Geissbuehler et al. Optical Nanoscopy 2012, 1:4http://www.optnano.com/content/1/1/4

ORIGINAL ARTICLE Open Access

Mapping molecular statistics with balancedsuper-resolution optical fluctuationimaging (bSOFI)Stefan Geissbuehler*, Noelia L Bocchio, Claudio Dellagiacoma, Corinne Berclaz, Marcel Leutenegger andTheo Lasser

Abstract

Background: Super-resolution optical fluctuation imaging (SOFI) achieves 3D super-resolution by computingtemporal cumulants or spatio-temporal cross-cumulants of stochastically blinking fluorophores. In contrast tolocalization microscopy, SOFI is compatible with weakly emitting fluorophores and a wide range of blinkingconditions. The main drawback of SOFI is the nonlinear response to brightness and blinking heterogeneities in thesample, which limits the use of higher cumulant orders for improving the resolution.Balanced super-resolution optical fluctuation imaging (bSOFI) analyses several cumulant orders for extracting molec-ular parameter maps, such as molecular state lifetimes, concentration and brightness distributions of fluorophoreswithin biological samples. Moreover, the estimated blinking statistics are used to balance the image contrast, i.e.linearize the brightness and blinking response and to obtain a resolution improving linearly with the cumulant order.

Results: Using a widefield total-internal-reflection (TIR) fluorescence microscope, we acquired image sequences offluorescently labelled microtubules in fixed HeLa cells. We demonstrate an up to five-fold resolution improvement ascompared to the diffraction-limited image, despite low single-frame signal-to-noise ratios. Due to the TIR illumination,the intensity profile in the sample decreases exponentially along the optical axis, which is reported by the estimatedspatial distributions of the molecular brightness as well as the blinking on-ratio. Therefore, TIR-bSOFI also encodesdepth information through these parameter maps.

Conclusions: bSOFI is an extended version of SOFI that cancels the nonlinear response to brightness and blinkingheterogeneities. The obtained balanced image contrast significantly enhances the visual perception ofsuper-resolution based on higher-order cumulants and thereby facilitates the access to higher resolutions.Furthermore, bSOFI provides microenvironment-related molecular parameter maps and paves the way for functionalsuper-resolution microscopy based on stochastic switching.

Keywords: Fluorescence microscopy, Super-resolution, Stochastic switching, Sofi, Cumulants, Balanced contrast,molecular statistics, Functional imaging

BackgroundThe spatial resolution in classical optical microscopesis limited by diffraction to about half the wavelengthof light. During the last two decades, several super-resolution concepts have been developed. Based on theon-off-switching of fluorescent probes, these conceptsovercome the diffraction limit by up to an order of

*Correspondence: stefan.geissbuehler@epfl.chLaboratoire d’Optique Biomedicale, Ecole Polytechnique Federale deLausanne, 1015 Lausanne, Switzerland

magnitude (Huang et al. 2009). A straightforward methodconsists of digitally post-processing an image sequence ofstochastically blinking emitters acquired with a standardwide-field fluorescence microscope. Densely packed sin-gle fluorophores can be distinguished in time by usinghigh-precision localization algorithms, used for instancein photo-activation localization microscopy (PALM)(Betzig et al. 2006; Hess et al. 2006) and stochastic opticalreconstruction microscopy (STORM) (Heilemann et al.2008; Rust et al. 2006), or by analysing the statistics of

