## Abstract

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

## 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:

- 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* [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

*evenly spaced observations are needed per resolution volume*

^{D}*δs*. Equivalently, this condition reads

^{D}*ρTσ*= 1, in order to guarantee that, on average, there is one observation per elementary volume

*σ*= (

*δs*/2)

*. However, the condition*

^{D}*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:

*σ*≪

*S*) and temporal (

*T*) resolutions depends (i) on the ratio of the volume of the ROI

*S*to the desired spatial resolution, (ii) on the necessity to separate the ROI from the noisy background, via a minimal number of redundant observations

*r*that is an increasing function of the background noise intensity, (iii) on the risk of an incomplete coverage of the ROI (i.e. of stochastic aberrations) via the 5%-centile parameter

*θ*= 0.95, and finally (iv) on the dimensionality

*D*∈ {1, 2, 3} of the ROI via the constant

*γ*(see Table 1 for notations). The prefactor ln(

_{D}*S/σ*) in Eq. (1) can be significantly larger than 1, e.g. a cell of extension

*S*= 10

^{3}

*μ*m

^{2}contains 10

^{7}squares of area

*σ*= 10

^{−4}

*μ*m

^{2}(i.e. a typical PSF area), which leads to ln(

*S/σ*) = 16.

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.

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.

**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 10*μ*g/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*μ*g/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 · 10^{4}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 [25].**PALM quasi-homogeneous sample**Biotinylated silane (Methoxy Silane PEG biotin) was mixed with Methoxy Silane to produce a final concentration of 10^{3}molecules/*μ*m^{2}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 100*X*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 512*x*512 electron multiplier CCD camera. A total of 2 · 10^{4}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 [8].**TIRM experiments**In a recent work [24], 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 [24]. 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*) =*I*_{0}(*x*,*y*) exp(−*z/β*(*x*,*y*)), where*β*is the penetration length of the field, and*I*_{0}is proportional to the optical intensity of the field at the surface. In this context, the image refers to the determination of the maps*I*_{0}(*x*,*y*) and*β*(*x*,*y*). As a first test of the method, we consider a situation in which both*I*_{0}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 [27]).

#### 2.3. Statistics of events

We assume that localization events are distributed according to a
homogeneous Poisson process, such that the probability density
d*P* that an event occurs in an infinitesimal space of
volume d*s* within an elementary element of the ROI
reads *dP* = *ρ*d*s*
[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:

*θ*is the tolerated risk. We define

*r*as the threshold number of observations required to distinguish a ROI pixel from its noisy background;

*t*

_{0.05}refers to the minimal number of frames that guarantees, with 95% probability, that there is no stochastic aberration within the reconstructed ROI image.

#### 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 · 10

^{5}) and the number of events is large (10

^{7}). 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 *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

*p*

_{1,m}= 2 · 10

^{−3}which guarantees with 95% probability, that the estimate of

*p*

_{1,i}is precise with up to 10% error.

#### 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: $\U0001d547\left[{\widehat{F}}^{(t)}=M\right]=\left(\begin{array}{c}F\\ M\end{array}\right){p}_{0}^{(M-F)t}{\left(1-{p}_{0}^{t}\right)}^{M}$, where *p*_{0} =
1 − *p*_{1}, and *p*_{1}
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 −
*p*_{1})* ^{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 −

*p*

_{1}). With the limits

*p*

_{1}≪ 1 and 1 ≪

*F*, and for sufficiently high centiles (

*θ*< 0.1), we find that the centile of the imaging time reads:

*p*

_{1}≪ 1), there is almost surely at most one event per pixel and per frame, hence that 1 −

*p*

_{0}∼

*ρσ*.

