ARTICLES
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Resolution in optical nanoscopy (or super-resolution
microscopy) depends on the localization uncertainty and
density of single fluorescent labels and on the sample’s
spatial structure. Currently there is no integral, practical
resolution measure that accounts for all factors. We introduce
a measure based on Fourier ring correlation (FRC) that can be
computed directly from an image. We demonstrate its validity
and benefits on two-dimensional (2D) and 3D localization
microscopy images of tubulin and actin filaments. Our FRC
resolution method makes it possible to compare achieved
resolutions in images taken with different nanoscopy methods,
to optimize and rank different emitter localization and labeling
strategies, to define a stopping criterion for data acquisition,
to describe image anisotropy and heterogeneity, and even
to estimate the average number of localizations per emitter.
Our findings challenge the current focus on obtaining the best
localization precision, showing instead how the best image
resolution can be achieved as fast as possible.
The first and foremost law of conventional optical imaging science
is that resolution is limited to a value on the order of
λ
/NA, with
λ
equal to the wavelength of light and NA to the numerical aper-
ture of the imaging lens. Rayleigh and Sparrow captured this law
through empirical resolution criteria. These criteria were placed
on solid foundations by Abbe and Nyquist, who defined resolu-
tion as the inverse of the spatial bandwidth of the imaging system.
This diffraction limit, however, can be overcome by numerous
optical nanoscopy techniques, notably stimulated emission deple-
tion (STED)
1
, reversible saturable optical fluorescence transitions
(RESOLFT)
2
, the family of localization microscopy techniques
such as photoactivated localization microscopy (PALM), stochas-
tic optical reconstruction microscopy (STORM), ground state
depletion microscopy followed by individual molecule return
(GSDIM) and direct STORM
3–6
and statistical methods such as
blinking fluorescence localization and super-resolution optical
fluctuation imaging (SOFI)
7,8
.
These revolutionary developments raise the question: what is
resolution in diffraction-unlimited imaging? The resolving power
of the instrument is often coupled to the uncertainty of localizing
Measuring image resolution in optical nanoscopy
Robert P J Nieuwenhuizen
1
, Keith A Lidke
2
, Mark Bates
3
, Daniela Leyton Puig
4
, David Grünwald
5,6
,
Sjoerd Stallinga
1,7
& Bernd Rieger
1,7
single emitters, that is, point sources. The closely related two-
point resolution can be given a precise meaning in the context
of localization microscopy
9
, thus generalizing the Rayleigh cri-
terion of conventional microscopy. These concepts characterize
the resolution in images in which the structure of interest can be
defined by a limited number of molecules—such as images of the
nuclear pore complex
10
—or when investigating the relative posi-
tion of different molecules
11
. However, if more-or-less continu-
ous structures with a large number of potential labeling sites are
imaged—for example, actin filaments or organelle membranes—
then it is clear that the average density of localized fluorescent
labels must also play a role. As early as the first demonstration
of localization microscopy for cell imaging
3
, it was noted that
“both parameters—localization precision and the density of ren-
dered molecule—are key to defining performance….” The effects
of labeling density and photoswitching kinetics on resolution
have since been investigated experimentally
12,13
. Recently, an
estimation-theoretic resolution concept was presented
14
that
combines both labeling density and localization uncertainty using
an a priori model of the sample.
We conclude from all prior work that neither the average den-
sity of localized molecules needed for random Nyquist sampling
nor the localization uncertainty alone is a suitable measure to
characterize the resolution. In addition, the resolution depends
on a multitude of other factors such as the link between the label
and the structure, the underlying spatial structure of the sample
itself and the extensive data processing required to produce a final
super-resolution image comprising, for example, single-emitter
candidate selection and localization algorithms. Ultimately, only
an integral, image-based resolution measure not depending on
any a priori information is suitable for determining what level of
detail can be reliably discerned in a given image.
