REVIEW

Statistical filtering in fluorescence microscopy and fluorescencecorrelation spectroscopy

Radek Macháň & Peter Kapusta & Martin Hof

Received: 23 January 2014 /Revised: 7 May 2014 /Accepted: 13 May 2014# Springer-Verlag Berlin Heidelberg 2014

Abstract We review the principles and applications of statis-tical filtering in multichannel fluorescence microscopy. Thisalternative approach to separation of signals from individualfluorophore populations has many important advantages, es-pecially when spectral and/or temporal overlap, or the com-plicated nature of those signals, makes their discrimination orsorting impossible by means of hardware. This situation istypically encountered for biological samples. This review ofwell established statistical filtering techniques and of emerg-ing, very promising new methods of analysis reveals remark-able progress in bioanalytical applications of fluorescencemicroscopy.

Keywords Fluorescence lifetime correlation spectroscopy .

Fluorescence spectral correlation spectroscopy . Filteredfluorescence correlation spectroscopy . Raster lifetime imagecorrelation spectroscopy . Time-correlated single photoncounting

Introduction

Fluorescence microscopy has, in recent decades, become anindispensable tool in science, especially in biological,

biochemical, and biophysical research. It owes its popularitymainly to its relative simplicity; high sensitivity, which en-ables detection and characterisation of individual molecules;chemical specificity (a single chemical species bearing thefluorescent moiety can be observed selectively); and non-invasiveness, which enables observation of processes in livingcells or multicellular organisms. Besides the most genericform of fluorescence microscopy, fluorescence intensity im-aging, there is a large selection of more advanced methods ofdata acquisition and/or analysis, which provide deeper quan-titative insight into the structure and dynamics of thesystem under study. To give a few examples, we canmention determination of diffusion coefficients and con-centrations of molecules (fluorescence correlation spec-troscopy, FCS [1, 2] and raster image correlation spec-troscopy, RICS [3, 4]), mapping the distribution ofexcited state lifetimes of emitters in different sampleregions, interpreted, depending on the nature of thefluorophore used, for example, in terms of environmentpolarity or concentration of particular ions (fluorescencelifetime imaging, FLIM) [5–7], mapping of Förster res-onance energy transfer (FRET) efficiency (“proximityimaging”) [8, 9], or mapping the rotational mobility ofmolecules (fluorescence anisotropy imaging) [10–12].

The chemical specificity enabling selective observation ofparticular chemical species (either inherently fluorescent ortagged with a label) is crucial in most applications of fluores-cence microscopy. Further possibilities emerge when multi-channel detection (frequently dual-channel) is used to char-acterise simultaneously two or more populations of mol-ecules bearing distinct fluorophores. Colocalisation anal-ysis in fluorescence imaging [13, 14] or quantitativecharacterisation of molecular interactions by dual-colour fluorescence cross-correlation spectroscopy (DC-FCCS) [15–17] are typical examples of the use ofmultichannel detection in fluorescence microscopy.

Radek Machán and Peter Kapusta contributed equally to thiscontribution.

ABC Highlights: authored by Rising Stars and Top Experts.

R. MacháňDepartment of Biological Sciences and Centre for BioImagingSciences, National University of Singapore, 14 Science Drive 4,Singapore 117546, Singapore

P. Kapusta :M. Hof (*)J. Heyrovský Institute of Physical Chemistry of ASCR, Dolejškova3, 182 23 Prague, Czech Republice-mail: hof@jh-inst.cas.cz

Anal Bioanal ChemDOI 10.1007/s00216-014-7892-7

Separation of detected emission from different fluorophorespecies into the respective detection channels typically relieson compounds having different emission spectra with mini-mum overlap. In the ideal case of non-overlapping spectra,each individual photon can be unambiguously assigned to oneof the detection channels (thus assigned to one of thefluorophore species) on the basis of its wavelength. In realitythere will always be some spectral overlap resulting in somecrosstalk between the detection channels; however, it can bekept reasonably low by selecting fluorophores with sufficient-ly spectrally separated emission bands and using appropriateemission filters. Although very straightforward and widelyused, separation based on non-overlapping (or minimallyoverlapping) emission spectra has its limitations. First, theneed to select fluorophores with sufficiently small spectraloverlap may, in some situations, be restrictive (for examplewhen special environment-sensitive fluorophores must beused). In addition, in advanced confocal techniques such asconfocal DC-FCCS, the mismatch in size and position of theconfocal volumes corresponding to different spectral regionsrequires tedious corrections if quantitative results are to beobtained [16, 18]. It is, therefore, logical to ask whether wereally need to be able to assign each photon unambiguously toone of the detection channels or whether any less stringentrequirement may not be sufficient to perform multichannelfluorescence microscopy successfully. The answer is that inmany situations it is sufficient to quantify the probability that adetected photon belongs to a particular fluorophore species.Statistical filtering can be used in such situations to performthe task for which hardware sorting (e.g. by optical filters) isused in the prevalent situation of non-overlapping spectra.

In this contribution we review the principles and applica-tions of statistical filtering as an alternative approach to sep-aration of signals from individual fluorophore populations inmultichannel fluorescence microscopy. The method was orig-inally developed by Jörg Enderlein for time-domain measure-ments and was demonstrated by separation of FCS curves oftwo fluorophores on the basis of different decay times [19].Initially called time-resolved FCS, it later became known asfluorescence lifetime correlation spectroscopy (FLCS) [20].FLCS is by far the best known and most widely used appli-cation of statistical filtering, and an overview of applicationsof FLCS can be found elsewhere [21]. However, other tech-niques based on statistical filtering have emerged recently, forexample fluorescence spectral correlation spectroscopy(FSCS) [22] or a generalised concept using multi-parameterstatistical filters called filtered fluorescence correlation spec-troscopy (fFCS) [23–25]. We will focus here on the generalunderlying concept of statistical filtering and discuss its pos-sibilities and limitations. Besides its already well-establisheduse in FLCS, we will outline other applications of statisticalfiltering in fluorescence microscopy, which show how univer-sal and powerful the method is.

Principles of statistical filtering

The main advantage of statistical filtering is its ability to separateoverlapping signals that cannot be sorted by hardware, as isillustrated in the following examples. Note that this featureenables us to separate signals from fluorophoreswith very similaremission properties, for example from populations of an envi-ronmentally sensitive probe embedded in a heterogeneous sam-ple. The differences between populations in terms of fluores-cence lifetime and/or emission spectra are too subtle for unam-biguous hardware sorting, but typically large enough to enableefficient signal separation by means of statistical filtering.

As an example, let us consider the emission of two wellknown fluorophores, green fluorescent protein (GFP) andAlexa Fluor 488 (AF488). Because their excitation spectraoverlap completely, excitation with any suitable wavelengthwill excite both with practically identical efficiency.Therefore, their fluorescence decays or spectra cannot be tem-porally separated by use of different excitation wavelengths, asis done in experiments involving pulsed interleaved excitation(PIE [26]) or alternating laser excitation (ALEX [27]). Whenthe emission of this mixture is detected, superimposition isalways obtained: in time resolved measurements by means oftime correlated single photon counting (TCSPC) this is a multi-exponential decay curve (Fig. 1a); in spectral measurementsthis is a single emission band (Fig. 2a) caused by substantialoverlap of the component spectra. In both cases, there areunavoidable, additional signal components also. When thesignal is recorded photon by photon, assignment of a photonto a particular fluorophore by hardware is not possible becausethere is no simple criterion that unambiguously distinguishesphotons emitted by GFP molecules from those emanating fromAF488. However, the component decay curves (or referencespectra) are unique fingerprints of the emitters under particularconditions of the experiment. Statistical filtering is based onefficient use of this prior information. For instance, the decaycurves in Fig. 1a are expressing the time-dependent probabili-ties of detecting photons at different delay times after theexcitation pulse. AF488 has a longer lifetime than GFP. Quitegenerally, the later a photon arrives, the more likely it wasemitted by AF488. Owing to the different decay patterns, therelative probabilities of detecting GFP and AF488 photons arefunctions of the delay time and also of the composition of theobserved decay. None of the photons can be unambiguouslyassigned to a single fluorophore species. Nevertheless, as willbe shown below, for every photon it is possible to determine theexact statistical weight (as a function of delay time) it contrib-utes to, e.g., the autocorrelation function (ACF) of GFP andsimultaneously to the ACF of AF488.