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 *p*_{0}) 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
− *p*_{0}. 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
Δ*t* that is required to detect an event within this
pixel *i* is of the order of the total number of
pixels, i.e. Δ*t* ∝ *F*. As
Δ*t* ≪ *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

*σ*≪

*S*. The latter relation corresponds to the centile of the coupon collector’s problem when

*r*copies of each coupon need to be collected (see [20, 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.
*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*: ${q}_{j}^{(t)}=\U0001d547\left({M}_{m}^{(t)}=j\right)$. 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:

*r*≥ 2 times is: ${q}_{r}^{(t)}=1-{\sum}_{j=0}^{r-1}{q}_{j}^{(t)}$. Under the long-time limit 1 ≪

*t*,

*t*!/(

*t*−

*j*)! ∼

*t*, the absorption probability ${q}_{r}^{(t)}$ tends to 1 and:

^{j}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*, we approximate the probability ${q}_{r}^{(t)}$ by its long-time behavior from Eq. (7) (which is valid for

_{θ}*θ*is sufficiently small or for

*F*sufficiently large) to obtain:

*θ*)

^{1/F}∼

*θ/F*in the limit

*θ*≪ 1, we obtain from Eq. (8):

*r*≪

*F*, leads to:

*p*

_{0}≪ 1), ln(

*p*

_{0}) = ln(1 − (1 −

*p*

_{0})) = −(1 −

*p*

_{0}) = −

*p*

_{1}= −

*σρ*, we obtain the result of 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
*p*_{0} (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:

*M*

_{1}≥

*rp*

_{0,i}=

*q*) is given in Sec. 3.1 and

*p*

_{1,i}= 1 −

*p*

_{1,0}in the small pixel size limit. From Eq. (11), we can numerically solve Eq. (2) to obtain the value of the centile time

*t*.

_{θ}As justified by the analysis live cell experiments (see Sec. 4.2), we
focus on the case of a hitting rate *p*_{0}
that is exponentially distributed:

*p*

_{0,m}is the minimal probability

*p*

_{0}per pixel within the ROI and

*λ*quantifies the dispersion from

*p*

_{0,m}. Our numerical simulations indicate that the image completion time behaves as:

*C*

_{1}> 0 is a function of both

*λ*and

*p*

_{0,m}, but which is relatively small compared to ln (

*F/θ*) for a large range of parameters. The coupon-collector scaling is robust to the presence of strong inhomogeneities within the ROI (e.g.

*λ*∼ 0.2). Equation (13) corresponds to Eq. (5) in which we identify

*p*

_{0}to the minimal probability

*p*

_{0,m}.

We point out that a naive application of Eq. (5) after estimation of
*p*_{0} 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:

*M*

_{1}≥

*rp*

_{0,i}=

*q*) is given by Eq. (7). Assuming that each pixels are independent, the image completion time

*t*is the solution of the following equation:

_{θ}*ψ*(

*q*) ∝ exp(−(

*q*−

*p*

_{0,m})/

*λ*). We numerically solve Eq. (15) to obtain the value of

*t*for a large array of values of

_{θ}*F*and

*r*. We then deduce a three-parameter linear fit of the surface:

*C*

_{1},

*C*

_{2}and

*C*

_{3}in the phase plane

*p*

_{0,m}and

*λ*. We observe that

*C*

_{3}and

*C*

_{2}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
*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*where

_{θ}/μ*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

*γ*

_{1}= 0 predicted in [31]; our simulations further indicate that

*γ*

_{2}≈ 2 and

*γ*

_{3}≈ 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 ${M}_{j}^{(t)}$.

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

*p*

_{1}≠ 0, the imaging completion time is also given by Eq. (5). However, the 1/(1 −

*p*

_{0}) 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:

*j*

_{1}, . . .

*j*) such that ${\sum}_{m=1}^{r}m{j}_{m}=j$, and (ii) ${j}_{u}={\sum}_{m=1}^{r}{j}_{m}$ is the total number of adsorption events. The image completion probability reads ${q}_{r}^{(t)}=1-{\sum}_{j=0}^{r-1}{q}_{j}^{(t)}$.