Here we propose an image-resolution measure that can be com-
puted directly from experimental data alone. It is centered on the
FRC (or, equivalently, the spectral signal-to-noise ratio), which is
commonly used in the field of cryo-electron microscopy (cryo-
EM) to assess single-particle reconstructions of macromolecular
complexes
15–17
. We have used the FRC resolution to analyze the
trade-off between localization uncertainty and labeling density,
1
Quantitative Imaging Group, Delft University of Technology, Delft, The Netherlands.
2
Department of Physics and Astronomy, University of New Mexico, Albuquerque,
New Mexico, USA.
3
Department of NanoBiophotonics, Max Planck Institute for Biophysical Chemistry, Göttingen, Germany.
4
Division of Cell Biology, The Netherlands
Cancer Institute, Amsterdam, The Netherlands.
5
Department of Biochemistry and Molecular Pharmacology, University of Massachusetts Medical School, Worcester,
Massachusetts, USA.
6
RNA Therapeutics Institute, University of Massachusetts Medical School, Worcester, Massachusetts, USA.
7
These authors contributed equally to this
work. Correspondence should be addressed to B.R. (b.rieger@tudelft.nl) or S.S. (s.stallinga@tudelft.nl).
Received 27 decembeR 2012; accepted 21 maRch 2013; published online 28 apRil 2013; doi:10.1038/nmeth.2448
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and we have applied it to monitor resolution buildup during data
acquisition and to compare different localization algorithms.
Quantification of the spatial correlations in the image leading
to this resolution measure also provides a means to estimate the
average number of localizations per emitter contributing to the
image. Software for computing the FRC curve and the image
resolution for localization microscopy data is available in the
form of an ImageJ plugin and Matlab code at http://www.diplib.
org/add-ons/ and as Supplementary Software.
RESULTS
To compute the FRC resolution, we divide the set of single-emitter
localizations that constitute a super-resolution image into two
statistically independent subsets, which yields two subimages
f r
1
( )
and
f r
2
( )
, where
r
denotes the spatial coordinates. Subsequent
statistical correlation of their Fourier transforms
f q
1
( )
and
f q
2
( )
over the pixels on the perimeter of circles of constant spatial fre-
quency with magnitude q = |
q
| gives the FRC
16
FRC
circle
circle
*
( )
( ) ( )
| ( )| | (
q
f q f q
q
f q
q
f
=
∈
∑
∈
∑
1 2
1
2
2
qq
q
)|
2
∈
∑
circle
For low spatial frequencies, the FRC curve is close to unity; and
for high spatial frequencies, noise dominates the data and the
FRC decays to 0. The image resolution is defined as the inverse
of the spatial frequency for which the FRC curve drops below a
given threshold. We evaluated different threshold criteria used
in the field of cryo-EM
15,18–20
and found that the fixed threshold
equal to 1/7 ≈ 0.143 (ref. 18) is most appropriate for localization
microscopy images (Supplementary Fig. 1). The FRC resolu-
tion concept and the steps needed to compute it are illustrated
in Figure 1a. FRC resolution describes the length scale below
which the image lacks signal content; smaller details are not
resolved in the image. Resolution values will always be larger
than those based on localization uncertainty or labeling density
alone (Supplementary Fig. 2).
(1)(1)
Theoretical considerations and simulations
FRC resolution allows predictions to be made about the impact
of different imaging and sample parameters on the achievable
resolution; these predictions are based on the expectation value of
the FRC curve (Supplementary Figs. 3 and 4 and Supplementary
Note 1), which is given by
〈 〉 =
+ −
∈
∑
+ +
FRC
circle
( )
( | ( )| )exp( )
[ ( | (
q
Q N q q
q
Q N
y p s
y
2
4
2 2 2
2
qq q
q
)| )exp( )]
2
4
2 2 2
−
∈
∑
p s
circle
where N is the total number of localized emitters,
σ
is the average
localization uncertainty and
y ( )q
denotes the Fourier spectrum
of the spatial distribution of the fluorescent emitters. The param-
eter Q is a measure for spurious correlations due to, for example,
repeated photoactivation of the same emitter. Each emitter con-
tributing to the image is localized once for Q = 0 and in general
Q/(1 – e
−Q
) times on average, provided the emitter activation
follows Poisson statistics. Careful analysis of the spatiotempo-
ral correlations in the image and the emitter activation statistics
(including effects of photobleaching) can provide a way to esti-
mate Q and correct for its effect on image resolution as well as
to estimate the number of fluorescent labels contributing to the
image, as is discussed below.