© 2012 Geissbuehler et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

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the temporal fluctuations as exploited in super-resolutionoptical fluctuation imaging (SOFI) (Dertinger et al. 2009;Dertinger et al. 2010). SOFI is based on a pixel-wiseauto- or cross-cumulant analysis, which yields a resolu-tion enhancement growing with the cumulant order inall three dimensions (Dertinger et al. 2009). Uncorre-lated noise, stationary background, as well as out-of-focuslight are greatly reduced by the cumulants analysis. WhilePALM and STORM are commonly based on a frame-by-frame analysis of images of individual fluorophores, SOFIprocesses the entire image sequence at once and thereforepresents significant advantages in terms of the number ofrequired photons per fluorophore and image, as well asin acquisition time (Geissbuehler et al. 2011). Localizationmicroscopy requires a meta-stable dark state for imag-ing individual fluorophores (van de Linde et al. 2010). Incontrast, SOFI relies solely on stochastic, reversible andtemporally resolvable fluorescence fluctuations almostregardless of the on-off duty cycle (Geissbuehler et al.2011). The main drawback of SOFI is the amplificationof heterogeneities in molecular brightness and blinkingstatistics which limits the use of higher-order cumu-lants and therefore resolution. In this article, we revisitedthe original SOFI concept and propose a reformulationcalled balanced super-resolution optical fluctuation imag-ing (bSOFI), which in addition to improving structuraldetails opens a new functional dimension to stochasticswitching-based super-resolution imaging. bSOFI allowsthe extraction of the super-resolved spatial distribution ofmolecular statistics, such as the on-time ratio, the bright-ness and the concentration of fluorophores by combiningseveral cumulant orders. Moreover, this information canbe used to balance the image contrast in order to compen-sate for the nonlinear brightness and blinking response ofconventional SOFI images. Consequently, bSOFI enableshigher-order cumulants to be used and thereby achieveshigher resolutions.

MethodsTheory and algorithmSOFI is based on the computation of temporal cumu-lants or spatio-temporal cross-cumulants. Cumulants area statistical measure related to moments. Because cumu-lants are additive, the cumulant of a sum of independentlyfluctuating fluorophores corresponds to the sum of thecumulant of each individual fluorophore. This leads to apoint-spread function raised to the power of the cumu-lant order n and therefore a resolution improvement of√n, respectively almost n with subsequent Fourier fil-

tering (Dertinger et al. 2010). So far, SOFI has beenused exclusively to improve structural details in an image(Dertinger et al. 2009; Dertinger et al. 2010). Informa-tion about the on-time ratio, the molecular brightnessand the concentration has to our knowledge never been

exploited before and therefore represents a new potentialfor super-resolved imaging.In the most general sense, the cumulant of order n

of a pixel set P = {�r1, �r2, . . . , �rn} with time lags �τ ={τ1, τ2, . . . , τn} can be calculated as (Leonov and Shiryaev1959)

κn

(�r = 1

n

n∑i=1

�ri; �τ)

=∑P

(−1)|P|−1(|P| − 1)

∏p∈P

⟨∏i∈p

I( �ri, t − τi)

⟩t

,(1)

where 〈. . .〉t stands for averaging over the time t. P runsover all partitions of a set S = {1, 2, . . . , n}, which meansall possible divisions of S into non-overlapping and non-empty subsets or parts that cover all elements of S. |P|denotes the number of parts of partition P and p enu-merates these parts. I(�ri, t) is the intensity distributionmeasured over time on a detector pixel �ri. We used thecross-cumulant approach without repetitions to increasethe pixel grid density and eliminate any bias arising fromnoise contributions in auto-cumulants (Geissbuehler etal. 2011). A 4x4 neighborhood around every pixel wasconsidered to compute all possible n-pixel combinationsexcluding pixel repetitions. For computational reasons,we kept only a single combination featuring the shortestsum of distances with respect to the corresponding outputpixel �r = 1

n∑n

i=1 �ri. For even better signal-to-noise ratios,it would be beneficial to average over multiple combina-tions per output pixel. The heterogeneity in output pixelweighting arising from the different pixel combinationshas been accounted for by the distance factor as describedin (Dertinger et al. 2010).Considering a sample composed of M independently

fluctuating fluorophores and assuming a simple two-stateblinking model (with characteristic lifetimes τon, τoff) withslowly varying molecular parameters compared to the sizeof the point-spread function (PSF), the cumulant of ordern without time-lags can be interpreted as

κn( �r) ∝M∑k=1

εnkUn(�r − �rk)fn(ρon,k)

≈ εn(�r)fn(ρon; �r)M∑k=1

Un(�r − �rk)(2)

with ε(�r) the spatial distribution of the molecular bright-ness and ρon(�r) = τon(�r)