Figure 1 illustrates the principle of time-domain statisticalfiltering for separation of signals from two fluorophores differ-ing in their lifetimes. Although almost all published applica-tions of time-domain statistical filtering have been based on

R. Macháň et al.

Fig. 1 Simulated fluorescence decay curve, TCSPC patterns, and thecorresponding FLCS filter functions. (a) The decay curve plotted with ablack solid line is a TCSPC histogram of 2×106 photons collected in ahypothetical FLCS measurement of a sample labelled with both GFP andAF488. Such a decay curve is one of the primary outputs of any FLCSmeasurement. Assuming that other contributions are negligible, the ex-perimental data must be a linear combination of three TCSPC patterns:decays of GFP (green) and AF488 (blue) and an offset (background)because of detector artefacts (red). (For simplicity, GFP and AF488patterns are simulated here as single exponential decays with lifetimesof 2.1 ns and 4.1 ns, respectively. The actual lifetime values and usualdeviations from single-exponential character of these decays have littlerelevance for this illustration. Only the difference of decay shapes isimportant.) These decay curves (TCSPC histograms) can, in principle,be obtained with good signal-to-background ratio in independent mea-surements of control samples containing only one of the labels. The third(in TCSPC ubiquitous) pattern is well approximated in practice by asimple horizontal line. This background is caused by detector afterpulsingand dark counts, evenly distributed in TCSPC histogram channels cov-ering the 50 ns (fluorescence) time scale (discussed in the section “Re-moval of background and parasitic signal components”). The composi-tion of the “experimental” decay curve is 30 % GFP fluorescence, 60 %AF488, and 10 % background counts. (b) FLCS filter functions corre-sponding to the three patterns (signal components) contained in theexperimental decay curve. These functions are calculated by use ofEq. (4) from the decay curve itself and from the area-normalized patterns.Note: this illustrative example was inspired by a previously published in-vivo FLCS study [28]

Fig. 2 Simulated fluorescence emission spectrum, spectral patterns, andcorresponding FSCS filter functions. (a) The emission spectrum plottedwith a black solid linewould be obtained by using a 128-channel spectralEM-CCD detector in a hypothetical FSCS measurement of the samesample as in Fig. 1. Such a spectrum is one of the primary outputs ofthe FSCS measurement. Assuming that the fluorescence of GFP andAF488 dominate the detected signal, the experimental data must be alinear combination of three spectral patterns—the GFP and AF488 emis-sion and an uneven background contributed by the detector. The spectraof GFP (green line) and AF488 (blue line) illustrated here are from theSpectra Database hosted at the University of Arizona [29]. Such spectralpatterns can, in principle, be obtained with good signal-to-backgroundratio in independent measurements of control samples containing onlyone of the labels. In this particular example, the third pattern is a tiltedbaseline that can be easily obtained with a closed shutter in front of theEM-CCD detector. This baseline is caused by the so-called readout-induced charge, when the EM-CCD is used with high EM gain and fastreadout speed, as required for FSCS. (b) FLCS filter functions corre-sponding to the three spectral patterns (signal components) contained inthe measured signal. These functions were calculated by use of Eq. (4)from the sample’s emission spectrum and from the area-normalizedspectral patterns. Note: for simplicity, the abscissa of these plots repre-sents a linear wavelength scale, as usual when plotting experimentalspectral data. In a real FSCS experiment utilising a spectral EM-CCDdetector, the x-axis values are pixel numbers and the wavelength calibra-tion is non-linear. Furthermore, the spectral intensity values plotted hereare corrected for wavelength-dependent variations in spectral sensitivity.In practice this correction in not necessary for FSCS. This example wasinspired by Refs. [22] and [28]

Statistical filtering in fluorescence microscopy and FCS

differences in lifetimes, any difference in fluorescence intensitydecays can be utilized for separation of signals from twofluorophore populations by statistical filtering [21]. It has, forexample, been shown that signals from two fluorophore popula-tions with identical lifetimes but different rotational correlationtimes (resulting in differences between the decay shapes causedby fluorescence anisotropy) can be separated by statistical filter-ing [23]. The difference between the decays can, in principle, bearbitrarily subtle; however, the more similar the decays are, themore photons are needed to achieve reliable separation of signals(Fig. 3, caption). Therefore, robust differences in decay shapes,for example those resulting from significantly different lifetimes,are required inmany applications of statistical filtering, especiallyin biological applications in which the number of photons thatcan be collected is limited by the stability of the samples. Whilethe theory is presented here in a general form applicable to anydifferences in decay shapes, differences in lifetimes are used inthe illustrative examples.

In a similar way, the reference spectra in Fig. 2a areexpressing wavelength dependent probabilities of detecting aphoton from the respective compound. The GFP spectrum isslightly blue-shifted compared with that of AF488. Quitegenerally, a blue photon in the experimentally obtained spec-trum was more likely to have been emitted by GFP. Owing tothe different spectral patterns, the relative probabilities ofdetecting GFP and AF488 photons depend on the observationwavelength and of course on the composition of the mixture.By composition we mean the abundance of photons withdifferent spectral patterns in the experimental spectrum.When a single photon with a specific wavelength is detected,clear assignment to a specific fluorophore is not possible, butthe calculated filter function determines (on the basis of thewavelength) the statistical weight of this photon entering thecalculation of, e.g., an ACF.

Patterns

In agreement with the terminology introduced in the originalworks on statistical filtering [19, 30], by patterns we mean thecharacteristics which distinguish the individual fluorophorespecies. Thus, the two fluorescence intensity decay curves andthe histogram of dark counts and afterpulses shown in Fig. 1a,after normalizing each to a unit area, will be the patterns ofGFP, AF488, and background, denoted p(GFP), p(AF488), andp(Bckg), respectively. Reference spectra from Fig. 2a (after areanormalisation) can be denoted and used in the same way. Themathematical procedures of statistical filtering are indepen-dent of the nature of the patterns; therefore, when we use theterm pattern in the following explanation of statistical filteringand in the formulas, readers may imagine the term denoteseither fluorescence intensity decays or emission spectra ac-cording to their preferences.

Let us consider the general situation of M signal compo-nents with different patterns p(k), k=1, 2, …, M. These signalcomponents are contributed by a variety of fluorophore pop-ulations and by the instrument itself (e.g. detector artefacts,scattered excitation light). Patterns are arrays (row vectors) ofvalues, p(k)=pj

(k), where the index j=1, 2, …, N refers toTCSPC channel or spectral detection channel (pixel) number.The detection system determines and stores the correspondingj value for each detected photon (count). Sorting the countsrecorded during a specific acquisition time according to their jwe obtain Ij, (a TCSPC histogram or emission spectrum)which is a linear combination of the individual patterns:

I j ¼Xk¼1

M

w kð Þp kð Þj ð1Þ

where w(k) is the amplitude of the kth pattern, p(k). Because thepatterns were normalized to unit area, w(k) corresponds to thenumber of counts contributed by the kth signal component.