Analytical expressions for the resolution can be derived for
particular object types (such as line pairs) often used in resolu-
tion definitions (Supplementary Note 2). The resolution R for
an image consisting of two parallel lines with a cosine-squared
(2)(2)
Split
Fourier
transform
Fourier
transform
q
a
100 nm
c
Localization uncertainty (nm)
Labeling density (10
4
per µm
2
)
d
Localization uncertainty (nm)
R (nm)
R = 2
b
100 nm
2
= e/6
0 0.05 0.10 0.15 0.20 0.25
0
0.2
0.4
0.6
0.8
1.0
Spatial frequency (nm
−1
)
q
res
FRC
Threshold
Expected
Correlation between two rings
1.00
0.75
0.50
0.25
50 10 15 20 25
60
50
40
30
20
10
0
5
10
0
20 nm
60 nm
80 nm
1 min
5 min
10 min
20 min
30 min
100 nm
40 nm
Figure 1
|
The FRC principle and trade-off between localization
uncertainty and labeling density. (a) All localizations are divided into
two halves, and the correlation between their Fourier transforms over
the perimeter of the circle in Fourier space of radius q is calculated for
each q, resulting in an FRC curve indicating the decay of the correlation
with spatial frequency. The image resolution is the inverse of the spatial
frequency for which the FRC curve drops below the threshold 1/7 ≈ 0.143,
so a threshold value at q = 0.04 nm
−1
is equivalent to a 25-nm resolution.
Error bars indicate theoretically expected s.d. (Supplementary Note 1).
(b) Simulated localization microscopy image of a line pair with mean
labeling density
ρ
= 2.5 × 10
3
per µm
2
in the area occupied by the lines
and localization uncertainty
σ
= 7.6 nm (line distance 70 nm, cosine-
squared cross-section). (c) Constant resolution in theory (lines) and
simulation data (circles) for line pairs as in b as a function of localization
uncertainty and labeling density. Regions of localization uncertainty–
limited resolution (blue) and labeling density–limited resolution (yellow)
are separated by the red line
ρσ
2
= e/(6
π
). (d) Simulation results for
localization uncertainty versus image resolution for different fixed total
measurement times. Camera frame rates were varied to match the on-times
of the emitter. The minima of the curves fall on the line R = 2
πσ
that
separates the yellow region (not enough emitters localized) from the blue
region (emitters not localized precisely enough).
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cross-section and mean labeling density
ρ
in the area occupied
by the lines is
R
W
=
2
6
2
ps
prs( )
where W(x) is the Lambert W-function
21
. Two regimes can be
identified in which changes in either labeling density or localiza-
tion uncertainty have the most impact on improving the resolu-
tion. At the boundary between these regimes, the relative gains
in resolution due to changes in either quantity are equally large.
This trade-off occurs at R = 2
πσ
(Supplementary Note 3), which
corresponds to
rs
p
2
6
0 14= ≈
e
.
The region
ρσ
2
< e/(6
π
) is labeling-density limited, whereas
ρσ
2
> e/(6
π
) is localization-uncertainty limited (Fig. 1b,c). The
exact boundary between the two regimes depends on the underly-
ing object, so the boundary value for the two-line example serves
only as a rule of thumb (Supplementary Note 3). For example,
for M parallel lines, we obtain a value e/(3
π
M). From this it may
be inferred that the trade-off occurs for a value smaller than 0.14
for any intricate but irregular object structure.