τon(�r)+τoff(�r) the on-time ratio. U(�r) is

Geissbuehler et al. Optical Nanoscopy 2012, 1:4 Page 3 of 7http://www.optnano.com/content/1/1/4

the system’s PSF and fn(ρon; �r) is the n-th order cumulantof a Bernoulli distribution with probability ρon:

f1(ρon; �r) = ρon

f2(ρon; �r) = ρon(1 − ρon)

...

fn(ρon; �r) = ρon(1 − ρon)∂fn−1∂ρon

(3)

Assuming a uniform spatial distribution of moleculesinside a detection volume V centered at �r, we may furtherapproximate

M∑k=1

Un(�r − �rk) ≈ EV{Un(�x)}N(�r), (4)

where EV {Un(�x)} = 1/V∫V Un(�x)d�x is the expectation

value ofUn(�x) or the n-thmoment ofU(�x) (see (Kask et al.1997) for some examples) andN(�r) denotes the number ofmolecules within the detection volume V. Finally, we canwrite

κn(�r) ≈ EV{Un(�x)}N(�r)εn(�r)fn(ρon; �r). (5)

Based on at least three different cumulant orders andapproximation (5), it is possible to extract the molecularparameter maps ε(�r), N(�r) and ρon(�r) by solving anequation system, or by using a fitting procedure. Forexample, the cumulant orders two to four can be used tobuild the ratios

K1(�r) = EV{U2(�x)} κ3

EV{U3(�x)} κ2

(�r)

= ε(�r) (1 − 2ρon(�r))

K2(�r) = EV{U2(�x)} κ4

EV{U4(�x)} κ2

(�r)

= ε2(�r) (1 − 6ρon(�r) + 6ρ2

on(�r))

(6)

and to solve for the molecular brightness

ε(�r) =√3K2

1 (�r) − 2K2(�r), (7)

the on-time ratio

ρon(�r) = 12

− K1(�r)2ε(�r) , (8)

and the molecular density

N(�r) = κ2(�r)ε2(�r)ρon(�r) (1 − ρon(�r)) . (9)

The spatial resolution of the estimation is limited by thelowest order cumulant, i.e. the second order in this case.However, the presented solution is not unique. Basicallyany three distinct cumulant orders could have provideda solution. Furthermore, the method is not limited to a

two-state system; it can be extended to more states as longas the differences in fluorescence emission are detectable.Additional details on the analytical developments as wellas a theoretical investigation of the estimation accuracyof the different parameters under different conditions aregiven in the Additional file 1.In order to correct for the amplified brightness

and blinking heterogeneities without compromising theresolution, the cumulants have to be deconvolved first.For this purpose, we used a Lucy-Richardson algorithm(Lucy 1974; Richardson 1972) implemented in MATLAB(deconvlucy, The MathWorks, Inc.), which is an itera-tive deconvolution without regularization that computesthemost likely object representation given an image with aknown PSF and assuming Poisson distributed noise. Apartfrom the estimate of the cumulant PSF and the speci-fication of a maximum of 100 iterations, all argumentshave been left at their default values. Assuming a perfectdeconvolution without regularization, the result could beinterpreted as

κn(�r) ≈ εn(�r)fn(ρon; �r)M∑k=1

δ(�r − �rk). (10)

Taking then the n-th root linearizes the brightnessresponse without cancelling the resolution improvementof the cumulant. To reduce the amplified noise andmasking residual deconvolution artefacts, small values(typically 1-5% of the maximum value) are truncated andthe image is reconvolved with U(n�r). This leads to a finalresolution improvement of almost n-fold compared to thediffraction-limited image, which is physically reasonablesince the frequency support of the cumulant-equivalentoptical transfer function (OTF) is n-times the support ofthe system’s OTF (cf. (Dertinger et al. 2010)). In contrastto Fourier reweighting (Dertinger et al. 2010), which isequivalent to a Wiener filter deconvolution and recon-volution with U(n�r), we explicitly split these two stepsand use an improved but computationally more expen-sive deconvolution algorithm that is appropriate for thesubsequent linearization.Since the cumulants are proportional to fn(ρon; �r), which

contains n roots for ρon ∈ [0, 1], there might still behidden image features in these brightness-linearizedcumulants (result after deconvolution, n-th root andreconvolution with U(n�r)). However, using the on-ratiomap ρon(�r), it is straightforward to identify the struc-tural gaps around the roots of f n and fill them in withthe brightness-linearized cumulant of order n-1. To thisend, the locations where f n approaches zero are trans-lated into a weighting mask with smoothed edges aroundthese locations. The thresholds have been defined by com-puting the crossing points of

