Exploring the physical bounds on light microscopy has been a mainstay activity in theoretical optics. In classical [
1,
2] and in most modern treatments [
3] the limits on resolution have been derived using deterministic field theories for optical diffraction. This ignores photon shot noise, the dominant noise source in perhaps most modern microscopy applications. Digital image processing algorithms that take noise into account can achieve subpixel superresolution by combining multiple conventional images of a single sample or scene [
4,
5]. However, the poor performance of these techniques in recovering high spatial frequency information led to the recognition that prior information about image structure is generally essential for attaining superresolution [
6]. Yet, existing theories do not provide physical limits on image estimation based on the noise and prior information available from specific measurement schemes.
Here we study the physics of superresolution imaging by stochastic localization of single molecules, as in photoactivated localization microscopy (PALM) and stochastic optical reconstruction microscopy (STORM), which achieve superresolution via the spatial localization of single fluorescent molecules [
7–
10]. We stress that stochastic localization is a distinct measurement and computational analysis procedure that exploits the prior knowledge the active fluorophores are dilute. Previous theories of superresolution from digital image processing do not describe the specific form of prior information that has allowed stochastic localization techniques to successfully achieve tenfold or better superresolution beyond the conventional diffraction limit. Our theory provides unified performance limits for stochastic localization, conventional microscopy, and the entire class of digital image processing algorithms for subpixel superresolution.
Our work builds on earlier applications of estimation theory that amended classical notions of resolution by including fluctuations in the propagating fields [
11,
12]. Instead, we consider photon statistics and model the stochastic activation and measurement of individual molecules. Our resolution measure, termed here the information transfer function (ITF), limits how well one can determine a specimen’s features at each spatial frequency. The ITF thus replaces the modulationtransfer function (MTF) but with an expanded range of explanation, covering conventional and stochastic localization microscopy. Importantly, for conventional microscopy the ITF reduces to the square of the MTF, scaled by the number of observed photons, thereby preserving physicists’ basic intuitions about imaging. Overall, our formalism unifies longstanding work on the spatial filtering properties of optical imaging with recent efforts to describe the limits imposed by photon shot noise.
We consider an image taken from a specimen of
m fluorophores, all of the same color and located in the sample plane at {
x_{j}}. The experimental data are the counts, {
N_{l}}, of photons detected at pixel
l which is located in the image plane at
y_{l},
l = 1…
L. The probability,
P, of observing the data is
where
p_{j} is the intensity of the
jth emitter, Δ
y_{l} is the pixel area, and the pointspread function (PSF),
h, is normalized,
. We idealize the detector by assuming uniformly distributed pixels with infinitesimal area, Δ
y_{l} → 0, so that
N_{l} {0,1}. This neglects the effect of finite pixel size. In this limit we can rewrite
Eq. (1) as a function of the locations in the image plane, {
z_{k}}, of
N = ∑
_{l}N_{l} detected photons:
P({
N_{l}}{
x_{j}}) =
P({
z_{k}}
N, {
x_{j}})
P(
N), where
P(
N) is the probability that
N photons are detected, and
This satisfies ∫
P({
z_{k}}
N, {
x_{j}})
dz_{1} …
dz_{N} = 1. Localization microscopy finds estimates, {
$\widehat{x}$_{j}}, of {
x_{j}} based on the observations, {
z_{k}}.
To study the limits of this technique, we use a key tool from estimation theory, the CramérRao lower bound [
13] (CRLB), which bounds the accuracy of any estimator, {
_{i}}, of a set of parameters, {
q_{i}}: [
E − J^{−1}] ≥ 0. Here
E_{ij} (
_{i} −
q_{i})(
_{j} −
q_{j})*
is the estimator covariance matrix,
is the Fisher information matrix, and
_{i} is assumed unbiased, i.e.,
_{i} =
q_{i}. The angular brackets,
···
, denote averages over the stochastic locations of detected photons, distributed according to
Eq. (2). The CRLB states [
E −
J^{−1}] is positive semidefinite, implying that the diagonal elements (the mean squared errors) obey
E_{ii} =

_{i} −
q_{i}
^{2} ≥ [
J^{−1}]
_{ii}. This inequality bounds estimator accuracy for any set of image parameters, such as the emitters’ locations (
q_{i} x_{i}) [
14,
15], or, as in our theory, the realspace [
q_{x} I(
x)] or spatial frequency [
q_{k} I(
k)] components.
