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The inset shows a close-up view of the region marked by the black rectangle. As can be seen, the antibunching and blinking effects are reduced in magnitude. Only photon pairs from the same emitter are correlated. It should be noted that in order to determine the distribution of time delays of photon pairs, all photons detected at a certain pixel are taken into account and considered as pairs, not just consecutive ones.

One can then apply appropriate fitting of the second-order correlation function to recover the position of the emitters with high resolution. The pixels of the detector used should be smaller than the spatial extent of the point spread function of the imaging system used. In the following we restrict ourselves to a discussion of the blinking signature, as it usually shows larger photon count rates.

Still, in experiments it will be most sensible to evaluate both quantities simultaneously. The combined information from both independent localization signatures will result in reduced localization error. In Fig. The positions of the emitters are marked by black crosses. The left panel shows the emitted intensity.

The emission from both emitters overlaps strongly and their individual position cannot be identified directly. A strong directional dependence is apparent. Black crosses mark the position of the emitters. In the presence of background noise or a second emitter, the distribution will change.

The spatially resolved total fluorescence signal intensity amounts to:. This is the fitting function. This property of the light field is routinely used to identify single photon sources In very good approximation, the same value of g 2 0 holds true for N single-photon emitters even if blinking is present Therefore, the value of the equal-time correlation function at zero delay may be used to estimate the number of emitters present, provided that the temporal resolution of the detector is sufficient to resolve the correct behavior at short delays.

It should be noted that number estimation becomes unreliable for large numbers of fluorophores as the values g 2 0 become close to one. Therefore, for wide field imaging, QUEST can be performed using iterative segmented spatial mapping of g 2 0 such that the QUEST algorithm estimates the number of emitters inside a region of interest and if it is too large to yield reliable results automatically subdivides it into two smaller regions of interest containing fewer emitters. Repeating this procedure until each segment contains a small number of emitters and consecutive reconstruction of the emitter locations in each individual segment will then provide the full list of emitter positions.

It should be noted that this segmented approach can be used to automatically investigate several region which contain the same emitters, but are slightly shifted with respect to each other. This technique allows for more robust emitter localization for emitters of varying brightness.

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We first test the QUEST localization technique on simulated data for two emitters placed at varying distances. In order to show the robustness of QUEST to noise, we choose a large dark count rate of about counts per second that is comparable to the photon emission rate per emitter. We investigate distances both larger and shorter than the diffraction limit.

The color plots show the total emitted fluorescence as acquired in typical experimental approaches such as total-internal-reflection fluorescence microscopy. White crosses mark the real positions of the emitters and black crosses show the reconstructed positions using QUEST for 20 imaging cycles. Magnified views of the boxed sections are shown in the right panel. White crosses mark the real positions of the emitters. It is worth noting that for small emitter distances, the reconstructed fluorophore positions are not distributed symmetrically around the real position, but show a bias towards the outward direction.

This effect is clearly demonstrated in Fig.

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This effect can in principle be avoided by specialized fitting procedures or maximum likelihood approaches. The total emitted fluorescence is shown in Fig. Only a single large spot can be seen, from which the positions of individual emitters cannot be identified reliably. In analogy to the case of two emitters shown in Fig. White crosses denote the real positions of the fluorophores. Black crosses represent the reconstructed fluorophore positions.

Simulation results are shown in Fig. Taking the dark count rate of The positions are determined well with a mean localization error of As can be seen in Fig. These points have been set to a value of 1 in the map and are not considered during the fitting procedure. Accordingly, it is interesting to investigate whether it is beneficial to have longer integration times and include regions with low photon count rates while applying QUEST.

To this end, we again investigate the mean localization error for the two fluorophores at varying distances already shown in Fig. Each of the emitters and the background noise contribute approximately the same number of detection events to the photon number.

Results are shown in Fig. The onset shifts to smaller photon numbers for larger distances. Increasing the photon number to even larger values does not improve the reconstruction significantly. The results may even become worse. Dedicated statistical approaches may improve the treatment of these pixels, but are out of the scope of the present manuscript. Mean localization error in dependence on the number of photons used as an input for QUEST for two fluorophores placed at varying distances.

A clear threshold for efficient reconstruction can be identified for every distance. Its onset is marked by dashed lines. Another factor to consider is the influence of the background noise level on localization accuracy. An exact treatment is way beyond the scope of this manuscript, but the general effect can be explained well qualitatively. As an example, Fig. For all data points, the same basic simulation of one million signal photons has been used and superposed with different noise datasets with a total number n Noise of noise counts and the same total duration as the signal datasets.