∣∣fn∣∣1/n and∣∣fn−1

∣∣1/(n−1).

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This mask is then applied on the n-th order brightness-linearized cumulant and its negation is applied on ordern-1 (see Additional file 1 for further details). The resultis a balanced cumulant image. It should be noted thatthe cancellation of fn(�r) by division is possible but notrecommended, because it amplifies noisy structures in thevicinity of the roots of f n. The combination of multiplecumulant orders in a balanced cumulant image results ina better overall image quality. However, it is also possiblethat the on-ratio varies only slightly throughout the field ofview, such that a combination of multiple cumulant ordersis not necessary. Figure 1 illustrates the different steps ofthe algorithm based on a simulation of randomly blink-ing fluorophores, arranged in a grid of increasing densityfrom right to left, increasing brightness from left to rightand increasing on-time ratio from top to bottom.

ExperimentsIn order to verify the concept experimentally, we useda custom-designed objective-type total internal reflec-tion (TIR) fluorescence microscope with a high-NA oil-immersion objective (Olympus, APON 60XOTIRFM, NA1.49, used at 100x magnification), blue (490nm, 8mW, epi-illumination) and red (632nm, 30mW, TIR illumination)laser excitation and an EMCCD detector (Andor Luca S).We imaged fixed HeLa cells with Alexa647-labelled alphatubulin and used a chemical buffer containing cysteamineand an oxygen-scavenging system (Heilemann et al. 2008)(see Additional file 1 for further details) to generatereversible blinking and an increased bleaching stability.

The blue laser excitation was used to accelerate the reacti-vation of dark fluorophores and to reduce the acquisitiontime. For data processing, 5000 images acquired at 69frames per second (fps) were divided into 10 subsequencessignificantly shorter than the bleaching lifetime to avoidcorrelated dynamics among the fluorophores (Dertingeret al. 2010). The final bSOFI images are obtained byaveraging over the processed subsequences.

Results and discussionFigure 2 compares the performance of bSOFI with con-ventional SOFI and widefield fluorescence microscopy.The peak signal-to-noise ratio (pSNR; noise measuredon the background) in a single frame was 20-23dB forthe brightest molecules, which is rather low for per-forming localization microscopy, but more than suffi-cient for SOFI (Geissbuehler et al. 2011). Additionally,we observed significant read-out noise at this acquisi-tion speed (fixed-pattern noise in the average image,Figure 2a,i), which was effectively removed in the cross-cumulants analysis (Figure 2b-e and j,k). The estimatedmolecular on-time ratio (c,k), brightness (d) and den-sity (e) are shown overlaid with the 5th order balancedcumulant as a transparency mask. Due to the overem-phasis of slight heterogeneities in molecular brightnessand blinking, the dynamic range of the conventional 5thorder SOFI image (b and j), where values above 1% ofthe maximum are truncated, is too high to be representedmeaningfully. Figures 2f-h are the projected profiles ofthe widefield, SOFI, Fourier reweighted SOFI (Dertinger

Figure 1 bSOFI algorithm. Flowchart to illustrate the different steps of the bSOFI algorithm. (a) Raw data. (b) Cross-cumulant computation up toorder n according to equation (1) without time lags. (c) Cumulant ratios K1 and K2 according to equation (6). (d) Deconvolved cumulant of order n.(e) Solution for the spatial distribution of the molecular brightness ε , on-time ratio ρon and density N using equations (7-9). (f) Balanced cumulant oforder n, obtained by computing the n-th root of the deconvolved cumulant, reconvolving with U(n�r) and filling in the structural gaps around theroots of f n with a lower-order cumulant. (g) Color-coded molecular parameter maps overlaid with a balanced cumulant as a transparency mask.

