes of the field intensity . Using holographic microscopy and a digital reconstruction of the imaged volume, the position of each particle is determined with nanometer-range accuracy by 3D point-spread-function superlocalization algorithm; the intensity of the scattered light is recorded to reveal the local optical intensity at the location of each particles. In a Total Internal Reflection Microscopy (TIRM) configuration, the optical intensity of the electromagnetic field at a height z above the glass surface can be modelled as I(x, y, z) = I0(x, y) exp(−z/β(x, y)), where β is the penetration length of the field, and I0 is proportional to the optical intensity of the field at the surface. In this context, the image refers to the determination of the maps I0(x, y) and β(x, y). As a first test of the method, we consider a situation in which both I0 and β are homogeneous within the whole field of view.
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.
- The box-filling method (BFM) consists of dividing the ROI into S pixels of equal area (2D) or volume (3D, [25,26,30]), which can therefore be expressed as the ratio of the total volume by the resolution volume: F = S/σ. Each new event falls within a specific pixel, thereby increasing by one the cumulative number of observations of this pixel.
- The patch method (PM) consists in representing each detection as a circular patch of radius σ. Generally, the spatial extension corresponds to the spatial uncertainty associated to the localization procedure (typically 10nm ).
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 . 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 ).
We further assume that the local density of events ρ is time-independent, thereby neglecting fluorophore bleaching . 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 , where is the cumulative number of measures of the pixel j. For example if the pixel j has never collected any event after t frames, and 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: ; 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 (Tt = 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.
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 ρmin = 1.5·10−7 nm−2.frame−1 and to a mean density of events ρmin = 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 p̂1,i = 1 − p̂0,i where p̂0,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
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: , 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 tθ = 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 tθ ∼ (F/μ) × ln(F/θ) . 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 Tnear−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 reads20, 29]). For relatively large values of r, our prediction Eq. (5) no longer agrees with the numerical solution of Eq. (2) (see Fig. 3(a)). This is to be expected, since the following proof holds only for small values of r.
Proof We consider the case where the number of observations of a pixel at each frame is either 0 or 1 (i.e. pk = 0 for all k > 1). We define the probability that the pixel m has been observed a number j times at the time t: . Successive observations are considered as independent in time, hence , 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 . 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 . We are interested in the centile time tθ given by the condition: . In order to obtain a simple explicit expression for tθ, we approximate the probability by its long-time behavior from Eq. (7) (which is valid for θ is sufficiently small or for F sufficiently large) to obtain:Eq. (8): Eq. (5).
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:Eq. (11), we can numerically solve Eq. (2) to obtain the value of the centile time tθ.
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:Equation (13) corresponds to Eq. (5) in which we identify p0 to the minimal probability p0,m.
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:Eq. (7). Assuming that each pixels are independent, the image completion time tθ is the solution of the following equation: Eq. (15) to obtain the value of tθ for a large array of values of F and r. We then deduce a three-parameter linear fit of the surface: Fig. 4, we represent the fitted values of C1, C2 and C3 in the phase plane p0,m and λ. We observe that C3 and C2 are close to 1, which proves the agreement with the coupon-collector scaling defined in Eq. (5).
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 nθ of the number of circular patches required to cover a circle  or a square . In the small patch limit σ/S ≪ 1, we expect that tθ ∼ nθ/μ where μ is the number of events per frame. Therefore, following  and , 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 ; 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 γD = −1 in Eq. (1). Such discrepancy at second order in the ratio σ/S ≪ 1 between Eq. (1) and Eq. (5) is discussed in .
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 , leading to P(∊) = 1 − exp(−ρt ∊D/Ω), where Ω = πD/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 pk = (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  and simulated evolutions of the cumulative number of events .
We define the mean and variance of the number of observations per pixel per frame as and , respectively. We assume that the set of probabilities pk, 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 tθ is defined by the relation . Provided that p1 > 0 and under the constraint that , the set of indexes that maximizes the exponent ju in Eq. (17) is (j, 0, . . . 0). Moreover, the index j = r − 1 maximizes the exponent ju = j1 = j. At the leading order in t ≫ 1, Eq. (17) reads:Eq. (7). Similar arguments as the ones to prove Eq. (5) lead to the coupon-collector scaling in the presence of a random number of observations per frame.
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 .
Secondly, when the ratio of the mean density of events within the ROI (ρROI) is comparable to the mean density of observations within the background (ρBG), 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 pi,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:Eq. (3), we obtain the inequality 1.96pi,1/r1/2 < Δpi,1 where Δpi,1 is the characteristic difference in the observation probabilities between the ROI and the background. As the observation probabilities are proportional to the local densities of observation, the later inequality leads to Eq. (19).
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: , 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:34]. With the limit of a large number of observations r ≫ F, the expansion of the error function around 0 provides the following approximate expression: 10]. Equation (21) does not follow the coupon-collector scaling, but rather scales linearly with F, which is expected since the effects of the localization randomness are averaged out when the required redundancy per pixel is large. Importantly, combining due to the Eq. (19) and Eq. (21), we expect that the image completion time diverges quadratically in the limit of a vanishing signal to noise ratio.
4. Comparison to experiments
4.1. Comparison to TIRM experiments
We analyze the TIRM experimental data from  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 pk = Akc exp(−k/kc), where Akc = 1/(1 − exp(−1/kc)), and kc = 2.9 (see Fig. 5(g).); we obtain the values ν = 1/(1 − exp(−1/kc)) = 3.4 and σ2 = 1/(cosh(1/kc) − 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.
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
- by a Gaussian distribution (p0 ∈ [0, 1]) in the PALM imaging of a quasi-homogeneous sample (see Fig. 8(a)).
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.
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 to the expected value of the centile time tθ 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 tθ = 760.
Our procedure for analyzing an imaging process in real time is the following: once t frames have been collected,
- estimate the total number of pixels within the ROI,
- estimate the probability that the image is complete – if this is higher than a desired confidence threshold 1 − θ, the imaging process can be stopped, otherwise, proceed 3
- compute the estimated image completion time . Perform additional frames and return to step 1 with the substitution .
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  to the detection of the electromagnetic field around nano-antennas by Brownian particles .
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.
A. M. M. and G. T. were supported by Université Paris Descartes. This research is also supported by the National Research Foundation, Prime Minister’s Office, Singapore and the Ministry of Education under the Research Centres of Excellence programme, Competitive Research Programme (CRP Award No. NRF-CRP11-2012-02) and Singapore Ministry of Education Academic Research Fund Tier 2 (MOE Grant No. MOE2016-T2-1-124).
We thank Xu Xiaochun (MBI Microscopy core) for designing the localization code of the PALM setup; Andrew Wong for proofreading; J.-F. R. thanks V. Studer, M. Coppey, and B. Hajj for enlightening discussion on the PALM technique and S. Tlili for comments on the manuscript.
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