Fig. 3 Illustration of the dependence of the reliability of retrieving thephotons from each signal component (w(k)) on the total number ofphotons collected from a mixture of two fluorophores. Photons originat-ing from a sample containing 30 % GFP, 60 % AF488, and 10 %background were simulated. The numbers of photons emitted by eachcomponent were kept in exact correspondence to the composition of themixture. Stochastic photon arrival times were generated in accordancewith the probability distributions given by the patterns (Fig. 1a). Plottednumbers of photons from GFP (green diamonds), AF488 (blue squares),and background (red circles) were obtained by applying the three respec-tive filter functions shown in Fig. 1b to the simulated photon ensemble.The simulations were repeated 20 times for each total number of photonsand the means and standard deviations are plotted in the figure. A smalloffset along the x-axis was introduced to the first three points to avoidoverlap of the long error bars. Note that for small total numbers ofphotons, the filtered numbers of photons corresponding to the individualfluorophores tend to suffer from larger relative errors than the filterednumbers of background photons. This is because differences between theshapes of the background pattern and any fluorophore pattern are largerthan those between the shapes of any pair of fluorophore patterns. In otherwords, fewer photons are sufficient to achieve correct separation ofbackground from fluorescence than are required for reliable separationof the signals from the two fluorophores from each other

R. Macháň et al.

The most straightforward and accurate way of obtainingpatterns is to measure them experimentally by using sampleswhich each contain only a single component of the mixture wewant to analyse by means of statistical filtering. Some patternsare readily available without any measurement (e.g. theTCSPC background described by a horizontal line); someneed application of more elaborate procedures, as will bedescribed later. It is important to ensure that all experimentalsettings (excitation intensity, photon count rate, etc.) andconditions (temperature, pH, ionic strength, local environmentof the probe, etc.) match, as closely as possible, those usedduring measurement of the mixture. This is particularly im-portant for environmentally sensitive probes, for which slightdifferences in experimental conditions may result in substan-tial differences in decay patterns. If that is satisfied, we obtainaccurate patterns, no matter how complex they may be (e.g.multi-exponential fluorescence intensity decays convolutedwith a broad instrument response function, IRF) and there isno need to make a-priori assumptions about their shape. Thisis the preferred way of obtaining patterns, unless made im-possible by the nature of the sample.

A typical situation when experimental recording of indi-vidual patterns is not possible is compounds in chemicalequilibrium. As an example we may consider a mixture oftwo fluorophore populations represented by two forms of thesame fluorophore coexisting in dynamic equilibrium. In sucha situation we cannot measure separately the patterns for eachcomponent. Patterns can be obtained only if we are justified inmaking specific assumptions about them. For example if thefluorescence intensity decay of the mixture can be describedby a linear combination of two exponential functions (anddecays of individual components can be expected to be mono-exponential), we can take the individual mono-exponentialdecays convoluted with the IRF of the instrument as thepatterns. For statistical filtering combined with FCS (the mostcommon application of statistical filtering), Humpolíčkováet al. showed that if the pattern for one of the components isunknown, the corresponding filtered ACF can be still obtain-ed, if the remaining patterns are known [31].

Obtaining accurate patterns of individual components canalso be a problem in live cell applications of statistical filter-ing, as a result of the intrinsically high heterogeneity andvariability of biological samples and high environmental sen-sitivity of some fluorophores. Because statistical filtering hasbeen so far used only scarcely for live cell studies, this issuehas not yet been addressed in depth in the literature. In the twomost notable live cell applications of statistical filtering[28, 32] (which we discuss below), patterns were con-structed from mono-exponential decays obtained bybiexponential fits to the intensity decay of the wholesample. The imperfection of the patterns thus obtainedis a possible cause of imperfect separation of the signalsfound in one of the studies [32].

Statistical filters

Calculating the statistical filter functions is, indeed, the mostcrucial step in the whole process and the quality of theresulting filters has a decisive effect on filtering performance.The fundamental requirement for a filter function of the kthcomponent, fj

(k), is that it should recover w(k), that is thenumber of photon counts contributed by the kth componentto the experimentally accumulated signal, Ij.X

j¼1

N

f kð Þj I j

* +¼ w kð Þ ð2Þ

Simultaneously fj(k) has to minimize the relative errors:

Xj¼1

N

f kð Þj I j−w kð Þ

!2* +ð3Þ

The brackets in Eqs. (2) and (3) mean averaging over aninfinite series of measurements (photon detection events). Ifsingle-photon detection in each channel j follows Poissonianstatistics (which true for well performed single-photoncounting measurements), the set of filter functions fj

(k) iscalculated by use of Eq. (4):

f kð Þj ¼ P⋅diag I j

� �−1⋅PTh i−1

⋅P⋅diag I j

� �−1� �kj

ð4Þ

where P is anM×Nmatrix of patterns, Pkj=pj(k), which means

each of the M rows contains an array of N pattern values.diag⟨Ij⟩

−1 is a diagonal N×N dimensional matrix containingelements ⟨Ij⟩

−1 along its diagonal; superscript T indicates atransposed matrix, dot means matrix multiplication, and[…]−1 means matrix inversion. Calculation of filter functionsby use of Eq. (4) is straightforward and all matrix operationsinvolved are supported by most of the mathematical softwarepackages used for data analysis. The result is anM×Nmatrix;the array of N values in each row is the filter function requiredfor component k [19, 30].

An example of statistical filters is shown in Fig. 1b; theseare filters calculated for the patterns and mixture intensitydecay shown in Fig. 1a. It is worth studying the shape of thestatistical filters in our example and discussing their meaningand relation to the corresponding emission patterns. Considerthat the shorter lifetime of GFP does not automatically meanthat photons arriving to the detector early after the onset of thelaser pulse (small j) are most likely to have been emitted byGFP molecules. The probability of detecting an AF488 pho-ton is also highest at the beginning of the decay; moreover thedecay curve example in Fig. 1 contains twice as many AF488photons as GFP photons. The key issue is that the ratio ofdetection probabilities is changing as the decays proceed, i.e.that ratio is a function of channel number, j.

Statistical filtering in fluorescence microscopy and FCS

It is obvious from Eq. (3) that filters fj(k) give the weights with

which a single photon detected in the jth channel contributes(simultaneously) to the three intensities, w(k), attributed to thethree signal components. A given weight value is related to theprobability that a photon falling into the jth channel has beenemitted with the kth pattern, but the filter function calculation hasmore to take into account. First, there are widely different num-bers of counts in the different channels. Weighting values forsmall j values have the largest effect, because the “early” photonsare much more frequent than those arriving later. According toFig. 1b, the counts registered near the beginning of the decaycontribute with almost triple weight to w(GFP), and at the sametime with negative weight to w(AF488). The latter may seemcounterintuitive or even unphysical. That the more abundantAF488 photons finally contribute their full amount is ensuredby the large positive weights of photons falling in the subsequentchannels. The same counts contribute with negative values tow(GFP), compensating for the much increased positive weight ofthe early photons. Because of the intensity decay, there are manyfewer of these “compensating” photons, hence the larger absolutevalues of fj

(k) in the range 7 to 15 ns. Another non-intuitivefeature of the statistical filters in Fig. 1b is the higher value offj(GFP) for photon arrival times larger than 25 ns, despite morephotons from AF488 than from GFP being detected in thoseTCSPC channels. In contrast with the behaviour of fj

(k) at shorterphoton arrival times, this feature of the statistical filters lacks anystraightforward explanation. The exact shape of fj

(k) is a result ofthe matrix pseudo-inversion in Eq. (4) and the requirement oftheir orthonormality to the respective patterns. The above pro-vided heuristic explanation of the shapes of the filters (althoughhelpful in gaining a basic understanding of the meaning ofstatistical filters) is, therefore, oversimplified and unable to pro-vide an explanation of all fj

(k) features.What is, on the other hand, very intuitive and logical, is the

fact that the sum of all statistical filters for each channel jequals 1:Xk¼1

M

f kð Þj ¼ 1 ð5Þ

Equation (5) actually represents the natural requirement ofthe unit contribution of each photon to the total detectedintensity Ij. It follows directly from Eq. (2) and from the

obvious relationship I ¼ ∑k¼1

M

w kð Þ ¼ ∑j¼1

N

I j .