In order to understand this kind of effect, it is important to keep in mind that QUEST does not map the absolute emitted intensity, but the overlap of the emission of different emitters and their relative intensities at different points. In the simplest case, background noise can be considered as an additional homogeneous emitter. Considering for example a single isolated emitter, one immediately sees that background noise may even enhance localization accuracy.

The normalized second-order correlation function for a single emitter is uniform all over space and thus yields no information about the emitter position at all. Adding some constant background noise will result in spatially varying relative contributions of signal and noise and thus result in a second-order correlation function that shows a peak at the emitter position and enhanced localization precision.

Similar considerations apply for larger numbers of emitters. With the exception of 0 noise photons, all data points are averaged over 30 realizations of different noise distributions. The peak signal intensity at a single pixel amounts to counts. For comparison the red line shows the ideal single emitter shot-noise limit that can be achieved using STORM in the absence of background noise.

Finally, it is instructive to discuss the performance of emittter number estimation based on the normalized second-order correlation function. The columns in Fig. Recognition is close to optimal for small emitter numbers, but even up to 10 emitters, the emitter number estimate distribution is moderately narrow and still centered on the real number of emitters. However, there is a slight trend towards overestimated emitter numbers for many emitters, which can be traced back to the range of values of the second-order correlation function that corresponds to a certain emitter number becoming smaller when placing more emitters.

However, this problem can in principle be overcome by using tailored sectioning approaches. Each column gives the relative frequency distribution for a fixed emitter number.

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For the simulations, an average photon number of photons per emitter has been assumed. Our approach is tailored for having low experimental requirements, is robust with respect to the experimental data acquisition process and is computationally efficient as it does not require an underlying grid. The experimental procedure amounts to recording a spatially resolved time-stream of photons under standard optical excitation conditions and the choice of fluorophores and optics used is not critical as long as the fluorophores are bright enough. The critical difference to standard STORM setups lies in replacing the CCD detector with a single photon fiber bundle camera, which requires no further changes to the experimental setup.

The method identifies the number of emitters present and does not require a priori knowledge of this number. The precision and speed of the method, however, strongly depend on the specific implementation of the recovery algorithm used and we want to emphasize that we only present the basic technique here which leaves plenty of opportunities for improvement in terms of data analysis.

First, the least-squares fitting approach here is easy to implement, but for real detectors with possibly asymmetric point spread functions a maximum-likelihood approach 9 also taking the total intensity distribution into account might be more suitable. We expect that a more specialized reconstruction approach may result in improved localization precision.

For example, as can be seen in Fig. This deviation may be corrected by tailored algorithms using weighted data. Further optimization might also be achieved by utilizing deep learning 22 to find optimal reconstruction parameters or by using compressive sensing 23 or other sparsity based algorithms with 24 or without 25 using an underlying grid. Also the photon count threshold used is a degree of freedom that may be optimized. For example approaches using weights depending on the photon count rates per pixel or using spatial regions of variable sizes for analysis are likely to outperform the basic least square fitting approach described here.

It should also be noted that the completely symmetric Gaussian point spread functions used in the simulations are the worst possible case for emitter localization. Any asymmetry in the point spread function will make it easier to reconstruct the position of the fluorophores. QUEST has the potential to perform super-resolution imaging at high frame rates up to real time imaging. The required integration times for QUEST will depend on the brightness and density of the fluorophore used. Acquisition times of few seconds seem reasonable for typical experimental parameters and sub-second data acquisition may become feasible for low background noise.

However, due to the complex data analysis involved, achieving such fast acquisition rates will require hardware based online data analysis e. Building upon such an architecture, further extensions might be considered. For example higher-order correlations are supposed to be even more sensitive to the emitter positions and taking cross-correlations between different pixels into account in a similar manner to SOFI 10 may result in reduced noise and acquisition times.

It should also be noted that QUEST is not necessarily limited to biological imaging and organic dyes. In some respects, quantum dots may be a better choice as the fluorescent emitter as tailoring their parameters allows some limited control over the blinking behavior and can reduce the disadvantageous low-frequency blinking Further, in order to utilize the fast frame rates available with ultrafast imagers, using emitters with matching photon emission rates would be advantageous.

Therefore, QUEST lends itself rather to imaging in fields such as semiconductor physics, where quantum dots may be used rather than biological imaging. Along the same lines, one drawback of QUEST is that its usefulness depends strongly on the detector architecture employed and for standard detectors such as CCD cameras, it will not be overly useful as it does not perform well compared to established techniques in the limit of very small photon count rates. It is instructive to compare our approach to other super-resolution techniques.