_{r}Following Eq. (2), the
centile time *t _{θ}* is defined by the relation
${\left({q}_{r}^{({t}_{\theta})}\right)}^{F}=1-\theta $. Provided that

*p*

_{1}> 0 and under the constraint that ${\sum}_{m=1}^{r}m{j}_{m}=j$, the set of indexes that maximizes the exponent

*j*in Eq. (17) is (

_{u}*j*, 0, . . . 0). Moreover, the index

*j*=

*r*− 1 maximizes the exponent

*j*=

_{u}*j*

_{1}=

*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:

*T*frames in terms of the mean number of observations

_{t}*r*in the pixel

*i*:

*r*=

*T*

_{t}*p*

_{i,1}; from Eq. (3), we obtain the inequality 1.96

*p*

_{i,1}/

*r*

^{1/2}< Δ

*p*

_{i,1}where Δ

*p*

_{i,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: ${M}_{j}^{(t)}~\mathcal{N}\left(t\mu /F,t{\mathrm{\Sigma}}^{2}/F\right)$, where *μ* and
Σ^{2} 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:

*r*≫

*F*, the expansion of the error function around 0 provides the following approximate expression:

*r*≫ ln(

*F/μ*). Our first order result

*t*∼ (

_{θ}*Fr*)/

*μ*agrees with the result of [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 [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*), where

_{c}*A*

_{kc}= 1/(1 − exp(−1/

*k*)), and

_{c}*k*= 2.9 (see Fig. 5(g).); we obtain the values

_{c}*ν*= 1/(1 − exp(−1/

*k*)) = 3.4 and

_{c}*σ*

^{2}= 1/(cosh(1/

*k*) − 1) = 17.

_{c}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 *p*_{0}. We
fit the distribution of *p*_{0} either

- by an exponential distribution in the STORM imaging of cells (see Figs. 2(e) and 7(a))
- by a Gaussian distribution $\psi ({p}_{0})\propto \text{exp}\left({\left({p}_{0}-{\nu}_{p}\right)}^{2}/\left(2{\sigma}_{p}^{2}\right)\right)$ (
*p*_{0}∈ [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 ${\widehat{{t}_{\theta}}}^{(t)}=\left({\widehat{F}}^{(t)}/{\widehat{\mu}}^{(t)}\right)\text{ln}\left({\widehat{F}}^{(t)}/\theta \right)$ 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*= 760.

_{θ}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 ${\widehat{{t}_{\theta}}}^{(t)}$. Perform ${\widehat{{t}_{\theta}}}^{(t)}-t$ additional frames and return to step 1 with the substitution $t\leftarrow {\widehat{{t}_{\theta}}}^{(t)}$.

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.

## Funding

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).

## Acknowledgments

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.

## References and links

**1. **C. W. McCutchen, “Superresolution in microscopy
and the Abbe resolution limit,” J. Opt. Soc.
Am. **57**,
1190–1192 (1967). [CrossRef] [PubMed]

**2. **E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular
fluorescent proteins at nanometer resolution,”
Science **313**,
1642–1645 (2006). [CrossRef] [PubMed]

**3. **Z. Liu, L. D. Lavis, and E. Betzig, “Imaging live-cell dynamics
and structure at the single-molecule level,”
Mol. Cell **58**,
644–659 (2015). [CrossRef] [PubMed]

**4. **L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution
fluorescence microscopy,” J. Cell.
Biol. **190**,
165–175 (2010). [CrossRef] [PubMed]

**5. **E. Betzig, “Proposed method for molecular
optical imaging,” Opt. Lett. **20**, 237 (1995). [CrossRef] [PubMed]

**6. **H. Wolfenson, G. Meacci, S. Liu, M. R. Stachowiak, T. Iskratsch, S. Ghassemi, P. Roca-Cusachs, B. O’Shaughnessy, J. Hone, and M. P. Sheetz, “Tropomyosin controls
sarcomere-like contractions for rigidity sensing and suppressing
growth on soft matrices,” Nat. Cell
Biol. **18**, 33
(2015). [CrossRef]