The shape of the statistical filters in the spectral domaindisplayed in Fig. 2b and their relationship to the correspond-ing spectral patterns (Fig. 2a) can be explained in a manneranalogous to that for the time-domain filters (Fig. 1). The GFPspectrum is slightly blue-shifted relative to the AF488 spec-trum. This means the ratio of the detection probability for GFPphotons to that for AF488 photons is highest at the blue end ofthe spectrum and, therefore, photons in that spectral region

contribute with the highest weight to w(GFP). The situation isreversed for wavelengths higher than approximately 514 nm.Here fj

(AF488)>fj(GFP) and fj

(GFP) assumes negative values tocompensate for its values larger than 1 at the blue end of thespectrum. Because more photons are detected in the regionwhere fj

(GFP) is negative than there are photons to be “com-pensated” for in the region where fj

(GFP)>1, the absolutevalues of filter functions in the former region are smaller thanin the latter region. The filter for the background has thelowest value in the region from 500–540 nm, correspondingto the highest fluorescence intensity, and increases towardboth ends of the spectrum. This increase does not only reflectthe diminishing numbers of fluorescence photons at thoseregions but also the dependence of the background on pixelposition. Again the obvious condition represented by Eq. (5)is valid also for the sum of the statistical filters.

Photon weights (filter function values) >1 and nega-tive weights have important consequences. If the datasetanalysed by statistical filtering contains informationabout a small number of photons only, the filteredamplitudes w(k) lack physical meaning, some being pos-sibly negative while others are possibly higher than thetotal number of photons detected. The statistical natureof the filtering process requires that the dataset analysedcontains information about a sufficiently large numberof photons. This is to enable reliable estimates of am-plitudes w(k) to be retrieved from a modified Eq. (2), inwhich averaging over an infinite series of measurementsis replaced by averaging over a finite dataset.

Figure 3 illustrates what is meant by a “sufficiently largenumber of photons” for the GFP–AF488 system shown inFig. 1. We simulated 10, 50, 100, 500, 103, 5×103, and 104

stochastic photon emissions from that sample. Thepartitioning of generated photons among the patterns presentin the signal was kept exactly corresponding to the composi-tion, that is 30 % GFP, 60 % AF488, and 10 % background(Fig. 1, caption). Simulated photon arrival times (TCSPCchannel number j, in the previous expressions) were random,however, obeying the probability distributions given bythe patterns. On application of the three filter functionsto the same random ensemble of generated photoncounts, one obtains (statistically) the number of photons(counts) belonging to the respective patterns (signalcomponents). Figure 3 shows that in the simulated real-istic case 5,000 photons are sufficient to determine thecomposition (that is w(k) values) with relative precisionbetter than 10 %.

The need to collect enough photons is one of the mostimportant limitations of statistical filtering and must be takeninto account when designing experiments and collecting datato be analysed by use of the method. This explains whystatistical filtering has been used predominantly in combina-tion with FCS. In a typical FCS measurement one collects a

R. Macháň et al.

huge number of photons (usually more than 106) which arethen used for calculating the correlation function(s).

Applications of statistical filtering in FCS

As has been already mentioned, application of time-domainfilters in FCS (a technique that became known as FLCS) washistorically the first application of statistical filtering and hasremained predominant [21, 33, 34]. Nevertheless, other typesof statistical filtering have recently been introduced to FCS.

Calculation of filtered ACFs is very straightforward; theonly difference from standard FCS is substitution of filteredamplitudes w(k) for the values of fluorescence intensity in thestandard formula for ACF. Thus, the filtered ACF for the kthcomponent of the mixture is given by the equation:

G k;kð Þ τð Þ ¼ w kð Þ tð Þw kð Þ t þ τð Þ� �w kð Þ tð Þh i2

¼XN

i¼1

XN

j¼1f kð Þi f kð Þ

j I i tð ÞI j t þ τð Þ� �XN

i¼1

XN

j¼1f kð Þi f kð Þ

j I i tð ÞI j tð Þ� �

ð6Þ

The brackets in Eq. (6) represent averaging over all thevalues of time, t.

By analogy with DC-FCCS, a cross-correlation function(CCF) between kth and mth components can be calculated:

G k;mð Þ τð Þ ¼ w kð Þ tð Þw mð Þ t þ τð Þ� �w kð Þ tð Þh i w mð Þ tð Þh i ¼

XN

i¼1

XN

j¼1f kð Þi f mð Þ

j I i tð ÞI j t þ τð Þ� �XN

i¼1

XN

j¼1f kð Þi f mð Þ

j I i tð Þh i I j tð Þ� �

ð7Þ

Fluorescence lifetime correlation spectroscopy

Because a recent comprehensive summary of published FLCSapplications can be found elsewhere [21], we will not ventureinto details here but merely briefly review the main types ofapplication and the most recent studies which have not beenincluded in the aforementioned review.

Being able to obtain ACFs of individual components of amixture and the CCF between them, some FLCS applicationsare similar to those of DC-FCCS [28, 35, 36]. For exampleChen and Irudayaraj measured interactions of molecules la-belled by fluorophores with overlapping emission and excita-tion spectra but with different lifetimes (GFP and AF488) insolution and in living cells. The latter is important as the firstexample of the separation, by FLCS, of signals from twofluorophores in living cells [28]. Use of FLCS instead ofDC-FCCS has two significant advantages: first, FLCS doesnot suffer from crosstalk (which requires negative controlexperiments in DC-FCCS) and, second, by using dyes withoverlapping spectra the authors avoided problems connectedwith differences in excitation and detection volumes (which

again require quite tedious corrections if quantitative resultsare to be obtained). Nevertheless, CCFs measured by FLCSare not free from potential artefacts. It has been shown thattemporal jitter in IRF, detection electronics dead time, orexcluded volume effects reduce the amplitudes of CCFs inFLCS [21, 23, 26, 37]. Those artefacts are the most likelyexplanation for the reported lower-than-expected values ofCCF amplitudes [36], including negative amplitudes observedin the absence of any real cross-correlation [21].

The last of the sources of artefacts (excluded volume) is notspecific to FLCS and is encountered in any FCCS experiment.However, it is negligible for most samples with the exceptionof large fluorescent particles (for example liposomes or fluo-rescent beads), in which case the presence of a particle in thedetection volume of the microscope significantly reduces theprobability of another particle entering the volume [21].Neither is the effect of detection electronics dead time specificfor FLCS. It is encountered in any FCCS experiment usingTCSPC detection and has been characterised in detail for DC-FCCS with PIE [26]. After detection of each photon thedetection electronics is occupied for some time (tens to hun-dreds of nanoseconds) and unable to process any other pho-ton. The simultaneous presence of more emitters in the detec-tion volume increases the probability of some photons being“overlooked”, resulting in underestimation of the actualamount of cross-correlation. (This is a special case of a varietyof TCSPC dead-time effects generally referred to as “pulsepile-up”.) The magnitude of the effect obviously increaseswith count rate. The real amount of cross-correlation in asample can be, thus, estimated by changing count rate permolecule (by changing excitation intensity) and extrapolatingthe dependence of CCF amplitude on count rate to zero [26].