Previous applications of the CRLB to emitter localization did not yield limits on how well entire images can be determined, nor formalism to supersede the PSF and MTF. To illustrate, consider a 2D scene composed of two equally bright emitters [
14]. We choose a coordinate system in which the emitters lie on the
x axis at (
x, y) = (±
d_{x}, 0).
J is diagonal in the basis given by the mean position and separation vector [, inset]. shows the corresponding CRLBs in units of the width of the Gaussian PSF, σ [
16]. Similar results hold for an Airy disk PSF, and when the two emitters are not equally bright [
16]. If
d_{x} σ,
J ≈
for each coordinate. When
d_{x} σ, one cannot unambiguously assign each photon to its correct emitter and
J_{dy,dy},
J_{dx,dx} both vanish. This is why in stochastic localization microscopy one typically uses a very dilute density of simultaneously active emitters, ρ
_{a} σ
^{−2}, to prevent emitters’ images overlapping.
This analysis extends to scenes of more emitters [], but in the general case localization accuracy varies with the scene’s details. The Fisher information for m emitters is a 2m × 2m matrix whose entries depend on the emitters’ 2m position coordinates. For m > 2 there is no general coordinate system diagonalizing J. A resolution measure that varies from sample to sample is inconsistent with traditional concepts of resolution and has limited practical utility.
We introduce a measure, the ITF, that gives resolution bounds for all specimens and generalizes the MTF to account for photon statistics. The ITF provides the CRLB for image estimation as a function of spatial frequency; this conception of resolution applies equally well to conventional or stochastic localization microscopy.
Consider an image scene, represented by a positive intensity function,
I(
x), normalized to have unit integral. In biological microscopy,
I may represent the spatial probability density of fluorescently labeled cellular or protein structures. We seek a bound on the squared error of an unbiased estimator,
Î(
k). The ITF,
F(
k), bounds the error in estimating the image frequency components:
E_{I(k)} 
Î(
k) −
I(
k)
^{2} ≥ 1/
F(
k). (In practice it may be impossible to define unbiased estimators for some image components, especially at high spatial frequencies [
16]. A generalization of the CRLB relates
F(
k) to the minimum variance of a biased estimator [
13]. Even when no unbiased estimator exists, the ITF provides an objective figure of merit for image resolution that is based on the measurement statistics.)
Physical intuition suggests the ITF should vanish at large
k and be proportional to
N, since each photon carries equal Fisher information. We first calculate the ITF exactly using three key simplifying assumptions. First, we ignore photons that cannot be assigned to an emitter due to overlap; the case of overlapping emitters will be considered below. This assumption simplifies the probability,
Eq. (2). If
n_{j} is the number of detected photons from the
jth emitter, and assuming the fluorophores’ locations are randomly sampled from the distribution,
I(
x), we have [
16]:
Second, assuming the PSF,
h, is Gaussian, we can compute the integral:
where
is the centroid of the observed photons’ locations, and
Ĩ_{nj} (
$\widehat{x}$_{j})
∫
dx_{j}I(
x_{j})
e^{−(nj/2σ2)(xj−$\widehat{x}$j)2} (
n_{j}/2πσ
^{2}) is a blurred version of the true image with blurring scale
[
16]. Third, we assume each emitter emits the same number of photons,
n_{j} =
n; numerical results show that a more realistic calculation with Poissondistributed
n_{j} has a nearly identical result [
16]. We find
Using the normalization R of the image and the PSF, ∫
d$\widehat{x}$Ĩ_{n}(
$\widehat{x}$) = 1, we invert the Fisher information matrix to compute the ITF:
where
A is the field of view area and ρ
_{e} is the total density of emitters; see . (Note ρ
_{e} appearing here is the density of all emitters observed in the experiment, whereas ρ
_{a} introduced above is the density of simultaneously activated emitters in a single time frame.)
This result is derived more explicitly in [
16] and shows that the ITF is an imageindependent, universal limit valid for any scene.
Equation (6) shows that superresolution hinges on the ability to have
n 1; in practice
n ~ 100−10
^{4} or more. In conventional microscopy, individual emitters may contribute more than one photon to the image, but since photons cannot be reliably assigned to the correct emitter their statistics are equivalent to the case
n = 1. Increasing the emitter labeling density, ρ
_{e}, reduces the error variance at all spatial frequencies but has little effect on the threshold frequency at which the ITF becomes vanishingly small [].
We can also compute the ITF exactly, with no restriction on the PSF, for conventional microscopy (
n = 1). In this case
Eq. (3) factorizes into a product of independent contributions from each photon, and the ITF is [
16]:
F(
k) =
N
h(
k)
^{2}. This result connects the ITF directly to the conventional MTF.