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0 100 200 300 400 500 6000

0.5

1

distance [nm]

inte

nsity

[a.u

.]

0 200 400 6000

0.5

1

inte

nsity

[a.u

.]

0 100 200 300 400

0.20.40.60.8

inte

nsity

[a.u

.]

0.2

1

NB

[a.u

.]

0.03

0.1

ρon,B

a

h

b

2 2

'

2 2

'

1

1'

1

1'

c

2 2

'

1

1'

ed 0.2

1

εB

[a.u

.]

0.03

0.1

ρon,B

i j k

i j k

292nm137nm 78nm

f

g

3 3'

3 3'

3 3'

80nm

Widefield SOFI5 SOFI5 FRW BC5

75nm

Figure 2 bSOFI demonstration. Experimental demonstration of bSOFI on fixed HeLa cells with Alexa647-labelled microtubules. (a) Summed TIRFmicroscopy image [Widefield]. (b) Conventional 5th order cross-cumulant SOFI [SOFI5]. (c-e) Color-coded molecular on-time ratio, brightness anddensity overlaid with the 5th order balanced cumulant [BC5]. (f-h) Profiles along the cuts 1-1’, 2-2’ and 3-3’. In yellow we added the correspondingprofiles when Fourier reweighting (cf. (Dertinger et al. 2010)) with a damping factor of 5% is applied on the 5th order cross-cumulant SOFI image.(i-k) Magnified views from the white insets highlighted in (a-c). Scale bars: 2μm (a-e); 500nm (i-k).

et al. 2010) and bSOFI images along the cuts 1-1’, 2-2’and 3-3’, respectively. The second profile, with two micro-tubule structures separated by 80nm, illustrates a situationclose to the Rayleigh criteria in the bSOFI case. This isconsistent with the measured full width at half maxi-mum (FWHM) of 78nm (Figure 2h). Although the Fourierreweighted SOFI features a FWHM of 75nm (Figure 2h),it does not resolve the two microtubules at 2-2’. Due tothe nonlinear brightness response only a single one is

visible (Figure 2g). When considering the effective widthof the microtubule of 22nm as well as a 15nm linkerlength, this translates into a bSOFI-equivalent PSF with64nm FWHM, respectively a 4.6-fold resolution improve-ment compared to widefield microscopy. The resolutionimprovement of the conventional 5th order SOFI imageis close to the expected factor

√5. With the red TIR

illumination, the excitation intensity decreases exponen-tially along the optical axis. Assuming a homogeneous

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illumination in the x-y plane, both the molecular bright-ness and the on-time ratio can be interpreted as a depthencoding because they are related to the illuminationintensity (van de Linde et al. 2011). Obviously, a depthencoding based onmolecular parameters can only be usedas a qualitative impression of depth information ratherthan real 3D imaging, because it does not resolve addi-tional structures in depth. Moreover, when looking atsmaller scales (Figure 2k), the depth impression of color-coded molecular parameters gets less evident, which canbe explained by the influence of local differences in thechemical microenvironment or by the stochastic nature ofindividual fluorophores.Although the used Lucy-Richardson deconvolution per-

formed well on our measurements, it is not specifi-cally adapted for cumulant images, because it assumes aPoisson-distributed noise model and an underlying sig-nal that is strictly positive. For the n-th order cumulant,the signal on a single image can vary between positiveand negative values according to the underlying on-ratios.Furthermore, initially Poisson-distributed noise leads toa modified noise distribution in the cumulant image.In our experiments, the local on-ratio variations weresmall, which proves to be unproblematic for a decon-volution with a positivity constraint, when the negativeand the positive parts are considered separately. How-ever, a deconvolution algorithm specifically adapted forcumulant images using an appropriate noise model mayimprove the results of balanced cumulants in the future.If the cumulants are computed for different sets of

time lags and the acquisition rate oversamples theblinking rate, it is also possible to extract absoluteestimates on the characteristic lifetimes of the differ-ent states. The temporal extent of the curve obtainedby computing the second-order cross-cumulant as a