The underestimation of CCF amplitudes caused by tempo-ral jitter in IRFs is the only one of the above mentionedsources of artefacts which is specific for FLCS. The effectresults in smearing of TCSPC histograms and has a detrimen-tal effect on the accuracy of statistical filtering. A method ofcorrection for the temporal jitter in IRFs has been proposedrecently, and has the potential to overcome this source ofartefacts in FLCS [38]. The authors of the method found thatthe (picosecond) response time of typically used SPAD detec-tors depends on the temporal distance from detection of thepreceding photon. This dependence is the dominating sourceof ps timing instability (jitter) of SPAD detectors. The time-tagged photon records also contain the photon detection timeon an absolute timescale, measured from the beginning of theexperiment. It is, therefore, possible to assign the temporaldistance from the preceding photon (tprec) to each de-tected photon. By calibration of IRF measurements, thedependence of measured picosecond delay (TCSPCchannel number) on tprec can be quantified. By usingthe known dependence of picosecond response delay ontprec it is possible to correct the TCSPC channel number

Statistical filtering in fluorescence microscopy and FCS

of each detected photon, so that smearing caused by thejitter is eliminated.

Other studies have used FLCS to suppress crosstalk in dual-colour cross-correlations, either using pulsed interleaved exci-tation [39] or a combination of one pulsed and one continuouswave laser. From the FLCS perspective, the latter excitationmodality results in a decay-like pattern for one of the emittersand a second, flat, background-like pattern for the otherfluorophore [40, 41]. This approach is useful for the (in biologycommon) fluorophore pairs of green and red fluorescent pro-teins, because affordable pulsed lasers in the spectral region(approx. 560 nm) needed for excitation of red fluorescentprotein and its variants such as mCherry are not yet available.

However, other applications of FLCS do not have a directDC-FCCS analogy, for example study of systems containing twopopulations of the same fluorophore with distinct lifetimes.Environment-sensitive probes which change their decay timeaccording to the properties of their surroundings are a typicalexample. The concept of obtaining ACFs and CCFs bymeans ofFLCS, and their use to study equilibrium dynamics between thefluorophore populations is explained by Gregor and Enderlein[20]. The dynamics of two forms of Tokyo green-II (a florescentdye which exists in either the protonated or deprotonated form,with different lifetimes) in different environments were system-atically studied by FLCS [42–45]. The method has also beenused to elucidate themechanism ofDNA compaction by cationiccompounds; DNA molecules were labelled by intercalating thefluorescent dye PicoGreen, which changes its lifetime in re-sponse to compaction of the DNA molecule [31, 46–48].

FLCS can be conveniently combined with approacheswhich can be called (using the term introduced by Bendaet al. [49]) “lifetime tuning”. The approaches are based onthe fact that the lifetime of fluorophores usually becomesshorter in the vicinity of conducting objects, for example thesemiconductor surfaces or metallic nanoparticles. FLCS withlifetime tuning has been used to separate the signals of mol-ecules freely diffusing in solution from those of moleculesattached to supported lipid bilayers on semiconductor surfaces[49] or to functionalized silver nanoparticles [50, 51].

Another very important application of FLCS is the suppres-sion of background and parasitic signal components (for exam-ple scattered light or autofluorescence of biological samples) inFCS data. We believe those FLCS applications to be of suchsignificance that we discuss them in the separate section“Removal of background and parasitic signal components”.

Fluorescence spectral correlation spectroscopy, filteredFCS and filtered dual-focus FCS

Statistical filtering in FCS is not limited to time-domain.Although it emerged much later than FLCS and has not yetbeen applied to any topic of interest in chemistry or biology,

FSCS is worth mentioning here as a direct analogy to FLCS.The method developed by Aleš Benda utilises spectrally re-solved (instead of time-resolved, TCSPC) detection. Spectraof individual components are used as patterns and spectraldetection channels are used in the same way as TCSPCchannels in FLCS (illustrated in Fig. 2). An imaging spectro-graph was used in the first experimental realisation of FSCS, inwhich fluctuating fluorescence emission spectra were projectedon to a single-photon-sensitive electron multiplying charge-coupled device (EM-CCD) camera. One-hundred and twenty-eight spectral channels, corresponding to 128 pixels on the EM-CCD chip, provided the multi-channel data in that experiment.An important outcome of that study is that despite completespectral overlap, as few as five or six spectral detection channelsare sufficient to separate signals from two fluorophores withdifferent emission spectral shapes. FSCS can be therefore im-plemented on commercially available microscopes equippedwith fast multichannel spectral detectors based on single photoncounting (for example avalanche photodiodes orphotomultiplier tubes), thus overcoming any limitations in timeresolution and sensitivity imposed by the EM-CCD camera[22]. Readers familiar with spectral unmixing in fluorescencespectral imaging may ask what is the relationship of statisticalfiltering in the spectral domain to the other algorithms used forspectral unmixing; we discuss this topic in the section“Applications of statistical filtering in fluorescence imaging”.

A generalisation of the principle of statistical filtering in FCSwas developed byClaus Seidel’s group and termed fFCS. It usesmulti-parameter fluorescence detection (several detection chan-nels for different spectral regions and/or polarisation) and glob-ally calculated statistical filters based onmulti-variable TCSPC-based patterns (sets of TCSPC histograms in all multi-variabledetection channels for each component). The experimentalrealisation of fFCS reported used only two polarisation-resolved channels; however, its extension to a greater numberof channels is straightforward. Results of simulations haveshown that components with identical lifetimes and differentrotational correlation times can be resolved by fFCS withpolarisation-resolved detection. This indicates that if multi-parameter detection is used with globally calculated statisticalfilters, very subtle differences in fluorescence characteristicscan be used to separate signals by statistical filtering [23–25].

Yet another type of application was recently developed byAleš Benda [52]. This is a modified experimental realisationof the dual-focus FCS (2fFCS) developed previously by JörgEnderlein’s group [53]. In the original design of 2fFCS, twoalternately pulsing lasers emit linearly polarized beams withpolarisation planes perpendicular to each other. Coupling thebeams into the same polarisation-maintaining single-modefibre ensures that the two beams are precisely overlapped.The excitation light then passes through a Nomarski prismplaced before the objective, which creates two separated, butstill slightly overlapping foci. The centre-to-centre distance

R. Macháň et al.

between them is constant and reproducible. Because of alter-nate pulsing of the lasers, signals from the two foci can beseparated by time-gating and then cross-correlated. The well-defined and stable distance between the foci serves as ameasure of distance (the term used by the authors is “intrinsicruler”). Unlike conventional FCS, 2fFCS is not prone to errorsresulting from refraction index mismatch, cover slide thick-ness variations, optical saturation, etc., and enables interpre-tation of the smallest changes of diffusion constant andcalibration-free determination of absolute diffusion coeffi-cients [54–56]. The same basic idea of 2fFCS was used byŠtefl et al. with the difference that excitation light was provid-ed by a single continuous wave laser and its polarisation washarmonically modulated by a fast electro-optical modulator.The detected fluorescence then contained a linear combinationof signals from the two foci with harmonically changingweights with which each of the foci contributed to the totalintensity. Because the periodically modulated signals fromeach focus followed a known temporal pattern, statisticalfiltering could separate their contributions and enabled calcu-lation of cross-correlation between fluorescence from the twofoci. An interesting feature of this method is the use ofTCSPC-based statistical filters with excitation by use of acontinuous wave laser, the polarisation of which is harmoni-cally modulated. This shows that fluorescence decays afterpulsed excitation are not the only possible patterns for time-domain statistical filtering, and that patterns for statisticalfiltering can be obtained by arbitrary periodic modulation ofthe excitation light [52].

Removal of background and parasitic signal components

Similarly to hardware optical filters, which can be used toprevent some unwanted signal components (for example pho-tons of elastic and Raman scattering) from reaching the detec-tors, statistical filters are able to separate background andparasitic signal components from the collected signal. Thewhole idea is very straightforward and based on the fact thatcharacteristic patterns, distinct from the patterns of fluorophorepopulations of interest, can be identified and assigned to straylight, detector thermal background, and other parasitic compo-nents. Identifying an additional signal pattern has importantconsequences. Because the spectrum or fluorescence decay ofthe total signal is treated as a linear combination of all identifiedpatterns, Eq. (4) provides dedicated filters for all components,including those we wish to remove. As a consequence, the filterfunctions corresponding to desired signal components becomemore specific and more accurate.