Both of these exact results are special cases of our main finding, which is a general bound on the ITF for any PSF. If the MTF is bounded by a Gaussian, 
h(
k) ≤
e^{−(2πkσ)2/2}, then we find [
16]:
where
C is a constant that depends on the image but not on
k,
A or ρ
_{e}.
We now consider how the ITF applies in the case the emitters’ images overlap, which is common when the density of active emitters, ρ
_{a}, approaches σ
^{−2}. Several computational procedures have recently been proposed for analyzing data from such crowded scenes to speed up superresolution image acquisition, but the fundamental limits on resolution and speed remain unknown [
17–
21]. We show that the spatial frequencydependent ITF is generally applicable and smoothly depends on ρ
_{a}.
To derive how the ITF depends on ρ
_{a}, we must specialize to images consisting of
m emitters. (More generally, the following model applies to images of
M emitters obtained in
M/m distinct rounds with exactly
m active emitters per round. Each round contributes independent information, so the total ITF is the sum of the ITF for each.) The image is
in real space, or
in the frequency domain. We can compute
F(
k) through its relation to the realspace Fisher information matrix [
13,
16]: 1/
F(
k)
. Conventional microscopy corresponds to ρ
_{a} σ
^{−2}. In this limit we may approximate the Poisson statistics of photon shot noise as additive Gaussian fluctuations in intensity, valid when many photons are detected per pixel [
16]. The result matches the form derived above for general images when
n = 1 photon is observed per emitter [, solid black curve].
We also reconsider superresolution microscopy with sparse emitters, ρ
_{a} σ
^{−2}. Here the Fisher information matrix is diagonal,
J_{ij} =
nJ^{(1)}δ
_{ij}, where
J^{(1)} is the Fisher information per photon for localizing a single emitter [
16]. The ITF,
, declines only quadratically with
k [, black dotdashed line]. Prior knowledge that the image structure consists of
m discrete emitters improves estimation as compared with a completely general image. Hence, the ITF here is strictly greater than the previous bound,
Eq. (7), derived for a general image with
n > 1. However, in both cases, the highest resolvable spatial frequency is proportional to
.
Besides providing a unified formalism for these two extremes, the ITF can quantify the tradeoff between image resolution and acquisition time as a function of the density of activated emitters. To do this, we computed the ITF for simulated specimens with emitters at a range of densities. The ITF smoothly transitions between the low and high emitter density regimes, . In livecell imaging, the time to acquire an image,
T, is a constrained resource. For fixed
T, the number of detected photons scales as
N B_{ρa}T, where
B is the average emitter brightness. There is thus a tradeoff between achieving superresolution, by working at low ρ
_{a}, and increasing the signaltonoise ratio by collecting more photons []. Anextremely low density of emitters is suboptimal, with
F(
k) falling off as ρ
_{a} (blue curve) for frequencies above the diffraction limit (vertical dashed line). Our simulations reveal the optimal active emitter density,
, that maximizes the ITF for each
k, . The sum of the ITF over all frequencies greater than the conventional diffraction limit, , is maximum around
, which provides sufficient separability for localizing neighboring emitters without wasting time by overly diluting the emitters.
In summary, we used estimation theory to calculate the ITF, a unified bound on image estimation accuracy, for stochastic localization and conventional fluorescence microscopy. For completely general images, the key parameter is the number of observed photons per emitter,
n. In conventional microscopy (
n = 1) the ITF reduces to the square of the MTF times the total number of observed photons. This result provides a performance bound for superresolution techniques that combine multiple, shifted images or photographs [
4,
5]. Such techniques can overcome undersampling due to finite pixels, but the fundamental resolution limit imposed by optical diffraction is described by the ITF for conventional microscopy [
6].
Overall, the ITF formalism reveals fundamental resolution limits on fluorescence microscopy, unifying the conventional concepts of resolution with those developed separately for stochastic localization microscopy. The ITF also fulfills the longstanding goal of characterizing how resolution depends on the unavoidable limitations imposed by photon statistics [
3,
11,
12]. Further, the ITF facilitates principled choices of experimental parameters that optimize imaging performance at any chosen length scale. Although improved localization and image estimation algorithms will surely continue to emerge [
17–
21], our work already provides the fundamental physical limits on the possible improvements. The ITF bounds the performance of any estimation procedure in fluorescence microscopy, regardless of emitter density.