The same trade-off as above may also manifest itself in the opti-
mization of image resolution, given a fixed total acquisition time
(
Fig. 1b,d). Suppose that the photon count per localization is
improved by increasing the on-times of the emitters while keeping the
emitters’ brightness and the number of simultaneously active emitters
(3)(3)
(4)(4)
constant: this then also reduces the total number of labels that can be
localized in a given acquisition time. Therefore, longer single-emitter
events yield more accurate localizations, but at the expense of a lower
recorded emitter density
3,22
. Again, the optimum is R = 2
πσ
, inde-
pendent of the object (Supplementary Note 3). Tuning the on-times
as described here may be done in the design phase of an experiment
by the choice of label or buffer composition.
Resolution buildup during data acquisition
To test and evaluate the FRC resolution measure, we imaged
tubulin networks in fixed HeLa cells labeled with Alexa Fluor
647 using localization microscopy (Fig. 2a and Online Methods).
The resolution improved with acquisition time (Fig. 2b–f), or,
equivalently, with the density of localized labels. The trade-off
point between the localization density and uncertainty limited
regimes lay at R = 2
πσ
= 61 nm. Therefore, the resolution values
for Figures 2b–e were labeling-density limited, and the trade-off
point was just crossed at the end of the data acquisition. Real-time
monitoring of the resolution buildup by real-time single-molecule
fitting algorithms
23
provides a much needed stopping criterion
for localization microscopy data acquisitions.
The FRC resolution concept is also sensitive to differences in
localization uncertainty (Fig. 2g–i). Maximum-likelihood estima-
tion (R = 58 ± 1 nm) is theoretically optimal
24
and is slightly better
than least-squares fitting (R = 60 ± 1 nm) and superior to centroid
fitting (R = 88 ± 2 nm). All specified uncertainties are computed
from 20 FRC resolution estimates obtained from different random
assignments of localizations to half data sets (s.e.m.). Because the
Figure 2
|
The effect of localization density and
data processing on resolution. (a) Localization
microscopy image of tubulin labeled with Alexa
Fluor 647 in a HeLa cell (R = 58 ± 1 nm for the
whole image, where uncertainty reflects s.e.m. of
20 random repeats of FRC resolution calculation).
Acquisition time was T = 12 min within 1.4 × 10
4
frames, the localization uncertainty was
σ
= 9.7 nm
after merging nearby localizations in subsequent
frames (Online Methods) and the density of
localizations was
ρ
= 6.0 × 10
2
per µm
2
.
(b–e) Magnified insets of two crossing filaments
(upper boxed region in a) constructed from
fewer time frames showing poorer resolution
(indicated by the distance between the blue arrows).
(f) Resolution (R) buildup during acquisition, with
R = 2
πσ
plotted in blue, showing a transition from
density-limited to precision-limited resolution.
(g–i) Reconstructions of lower boxed region in a by
different localization algorithms showing maximum-
likelihood estimation (g; MLE, R = 58 ± 1 nm), least
squares fitting (h; LS, R = 60 ± 1 nm) and centroid
fitting (i; CEN, R = 88 ± 2 nm). (j) Localization
microscopy image of the actin cytoskeleton (F-actin)
of a fixed HeLa cell labeled with phalloidin coupled
to Alexa Fluor 647 after correction for sample drift
of ~70–100 nm during acquisition. The image was
obtained from 5.0 × 10
4
frames in 8 min (
σ
= 8.0 nm,
ρ
= 8.2 × 10
3
µm
−2
,
ρσ
2
= 0.52, 2
πσ
= 50 nm).
(k–n) Magnified insets of reconstructions before
(k,l; left boxed region in j) and after drift correction
(m,n; right boxed region in j). Resolutions before
and after drift correction were R = 79 ± 2 nm and
R = 54 ± 1 nm, respectively. The arrows indicate
regions of sharper detail after drift correction.
a
R = 58 nm
4 µm
j
4 µm
k
l
Without drift correction
R = 79 nm
200 nm
g
400 nm
h i
e
T/8
T/4
T/2
R = 88 nm
CENLSMLE
R = 60 nm
b dc
f
0 0.2 0.4 0.6 0.8 1.0
40
80
120
160
Relative acquisition time
R (nm)
500 nm
500 nm
m
With drift correction
R = 54 nm
n
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effect of the parameter Q on the resolution
for this data set was found to be negligible,
it was not necessary to correct for it.