**7. **C. Bertocchi, W. I. Goh, Z. Zhang, and P. Kanchanawong, “Nanoscale imaging by
superresolution fluorescence microscopy and its emerging applications
in biomedical research,” Crit. Rev. Biomed.
Eng. **41**,
281–308 (2013). [CrossRef] [PubMed]

**8. **R. Changede, X. Xu, F. Margadant, and M. P. Sheetz, “Nascent integrin adhesions
form on all matrix rigidities after integrin
activation,” Devel. Cell **35**, 1–8
(2015). [CrossRef]

**9. **K. H. Biswas, K. L. Hartman, C.-H. Yu, O. J. Harrison, H. Song, A. W. Smith, W. Y. C. Huang, W.-C. Lin, Z. Guo, A. Padmanabhan, S. M. Troyanovsky, M. L. Dustin, L. Shapiro, B. Honig, R. Zaidel-Bar, and J. T. Groves, “E-cadherin junction formation
involves an active kinetic nucleation process,”
Proc. Natl. Acad. Sci. USA **112**,
10932–10937 (2015). [CrossRef] [PubMed]

**10. **A. R. Small, “Theoretical limits on errors
and acquisition rates in localizing switchable
fluorophores,” Biophys. J. **96**, L16–L18
(2009). [CrossRef] [PubMed]

**11. **S. van de Linde, S. Wolter, M. Heilemann, and M. Sauer, “The effect of photoswitching
kinetics and labeling densities on super-resolution fluorescence
imaging,” J. Biotechnol. **149**, 260–266
(2010). [CrossRef] [PubMed]

**12. **G. T. Dempsey, J. C. Vaughan, K. H. Chen, M. Bates, and X. Zhuang, “Evaluation of fluorophores
for optimal performance in localization-based super-resolution
imaging,” Nat. Methods **8**, 1027–1036
(2011). [CrossRef] [PubMed]

**13. **J. E. Fitzgerald, J. Lu, and M. J. Schnitzer, “Estimation theoretic measure
of resolution for stochastic localization microscopy,”
Phys. Rev. Lett. **109**,
1–5 (2012). [CrossRef]

**14. **M. Ester, H. P. Kriegel, J. Sander, and X. Xu, “A density-based algorithm for
discovering clusters in large spatial databases with
noise,” in *Proceedings of the 2nd
International Conference on Knowledge Discovery and Data
Mining* (AAAI Press,
1996), pp.
226–231.

**15. **K. I. Willig, R. R. Kellner, R. Medda, B. Hein, S. Jakobs, and S. W. Hell, “Nanoscale resolution in
GFP-based microscopy,” Nat. Methods **3**, 721–723
(2006). [CrossRef] [PubMed]

**16. **C. E. Shannon, “Communication in the Presence
Of Noise,” Proc. IEEE **86**, 447 (1998). [CrossRef]

**17. **R. P. J. Nieuwenhuizen, K. a. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in
optical nanoscopy,” Nat. Methods **10**, 557–562
(2013). [CrossRef] [PubMed]

**18. **M. Alexeyev, I. Shokolenko, G. Wilson, and S. Ledoux, “The maintenance of
mitochondrial DNA integrity,” Cold Spring
Harb. Perspect. Biol. **5**,
1–17 (2013). [CrossRef]

**19. **Z. Zhang, Y. Nishimura, and P. Kanchanawong, “Extracting microtubule
networks from superresolution single-molecule localization microscopy
data,” Molec. Biol. Cell **28**, 333–345
(2016). [CrossRef] [PubMed]

**20. **P. Erdos and A. Renyi, “On a classical problem of
probability theory,” Magyar Tudomanyos
Akademia Matematikai Kutato Intezetenek Kozlemenyei **6**, 215–220
(1961).