FCS data typically suffer from artefacts caused by uncor-related background (consisting of thermal noise of the detec-tors and photons of stray light) and detector afterpulsing, i.e.the false counts which follow (typically on the μs scale) real

photon-detection events as a result of transient effects inducedby the detected photons. Most avalanche photodiodes andphotomultiplier tubes (the common types of detector used inFCS) suffer to some extent from this artefact. The two sourcesof background affect the ACFs in following manners: uncor-related background lowers their amplitudes, thus, preventingaccurate determination of concentrations of fluorescent mole-cules of interest; and afterpulses (correlated on the μs scalewith the photons which caused them) add to the ACF anartificial correlation decay on the μs scale, which can be easilymisinterpreted as an indication of triplet-state dynamics orother photophysical processes.

Readers familiar with FCS are probably aware of anotherway of preventing afterpulses from distorting the ACFs,namely cross-correlation of signals from two detectors.Although successfully solving the problem of afterpulses,the approach does not deal with uncorrelated background(actually makes it worse by including all the afterpulse countsin it). Therefore, amplitudes of ACFs are lower than valuescorresponding to the actual fluorophore concentrations [57,58]. Statistical filtering also has the additional advantage ofnot needing more than a single detector.

The pattern corresponding to uncorrelated background anddetector afterpulsing is very simple—a straight line. Falsecounts corresponding to both effects are randomly distributedon timescales up to hundreds of nanoseconds (those being thetimescales covered by TCSPC decays). While this is quiteobvious for dark counts (thermal background), it is less intu-itive for the afterpulsing counts, which are clearly correlatedwith real photon detection. However, the characteristic time ofthis correlation is typically of the order of μs. On the time-scales of tens to hundreds of ns probed by TCSPC, afterpulsesessentially arrive randomly; in other words they are uncorre-lated in TCSPC histograms [59]. Because of its very simpleshape, no additional measurement is needed to obtain thebackground pattern. The amplitude corresponding to the back-ground can be directly obtained from the TCSPC histogram ofthe overall signal as the average intensity for channels inwhich fluorescence intensity can be regarded as havingdecayed completely (typically, channels preceding the excita-tion pulse, for example the channels in Fig. 1a correspondingto times ≤1 ns). As a result, the pattern of a fluorophore caneasily be obtained from a TCSPC decay containing a combi-nation of background and the fluorophore decay bysubtracting the constant background from values in allTCSPC channels. Note that this implication (however trivial)is highly important for practical realisation of FLCS, becauseall experimental recordings of patterns inevitably contain abackground component.

This method of background suppression in FLCS is amongthe earliest applications of statistical filtering. Since its publi-cation in 2005 it has become an integral part of FLCS [59](and also became the first application of statistical filtering to

Statistical filtering in fluorescence microscopy and FCS

living cells [60]). As such, it has been used inmost subsequentFLCS studies and in several FCS studies performed on instru-ments equipped with TCSPC. The large difference betweenshapes of the patterns for intensity decay and for the back-ground guarantees their correct separation even if the numberof detected photons is not high enough for reliable separationof two distinct intensity decays (Fig. 3). Because of its sim-plicity, robustness, and very significant positive effect on thequality of ACFs, background removal has become a standardpart of FCS data treatment for some researchers and thus hasthe potential to remain the most widespread application ofstatistical filtering [57, 61–69].

Other parasitic signal components which can be removedby time-domain statistical filtering contain elastic and Ramanscattering [30, 57, 58]. Scattering processes are characterizedby very short TCSPC decays (in practice identical to the shapeof the IRF), which usually differ substantially from those offluorescence emission. Scattering is especially pronounced forsamples containing low fluorophore concentrations and highconcentrations of macromolecules, supramolecular structures,or nanoparticles (which scatter light more efficiently thansmall molecules). An approximate pattern of the scatteringcan be obtained experimentally, for example by measuringlight reflected from a cover slip, using a neutral density filterinstead of an emission filter to adjust the count rate to valuescomparable with those recorded in the actual fluorescenceexperiment. However, because of the wavelength dependenceof detector response, it has been argued that for precise deter-mination of IRF shape, more elaborate procedures (and sam-ples) are necessary [70–73]. To obtain the pattern of the purefluorophore (free from the scattering component), afluorophore solution of higher concentration can be used toreduce the ratio of scattering to fluorescence to negligiblevalues [57, 58].

The contribution of autofluorescence, which may imposeproblems on FCS for some types of biological sample, can, inan analogous manner, be removed by filtering on the basis ofits different intensity decay compared with the fluorophore ofinterest. Recently, the azadioxatriangulenium dye was pro-posed specifically for FCS in the presence of high autofluo-rescence. It has a very long lifetime of more than 19 ns, whichmakes its emission easily distinguishable from any autofluo-rescence. The efficiency of FLCS in obtaining the correctvalues of concentration and diffusion coefficient of theazadioxatriangulenium dye in the presence of autofluorescentbackground was demonstrated by experiments in which rho-damine 123 was used to simulate the autofluorescence ofbiological samples [74, 75]. However, such large differencesin lifetimes between the fluorescence of interest and theautofluorescent background are not necessarily required fortheir successful separation by FLCS. Different shape of thefluorescence decays is sufficient. The autofluorescence decayis typically multiexponential and easily distinguishable from

the decay of the probe (which is frequently close tomonoexponential) even when their average lifetimes are notdramatically different [76]. An example of FLCS separation ofthe Oregon green signal from the autofluorescence of diatomfrustules can be found in Ref. [77]. In general, there is nolower limit on the difference between patterns needed toachieve correct separation by statistical filtering [21]; howev-er, the more similar the patterns are, the more photons arerequired to reliably separate the signals (Fig. 3, caption). Themaximum number of photons available in an experiment thusimposes a practical limit on the similarity of patterns whichcan be separated by FLCS. Analysis analogous to that shownin Fig. 3 can be performed to estimate the range of usability ofstatistical filtering for any particular pair of patterns.

Note that background removal by statistical filtering is notexclusive to time-domain filtering. An analogous approachwas also used to remove camera background in FSCS (Fig. 2)[22]. Similarly, elastic and Raman scattering can be removedin FSCS, on the basis of their different spectra.

Other applications of multi-dimensional fluorescence datain FCS

Statistical filtering as outlined in the section “Principles ofstatistical filtering” is not the only method that can utilise themulti-dimensional (either time-resolved or spectrally-resolved)raw data in FCS. In this section we briefly review alternativeapproaches to the treatment of multi-dimensional FCS data.Although they do not belong to applications of statistical filter-ing, they can be applied to the same datasets and may oftencomplement the analysis based on statistical filtering.

The oldest and conceptually the simplest among these istime-gated FCS [78–81]. In modern instrumentation, time-gated detection typically means application of time-gates totime-tagged photon records. That is, in statistical filtering,equivalent to use of a very simple filter—a step functionwhich assumes values of either 1 or 0. Spectrally resolvedFCS can be regarded as analogous with time-gated FCS in thespectral domain (such as FSCS is an analogy to FLCS). Inspectrally resolved FCS, fluorescence emission is projected byan imaging spectrograph on to an EM-CCD chip and eachACF is calculated from the signal detected by a specific rangeof pixels only, which corresponds to the emission band of theparticular component [82]. This means, again, that a step-likefilter function is used, and acts as an ideal hardware opticalfilter. The advantage of the spectrograph and EM-CCD-basedapproach over the traditional setup using dichroic mirrors andoptical filters is in its greater flexibility in selecting wave-length ranges for FCS analysis. However, unlike statisticalfiltering, it cannot adequately deal with strongly overlappingemission spectra. The choice of spectral ranges is necessarily atradeoff between the need to minimize the spectral crosstalk

R. Macháň et al.

and the need to utilise as many detected photons as possible.Rejection of all pixels containing significant overlap of emis-sion spectra of multiple components, implied by the formerconsideration, results in rejection of some parts of each emis-sion band, which at the same time goes against the latterconsideration. Time-gated FCS faces analogous problemswhendealing with overlapping decays. For example, the contributionof elastic and Raman scattering can be removed from the ACFby rejecting photons from TCSPC channels corresponding tothe IRF. However those channels also correspond to the highestfluorescence intensity and their rejection results in the loss of asubstantial fraction of fluorescence photons. Therefore, simplegating (i.e. application of step-like filter functions) is ideallysuited only for situations in which components with non-overlapping patterns must be separated. A common exampleof such a situation is DC-FCCS with PIE, where any overlapbetween decays is avoided by proper selection of the intervalbetween interleaved excitation pulses [26, 32].