Sample drift is a common annoy-
ance in optical nanoscopy, as motion has to be limited to a few
nanometers over typical acquisition times of many minutes. We
analyzed the drift in localization microscopy data of the actin
cytoskeleton of a fixed HeLa cell labeled with phalloidin coupled
to Alexa Fluor 647 (Fig. 2j) without the use of fiducial mark-
ers
25
. A drift of ~70–100 nm was found with this procedure
and corrected for. Computed resolution values before drift cor-
rection (Fig. 2k,l; R = 79 ± 1 nm) were much worse than those
after drift correction (Fig. 2m,n; R = 54 ± 1 nm), which is in
agreement with the apparent detail in the images (Fig. 2k–n).
For this data set also, the effect of Q was found to be negligible.
More experimental data are provided to show the validity of the
FRC resolution (Supplementary Figs. 5–7).
Estimation of the number of localizations per emitter
Multiple localizations per emitter due to, for example, repeated
photoactivations lead to spurious correlations between the two
image halves, resulting in overoptimistic resolution values. This
is particularly problematic for cases involving large numbers of
localizations per emitter, low localization uncertainties and low
labeling densities (Supplementary Figs. 8 and 9).
The FRC can be corrected for this effect by estimating the spu-
rious correlation parameter Q in equation (2). To that end, we
divided the numerator of the FRC by the weighted average of
the function exp(−4
π
2
σ
2
q
2
) over the distribution of localization
uncertainties. The parameter Q is proportional to the minimum
of that curve, which takes the form of a broad plateau if Q >> 1
(Online Methods). To test this method, we analyzed a two-color
image of tubulin labeled with both Alexa Fluor 647 and Alexa
Fluor 750 (ref. 26; Fig. 3a–c). The resolution values for Alexa
Fluor 647 and Alexa Fluor 750 without correction (25 ± 1 nm
and 34 ± 1 nm, respectively) were much lower than the resolution
derived from the cross-channel, that is, when taking the two color
images as data halves for the FRC (118 ± 2 nm). This difference
was due to the multiple localizations per emitter, which affect the
one-color FRC curves but not the cross-channel curve. The FRC
curves and attendant resolution values were much more simi-
lar after correction. The remaining differences in the calculated
resolution values reflected the differences in labeling density (the
density of localizations was 4.0 × 10
3
per µm
2
for Alexa Fluor
647 and 1.3 × 10
3
per µm
2
for Alexa Fluor 750) and localization
uncertainty (9.2 nm and 12 nm, respectively).
We checked the data sets of Figure 2a,j for spurious correla-
tions and found Q = 0.28 and Q = 0.33, respectively, which led
to corrected resolution values equal to 62 ± 2 nm and 66 ± 1 nm,
respectively. This means that neglecting to correct for spurious
correlations gave rise to an underestimation of the resolution
value by only several nanometers. These estimated values for
Q are smaller than the values for the data set of Figure 3 primarily
because Q scales with the data acquisition time, which is much
smaller in Figure 2a,j than the 1.4 × 10
5
frames in 39 min and
3.0 × 10
4
frames in 25 min for Alexa Fluor 647 and Alexa Fluor
750, respectively (Fig. 3). Other reasons for the discrepancy of
Q values may be found in differences in photobleaching behav-
ior and preprocessing for candidate selection of single-emitter
events (from false positives, for example). Finally, the density of
localizations for Figure 2j is close to 10
4
per µm
2
, 1–2 orders
of magnitude larger than the density in other data sets. In the
limit of high labeling density, the effects of spurious correla-
tions are negligible compared to the intrinsic image correlations
Figure 3
|
Spurious correlations from a two-color
localization microscopy image. (a) Overview image
of a tubulin network labeled with both Alexa
Fluor 647 (magenta) and Alexa Fluor 750 (green).