**21. **W. Feller, *An Introduction to Probability Theory
and Its Applications*, Vol. 2
(John Wiley & Sons Inc,
1968).

**22. **R. P. Stanley and H. S. Wilf,
“Generatingfunctionology,” Am.
Math. Monthly **97**, 864
(1990). [CrossRef]

**23. **P. Annibale, S. Vanni, M. Scarselli, U. Rothlisberger, and A. Radenovic, “Identification of clustering
artifacts in photoactivated localization microscopy,”
Nat. Methods **8**,
527–528 (2011). [CrossRef] [PubMed]

**24. **A. Martinez-Marrades, J.-F. Rupprecht, M. Gross, and G. Tessier, “Stochastic 3D optical mapping
by holographic localization of Brownian scatterers,”
Opt. Express **22**,
29191–29203 (2014). [CrossRef] [PubMed]

**25. **G. Shtengel, Y. Wang, Z. Zhang, W. I. Goh, H. F. Hess, and P. Kanchanawong, “Imaging cellular
ultrastructure by PALM, iPALM, and correlative
iPALM-EM,” Methods Cell Biol. **123**, 273–294
(2014). [CrossRef] [PubMed]

**26. **R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3D high- and super-resolution
imaging using single-objective SPIM,” Nat.
Methods **12**, 641
(2015). [CrossRef] [PubMed]

**27. **A. Triller and D. Choquet, “Surface trafficking of
receptors between synaptic and extrasynaptic membranes: and yet they
do move!” Trends in Neurosciences **28**, 133–139
(2005). [CrossRef]

**28. **P. Hall, *Introduction to the Theory of
Coverage Processes* (John Wiley & Sons
Australia, Limited,
1988).

**29. **D. J. Newman, “The double dixie cup
problem”, The American Mathematical
Monthly **67**,
58–61 (1960). [CrossRef]

**30. **B. Hajj, J. Wisniewski, M. El Beheiry, J. J. Chen, A. Revyakin, C. Wu, and M. Dahan, “Whole-cell, multicolor
superresolution imaging using volumetric multifocus
microscopy,” Proc. Natl. Acad. Sci.
USA **111**,
17480–17485 (2014). [CrossRef] [PubMed]

**31. **L. Flatto, “A limit theorem for random
coverings of a circle,” Israel J.
Math. **15**,
167–184 (1973). [CrossRef]

**32. **H. Zhang and J. C. Hou, “On deriving the upper bound
of alpha-lifetime for large sensor network,” in
Proceedings of the 5th ACM International Symposium on
Mobile Networking, 121–132
(2004).

**33. **H. Solomon, *Geometric Probability*
(SIAM, 1978). [CrossRef]

**34. **I. M. Ryzhik and I. S. Gradstein, *Tables of Series, Products and
Integrals* (Academic Press,
2015).

**35. **B. Efron, “Bootstrap methods: another
look at the jackknife,” Ann. Stat. **7**, 1–26
(1979). [CrossRef]

**36. **S. Wolter, A. Löschberger, T. Holm, S. Aufmkolk, M.-C. Dabauvalle, S. V. D. Linde, M. Sauer, and S. van de Linde, “RapidSTORM: accurate, fast
open-source software for localization microscopy,”
Nat. Methods **9**,
1040–1041 (2012). [CrossRef] [PubMed]

**37. **M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging
by stochastic optical reconstruction microscopy
(STORM),” Nat. Methods **3**, 793–795
(2006). [CrossRef] [PubMed]

**38. **J.-F. Rupprecht, “Dataset: Trade-offs between
structural integrity and acquisition time in stochastic
super-resolution microscopy techniques,”
figshare (07.09.2017) https://doi.org/10.6084/m9.figshare.4857137.

**39. **J.-F. Rupprecht, “Code: Trade-offs between
structural integrity and acquisition time in stochastic
super-resolution microscopy techniques,” figshare
(07.09.2017) https://doi.org/10.6084/m9.figshare.5350057.