Several approaches to analysis of time-resolved FCS datahave been developed by Ishii and Tahara. The first of these waslifetime-weighted FCS in 2010 [83]. The method uses TCSPCchannel of each photon as its weighting factor during calcula-tion of a lifetime-weighted ACF, which is then divided by thestandard ACF (to which each photon contributes with unitweight). A ratio of unity is obtained for a homogeneous samplecontaining a single population of fluorophores; the presence ofmultiple populations characterised by distinct lifetimes resultsin the ratio deviating from unity. Furthermore, the characteristictime of dynamics between the populations (for example thedynamics between conformational states of moleculescharacterised by distinct lifetimes) is indicated by the lag timecorresponding to the onset of the deviation from unity. Equation(8) is a general formula for the lifetime-weighted ACF GL(τ);T(t) is the photon arrival time (TCSPC channel number) of thephoton detected at time t.

GL τð Þ ¼ T tð ÞI tð Þ⋅T t þ τð ÞI t þ τð Þh iT tð ÞI tð Þh i2 ð8Þ

Later, the same authors, inspired conceptually by previouswork of Yang and Xie [84, 85], proposed analysis of time-tagged photon data by use of 2D correlation maps [86]; theyrecently developed the method further and called it 2D FLCS[87, 88]. Central to this method are 2D emission delay corre-lation maps. Each map is a two-dimensional histogram ofnumbers of detected photon pairs in which each axis corre-sponds to the TCSPC channel number of one of the photons inthe pair. Maps are constructed for individual values of macro-scopic lag time τ (or more exactly for individual rangesthereof) and contain numbers of photon pairs separated bythe given lag time τ. The 2D emission delay correlation mapscan be used, similarly to statistical filtering, to subtract back-ground (for example detector noise or elastic and Raman

scattering) and to extract fluorescence decays correspondingto a particular range of lag times [86]. The latter can beregarded to some extent as reversed FLCS; instead of con-structing FCS ACF from photons with a particular decaysignature, the method takes photons with a particular FCSlag time signature to construct a corresponding decay curve.In their further work, Ishii and Tahara converted the 2Demission delay correlation maps into 2D lifetime correlationmaps for each macroscopic lag time. Lifetimes of individualcomponents in the sample correspond to diagonal peaks in the2D lifetime correlation maps, whereas off-diagonal cross-peaks indicate dynamics between individual components.The lag time corresponding to the appearance of the cross-peaks reveals the time-scale of the dynamic interchange [87,88]. Although the method was developed primarily for time-resolved data, the authors revealed its more general nature.The analysis based on 2D correlation maps can be readilyapplied to datasets in which an arbitrary spectroscopic quan-tity (for example emission wavelength) assumes the functionof TCSPC channels in the outlined method [88].

Another interesting means of utilising lifetime informationin FCS was developed by Berland and Anthony and calledτFCS [89–92]. The method is based on global fitting offluorescence intensity decays with ACFs. Each componentin the sample is characterised by a specific fluorescencelifetime (or, in general, a specific distribution of lifetimes)and by a specific ACF (either with a single diffusion time orin general with a specific distribution of diffusion times, tripletstates, etc.). The total intensity decay and ACF measured forthe sample are linear combinations of decays and ACFs of allcomponents. Because both the decay and the ACF originatefrom the same sample, the coefficients for each component inboth of the linear combinations (decay and ACF) are notindependent; both need to correspond to the actual fractionof the particular component in the sample. That imposesconstraints on the free parameters in the global fitting ofintensity decays and ACFs and greatly increases accuracy ofthose parameters. Although very powerful for enhancing thereliability of fitting in FCS, it is less generally applicable thanstatistical filtering. The main limitation is that τFCS does notachieve actual separation of the signals; therefore, it allowsneither rejection of undesirable signal components nor analy-sis of cross-correlations between individual species.

Applications of statistical filtering in fluorescence imaging

Although, so far, it has been used almost exclusively in FCS,statistical filtering can be used to separate components in anytype of fluorescence data. Recently, time-domain statisticalfiltering has been applied to confocal images. Photons in eachpixel were filtered according to their TCSPC channels, andimages corresponding to individual components were

Statistical filtering in fluorescence microscopy and FCS

obtained, and subsequently analysed by RICS. The techniquewas named raster lifetime image correlation spectroscopy(RLICS) [32].

Its performance was tested on images recorded for livingcells expressing tandem constructs of fluorescent proteins(green GFP and red mCherry) forming a FRET pair.Because not all mCherry proteins were properly folded, notevery GFP molecule was involved in a FRET pair. Statisticalfilters were calculated for GFP using ideal fluorescence de-cays of non-quenched GFP and of FRET-quenched GFPconvoluted with IRF as patterns. The two lifetimes used toconstruct the patterns were determined by double-exponentialfitting of measured intensity decay of the sample containingthe tandem construct; the longer of the two lifetimes wasassumed to correspond to non-quenched GFP and the shorterto GFP in FRET pairs. Two GFP images were, thus, obtained;one corresponding to GFP molecules not undergoing FRET,the other to GFP molecules involved in FRET pairs withmCherry. Those images were cross-correlated with themCherry image and the cross-correlation was compared witha CCF between mCherry image and the total GFP imagecontaining both components. In accordancewith expectations,higher CCF amplitude was observed in the latter case (FRET-quenched GFP) and significantly lower amplitude in the for-mer. However, the CCF amplitude did not decrease to zero fornon-quenched GFP (as theoretically expected for perfect sep-aration of signals), suggesting that the complexity of thebiological sample compromises the accuracy of filtering.

RICS, a form of image correlation spectroscopy, is closelyrelated to FCS. Therefore, this particular application of statis-tical filtering can be viewed as a variation of the well-established FLCS. Nevertheless, we regard it as a conceptu-ally different utilisation of statistical filtering leading to newapplications of the method not only in the form of RLICS butalso in other types of quantitative analysis of fluorescencelifetime images.

By analogy with the suppression of parasitic signal com-ponents in FCS, statistical filtering can be also used to sup-press unwanted background in fluorescence images.Autofluorescence background is an important concern in im-aging of living cells, especially plant cells, because it overlapsspectrally with the emission of many common fluorescentlabels (including green fluorescent protein) and significantlyreduces image contrast [76]. Although, as far as we are aware,statistical filtering has not yet been used for this purpose,another method based on differences in fluorescence decaysbetween the autofluorescence and the probes of interest hasbeen successfully used. The method, called fluorescence in-tensity decay shape analysis microscopy (FIDSAM), analysesstandard time-domain fluorescence lifetime images (the sametype of data as needed in, e.g., RLICS) and fits the decay ineach pixel with a single exponential function. A good qualityfit is achieved for pixels containing a high fraction of probe

fluorescence (which is assumed to be a homogeneous popu-lation following a monoexponential decay). On the otherhand, the broad distribution of lifetimes of the autofluores-cence leads to poor fit quality for pixels containing highfraction of background. The signal in each pixel is thendivided by the goodness of fit estimator χ2. The intensity ofpixels with high autofluorescence content (having highχ2 as aresult of poor fit quality) is reduced, whereas the intensity ofpixels with high content of probe fluorescence (having χ2

close to 1) is little affected by this rescaling. The resultingimage has, thus, increased contrast between probe fluores-cence and the autofluorescence background [76, 93, 94].