The inset shows the quality of registration.
(b) Uncorrected FRC curves for the magenta
and green channels are higher than that for the
cross-channel because of spurious correlations
from repeated photoactivations of individual
emitters, which result in overly optimistic
resolution values (R = 25 ± 1 nm and 34 ± 1 nm,
respectively, compared to 118 ± 2 nm for the
cross-channel). Uncertainty reflects s.e.m.
of 20 random repeats of the FRC resolution
calculation. (c) FRC curves corrected for spurious
correlations all yield similar resolution values
(108 ± 1 nm for Alexa Fluor 647, 133 ± 2 nm
for Alexa Fluor 750, 121 ± 2 nm for the cross-
channel). (d–f) Scaled FRC numerator curves
showing a plateau for intermediate spatial
frequencies, which is used to estimate the
correction term and Q parameter. For this
correction (Online Methods) we used a mean
and width of the distribution of localization
uncertainties equal to 9.2 nm and 2.8 nm for
Alexa Fluor 647 and 12 nm and 2.0 nm for
Alexa Fluor 750.
0 0.01 0.02 0.03 0.04 0.05
−2
−1
0
1
2
3
Spatial frequency (nm
−1
)
log
10
(scaled FRC numerator)
Data
Smooth
Plateau
Alexa 647
Q = 10
d
0 0.01 0.02 0.03 0.04 0.05
−2
−1
0
1
2
3
Spatial frequency (nm
−1
)
log
10
(scaled FRC numerator)
Alexa 750
Q = 18
e
0 0.01 0.02 0.03 0.04 0.05
−2
−1
0
1
2
3
Spatial frequency (nm
−1
)
log
10
(scaled FRC numerator)
Cross-channel
Q = 0.5
f
0 0.01 0.02 0.03 0.04 0.05
0
0.2
0.4
0.6
0.8
1.0
Corrected FRC
Alexa 647
Alexa 750
Cross-channel
0 0.01 0.02 0.03 0.04 0.05
0
0.2
0.4
0.6
0.8
1.0
Spatial frequency (nm
−1
)
Spatial frequency (nm
−1
)
Uncorrected FRC
a
400 nm
4 µm
34 nm
118 nm
b
25 nm
R =
108 nm
133 nm
121 nm
c
R =
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(Supplementary Notes 1 and 2). We point
out that the correction method appears to
be quite sensitive to (the distribution of)
the localization uncertainty, and to any residual effects of drift,
and must therefore be applied with care.
The estimation of the average number of localizations per
emitter from the spurious correlation parameter Q also makes
it possible to count the actual number of fluorescent labels that
contribute to the overall image. Although such counting has been
demonstrated for irreversibly photoactivatable fluorophores
27,28
,
only a few studies have investigated the possibility of counting
with reversibly photoswitchable dyes
29,30
. One approach is based
on pair correlation functions
28,29
, but unlike our method, it
requires a model for the correlations in the spatial distribution of
the fluorescent labels. Neither do we require a calibration experi-
ment, in contrast to cluster kymography analysis, for example
30
.
A potential complication for our method is that deviations from
Poisson statistics of emitter activations due to photobleaching
may lead to overestimation and in some cases underestimation
(Q < 1) of the number of localizations per emitter (Supplementary
Note 1). The same caveat applies to alternative approaches
28,29
.
Control experiments on sparsely distributed labeled antibodies on
a glass surface indicated that photobleaching caused Q values for
the data (Fig. 3) to overestimate the true number of localizations
per emitter by a factor of 1.5 for Alexa Fluor 647 and 1.7 for Alexa
Fluor 750, even though the Q parameter was estimated much more
accurately (Supplementary Fig. 10 and Online Methods). Taking
into account all factors leads to an estimated number of localiza-
tions per molecule equal to 7 for Alexa Fluor 647 and 11 for Alexa
Fluor 750, which is in qualitative agreement with values reported
earlier
31
. We found the labeling densities to be 6.0 × 10
2
per µm
2
(Alexa Fluor 647) and 1.2 × 10
2
per µm
2
(Alexa Fluor 750).