function of time lag τ (corresponding to a centred second-order cross-correlation) before it approaches zero yieldsan estimate on the blinking period, provided that thetimeframe of the measurement includes many blinkingperiods. In our case however, with only 10 to 20 blink-ing periods within a measurement window of 500 frames(@69fps) and a low on-ratio, the temporal extent of thecorrelation curve rather corresponds to the characteristicon-time. Figure 3a,b show the resulting on- and off-timemaps overlaid with a 5th order balanced cumulant as atransparency mask. The reported on-times correspond tothe time position where the curve decreased to e-1 of thevalue at zero time lag. The off-time map is obtained bycalculating τoff = τon

(ρ−1on − 1

). The off-time map hardly

shows a dependency on the illumination intensity, whichis in line with the deep penetration into the sample of theblue activation light. In the present case, the lifetime of theoff-state is influencedmainly by the chemical compositionof the microenvironment surrounding the probe (van deLinde et al. 2011).For estimating the average on-time, we computed the

second-order cross-cumulant as a function of time lag andaveraged it over the x-y-plane and 10 subsequences of500 frames (Figure 3c). The fitted exponential curve has acharacteristic time constant of τon = 31ms.

ConclusionsbSOFI is an extended version of SOFI and shares itsadvantages of simplicity, speed, rejection of noise andbackground, and compatibility with various blinking con-ditions. Since the bSOFI-PSF shrinks in all three dimen-sions with increasing cumulant orders, bSOFI can easilybe extended to the axial dimension by acquiring multipledepth planes and performing the analysis in threedimensions. In contrast to SOFI, the bSOFI response

0.1

1.5

τ

5

80

τον,Β

a b

10−3

10−2

10−1

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [s]

κ 2(τ)

fit exp(−τ/τ

on) with τ

on=31ms

normalized <κ2(x,y;τ)>

x,y

10−3

10−2

10−1

100

−0.05

0

0.05

τ [s]

Res

.

c

[s]

[ms]

Figure 3 On- and off-times. Spatial distribution of the estimated on- (a) and off-times (b) overlaid with a 5th order balanced cumulant as atransparency mask. The images correspond to an average over 10 subsequences of 500 frames each. (c) Second-order cross-cumulant with differenttime lags, averaged over the x-y-plane and 10 subsequences of 500 frames each. An exponential fit to the measured curve is shown in black. Scalebars: 2μm.

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to brightness and blinking heterogeneities in the sam-ple is nearly linear, which allows higher resolutionsto be obtained by computing higher cumulant orders.The additional information on the spatial distribu-tion of molecular statistics may be used to monitorstatic differences and/or dynamic changes of the probe-surrounding microenvironments within cells and thusmay enable functional super-resolution imaging withmin-imum equipment requirements.

Additional file

Additional file 1: Additional details on the development of the theory,the algorithm, sample preparation protocols and a theoreticalinvestigation of parameter estimation accuracies.

Competing interestsThe authors declare that they have no competing interests.

AcknowledgementsThis research was supported by the Swiss National Science Foundation (SNSF)under grants CRSII3-125463/1 and 205321-138305/1. The authors would liketo thank Prof. Anne Grapin-Botton for the provided infrastructures used for thepreparation of the samples and acknowledge Arno Bouwens, Dr. MatthiasGeissbuehler, and Dr. Erica Martin-Williams for their constructive comments onthe manuscript.

Author’s contributionsSG and CD developed the theory and the algorithm, SG, NB, ML and TLconceived the study, NB and CB prepared the samples, SG and NB performedthe experiments and analyzed data, ML and TL supervised the project and SGwrote the manuscript. All authors discussed the results and implications andcommented on the manuscript at all stages.

Received: 28 January 2012 Accepted: 25 April 2012Published: 25 April 2012

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doi:10.1186/2192-2853-1-4Cite this article as: Geissbuehler et al.: Mapping molecular statistics withbalanced super-resolution optical fluctuation imaging (bSOFI). OpticalNanoscopy 2012 1:4.

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