A conceptually similar, but more versatile approach, calledpattern-matching analysis, has been developed and recently com-mercialized [95, 96]. The idea is to find and select arbitrary decaypatterns contained in the image, without making prior assump-tions. Specific decay patterns can be directly located in specificimage regions, for example a background pattern in dark areas.Finding unique decay patterns which are mixed with otherpatterns and distributed across image pixels is facilitated by theso-called decay diversity map. This is a 2D correlation plotshowing the distribution of decay heterogeneity (multi-exponentiality, i.e. deviation from single exponential behaviour)versus the average lifetime of the pixels. Distinct fluorophorepopulations then appear as more or less separated spots(correlations) in the map, and their characteristic decay curvesare then assigned as patterns. Moreover, patterns can be loadedfrom control measurements also, for example to define the“donor only” and “acceptor only” decay curves in an FLIMimage of FRETconstructs. In comparison with FLCSmathemat-ics (or with RLICS), the major difference is that no filter func-tions are calculated. Instead the TCSPC histogram of each imagepixel is decomposed into patterns; in other words, the patternamplitudes are fitted. As a final step, colours are assigned to eachpattern and their amplitudes are visualized in a false-colouredimage revealing the exact location of fluorophore populations. Avery similar approach has been used to separate autofluorescencefrom the fluorescent protein signal in living mice [97].

Pattern matching can be regarded as a time-domain analo-gy of linear spectral unmixing, for which a variety of algo-rithms exist in the spectral domain [98–101]. Alternatively,image-separation approaches based on phasor plots have beenused in the time and spectral domains [102–104].

A detailed comparison of available image separation tech-niques would exceed the scope of this review. We will, there-fore, limit this discussion to the main difference betweenstatistical filtering and the approaches mentioned above,which is that statistical filtering performs the separation atthe level of individual photons, by assigning to each photona set of weights w(k) with which it contributes to the signalfrom individual fluorophore populations, whereas in patternmatching or linear spectral unmixing the statistical averagingprecedes the actual separation step.

R. Macháň et al.

To illustrate better what we mean, let us consider a largestack of S image frames Imr (r=1, 2, …, S) acquired consec-utively within the same region of the sample with a very shortexposure time, so that each pixel in each frame contains, onaverage, a very small number of photons (for example a singlephoton or fewer on average). By applying statistical filteringto each single frame Imr in the stack we can separate it intoframes Imr

(k) corresponding to the individual fluorophorepopulations. To obtain any meaningful image Im(k) we thenneed to average a sufficient number of frames Imr

(k) in thestack. (The caption to Fig. 3 indicates that a specific number ofphotons is needed to retrieve any meaningful separation ofsignals; however, the statistical averaging necessary occursonly after the actual separation step, i.e. after weighting ofindividual photons.)

In pattern matching or linear spectral unmixing, the sepa-ration procedure is applied to an ensemble of photons (allphotons corresponding to a pixel of the image), which must besufficiently large to represent the underlying spectra or inten-sity decays with reasonable accuracy. Using the illustrativeexample outlined in the previous paragraph, we first need toperform the averaging over a sufficient number of frames inthe stack before applying the actual separation step to theaveraged image. The averaging step may, of course, be re-placed by acquiring a single image with a longer exposuretime, thus, performing a sort of “hardware averaging”.

It is obvious from our hypothetical example that for stan-dard fluorescence imaging it makes no difference in whichorder we perform the averaging and separation. Statisticalfiltering is, then, only one of several approaches availableand not necessary the ideal approach. In several special imag-ing modalities, however, it is important that separation pre-cedes the averaging step. Statistical filtering on the singlephoton level is, to the best of our knowledge, the only suitableapproach in imaging techniques which utilise fluorescenceintensity fluctuations to extract information about underlyingmolecular dynamics or aggregation states, for example vari-ous ICS modalities [105]. In RICS, for example, stacks ofimages are recorded with short exposure times to enableobservation of fast molecular dynamics. Two averaging stepsare then performed: the first is calculation of a spatial correla-tion function from each frame and the second is averaging ofthe spatial correlation functions throughout the whole stack [3,4]. When combining RICS with statistical filtering (as inRLICS), the separation has to be conducted on the level ofindividual frames in the stack, to obtain separate correlationfunctions for each fluorophore population. Another exampleis the superresolution optical fluctuation imaging (SOFI), asuperresolution technique using fluorescence intensity fluctu-ations to locate individual emitters [106, 107].

Note that the situation in FCS is analogous. Fast signalfluctuations have to be separated in a manner which maintainstheir high temporal resolution. Subsequent calculation of the

ACFs or CCFs is equivalent to averaging over a large numberof photons. This explains the so far exclusive use of statisticalfiltering for signal separation in FCS and its modalities.

Conclusions

The concept of statistical filtering was introduced in 2002 asthe mathematical basis of a method originally called time-resolved FCS and nowadays known as FLCS. Since then,FLCS has become quite a well-established FCS modality.The increasing availability of FCS instrumentation equippedwith time-resolved (TCSPC) detection and the implementa-tion of FLCS analysis in commercial FCS software havecertainly promoted proliferation of the technique. Removalof the background by FLCS, especially, has been used in avariety of FCS studies because of its simple implementationand robustness in suppressing two major sources of artefactsin FCS—detector afterpulses and uncorrelated background.More recently, statistical filtering has been used in contextsother than in FLCS. A multi-parameter generalisation ofFLCS (termed fFCS) has been developed [23, 25], andFSCS—a counterpart of FLCS in the spectral domain [22];time-domain filters have been applied to fluorescence imagesanalysed subsequently by RICS [32]. This work demonstratesthat statistical filtering is a general and broadly applicablemethod of data treatment, a point we wished to further em-phasise in this review.

One way of generalising the concept of statistical filteringis replacement of the time-domain patterns used in FLCS withspectral patterns or patterns based on any arbitrary combina-tion of spectroscopic characteristics. Further generalisation isachieved by treating any arbitrary fluorescence dataset bystatistical filtering, not just the intensity time traces used inFCS. Statistical filters (in either the time or spectral domains)can be used to separate fluorescence images or generic inten-sity time traces (such as those utilised in fluorescence burstanalysis [108]). The main limitation on applications of statis-tical filtering is the requirement of the mathematical treatmentfor a sufficient number of photons (as discussed in the section“Statistical filters” and illustrated in Fig. 3). To to reliablyseparate fluorescence images, sufficient amount of photonsper pixel are needed. To obtain separate intensity time traces,these need to be binned with sufficient amount of photons pertime bin. The concern about the number of photons is notrestrictive in applications of statistical filtering in FCS, be-cause FCS itself requires a large numbers of photons to furnishcorrelation curves with good statistics. That explains why theapplications of statistical filtering have remained restricted toFCS for so many years. However, on the basis of recentdevelopments we expect statistical filtering to be implementedin several other techniques of quantitative fluorescence

Statistical filtering in fluorescence microscopy and FCS

microscopy in the near future, with FCS continuing to be itskey field of application.

Acknowledgements Financial support was provided by Ministry ofEducation of Singapore (R-154-000-543-112 to R.M.) and by the CzechScience Foundation (P208/12/G016 to P.K. and M.H.). Moreover, theAcademy of Sciences of the Czech Republic is acknowledged for thePraemium Academie award (M.H.).

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