Resolution in 3D, anisotropic and heterogeneous content
The FRC resolution concept can be generalized and extended in
several ways. The first way addresses image anisotropy, which
may arise, for example, from line-like features in the image or
from differences between the axial and lateral resolving power
in 3D imaging
32
. Anisotropic image resolution can be described
similarly to FRC by correlating the two data halves in Fourier
space over a line in 2D (Fourier line correlation, FLC) or plane
in 3D (Fourier plane correlation, FPC) perpendicular to spatial
frequency vectors
q
. Spatial frequencies for which the FLC or FPC
is above the threshold in the image are resolved. The FPC for a 3D
image of a tubulin network labeled with Alexa Fluor 647 (Fig. 4)
using the bifocal method
33
shows clear anisotropy with filaments
oriented mostly in the xy plane along the x direction. Therefore,
the FPC is highest in the y direction, orthogonal to the filaments,
and worst in the z direction.
The FLC for part of the data set of Figure 2j shows that the
region of resolved spatial frequencies is anisotropic and high-
est in the direction orthogonal to the filaments (Supplementary
Fig. 11), as expected. Another way in which the FRC resolution
concept can be generalized targets local variations in the density
of the sample’s spatial structure. Local image resolution can be
obtained from resolution values of overlapping subimage patches
(Supplementary Fig. 12).
DISCUSSION
The FRC resolution concept can be naturally extended to STED,
imaging with an extended diffraction limit such as structured illu-
mination microscopy
34
, and conventional confocal and wide-field
imaging. It is possible not only to conceptually extend the FRC
method but also to measure the resolution directly from experi-
mental data. This stands in contrast to recently introduced unified
resolution concepts
35
, which provide only a rigorous theoretical
framework. The FRC resolution is most easily computed from
two images of the same scene that differ only in noise content.
The resolution then depends on the signal-to-noise ratio, spectral
image content and (effective) optical transfer function. The width
of the effective point-spread function replaces the role of the
localization uncertainty. In the limit of infinitely high signal-to-
noise ratio, the FRC resolution reduces to Abbe’s diffraction limit
(for the conventional fluorescence imaging modalities) or to the
limit that has been proposed for STED
36
(Supplementary Note 4
and Supplementary Fig. 13). We also applied the FRC concept
to time-lapse recordings in scanning electron microscopy in
single-electron counting mode (Supplementary Fig. 14). For any
extension of the FRC concept, systematic dependencies between
image halves due to, for example, fixed-pattern noise or common
alignment references must be prevented. Alignment references
have caused particular problems for the application of the FRC
concept in the field of single-particle cryo-EM
37
.
We envision that FRC resolution may be used for character-
izing and optimizing fluorescence labeling and data processing
Figure 4
|
3D resolution. (a) Representation
of a 3D localization microscopy image of a
tubulin network, with the axial coordinate
in false color. (b) Orthogonal slices of the
Fourier plane correlation (FPC). (c–e) Cross-
sections of the FPC for this data set in the
q
x
q
z
plane (c), q
y
q
z
plane (d) and q
x
q
y
plane (e),
with added threshold contours for FPC = 1/7
(black lines). The FPC clearly shows the
anisotropy of image content resulting from
the line-like structure of the filaments
(the highest image resolution is perpendicular
to the filaments) as well as from the anisotropy
in localization uncertainty (the lowest
resolution is in the axial direction).
b
q
x
q
z
q
y
x
y
a
0.4
0.8
1.2
0
2
µm
z (µm)
c
8
4
0
–4
–8
840–4–8
q
z
(µm
–1
)
q
x
(µm
–1
)
0.6
0.4
0.2
0
d
FPC
8
4
0
–4
–8
840–4–8
q
z
(µm
–1
)
q
y
(µm
–1
)
e
8
4
0
–4
–8
840–4–8
q
y
(µm
–1
)
q
x
(µm
–1
)
npg
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