It provides a tradoff between localization techniques such as STORM and super-resolution techniques based on unnormalized emitter noise properties. Regardless of whether the latter techniques rely on antibunching 29 , 30 or blinking such as SOFI, they are usually not localization techniques like STORM, but may be cautiously called imaging techniques because they provide wide field images of quantities such as cumulants, which allow for better resolution than fluorescence images. Both kinds of techniques are known to have advantages and disadvantages Localization techniques typically yield higher resolution, but also have more strict requirements.

The point spread function of the imaging system must be known in detail and simultaneous emission from overlapping emitters must be avoided, which renders data acquisition slow for large emitter densities. SOFI, on the other hand, has less strict requirements on knowledge of the imaging system and emitter properties.

Further, simultaneous emission from densely packed emitters is not a problem for SOFI, so it has the potential for substantially faster data acquisition rates compared to STORM and for straightforward implementations of 3D imaging. However, as SOFI is based on higher-order cumulants, the intensity scale in SOFI images is necessarily strongly non-linear which may result in emitters with low intensity not being recognized.

QUEST takes a position in the middle between these techniques. As the most important difference to SOFI, instead of cumulant images, QUEST records spatially resolved images of normalized second-order or higher-order correlation functions. Normalized correlation functions avoid the problem of emitters of weak intensity as is known from similar correlation-based super-resolution techniques that rely on confocal scanning Besides the effects of noise, every isolated emitter with the same emission profile will form exactly the same QUEST image.

The normalization ensures that weak emitters also leave a significant trace for densely packed emitters that will show up as an anisotropy in the QUEST image. However, this advantage comes at the drawback that the QUEST image does not directly provide information about emitter positions. The positions of emitters do not correspond directly to maxima or minima in the QUEST image and must be recovered via fitting procedures which require information about the imaging system. However, due to the usage of normalized correlation functions, it requires little information about emitter properties such as emitter number or brightness.

As a first approach, one may use iterative sectioning and include these quantities as fitting parameters, more sophisticated and precise strategies for emitter number estimation exist 33 , 34 and may be optimized for QUEST images. QUEST is therefore similar to localization techniques, but has the additional advantage that it does not require only a sparse subset of all emitters to be active at any time, which promises significantly faster data acquisition rates. In order to further understand the strengths and weaknesses of the present proposal and the differences compared to other super-resolution techniques, it is worthwhile to revisit the performance of QUEST in the presence of noise.

As shown and discussed in Fig. The reason for this is that QUEST utilizes the normalized second-order correlation function, which is ideally completely independent of the intensity at any pixel. However in order to achieve good emitter number estimation and precise results at the fitting stage, information about the value of g 2 at every pixel is required.

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This is a huge drawback for situations with strongly inhomogeneous illumination, such as having few emitters and regions in between with almost no signal intensity. In this case, a reasonable measurement of the value of g 2 is achieved quite quickly at the emitter positions, but it takes very long for the pixels with low count rates to accumulate enough statistics to achieve low-error measurements of g 2. As can be inferred from Fig. This also puts the quite large number of above photons needed to achieve good localization accuracy into perspective.

For smaller regions or homogeneously illuminated detector pixels, the performance will improve significantly. These non-standard effects on the localization accuracy already show that the signal-to-noise ratio is not an ideal quantity in order to estimate how well QUEST performs, not only because of the non-standard interplay between noise and localization accuracy, but also because the temporal properties of the background noise distribution are more important than its mean value or variance.

Considering a realistic implementation of the proposed technique, it will be necessary to find a suitable way to avoid the problem of regions with low intensity having a negative influence on localization accuracy. Neglecting pixels below some threshold intensity or using the mean intensity as a weight function are some basic possibilities, but their usefulness will depend strongly on the experimental conditions and are not optimal.

It seems to be a more suitable approach to use QUEST in combination with another technique, using an algorithm which first applies the standard super-resolution technique and provides estimates of the spatially resolved emitter density and then applies QUEST selectively to these areas, in order to determine whether the high-density areas consist of several overlapping emitters. However, due to the huge number of possible combinations of techniques and ways to implement them, this is a problem best solved by machine learning or similar techniques.

In order to visualize how QUEST works in general and to point out at which steps the addition of other techniques would be most beneficial, Fig. The steps, where optimization should take place are labeled as optional. Schematic flowchart for QUEST including the steps where optimization or application of complementary technique seems appropriate. Still, the performance of QUEST is based on photon pairs and should therefore be inferior to other standard techniques in the limit of low count rates.

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Accordingly, at current it seems to be most promising to apply QUEST as an extension to other imaging techniques, which work well for isolated emitters, but have problems with distinguishing overlapping emitters at larger count rates. To this end, one would construct a superresolution image using a standard technique, use real-time statistical analysis on this image in order to identify regions, which are likely to have unidentified overlapping emitters and then apply QUEST to these regions.

In order to be able to apply both the standard technique and QUEST to the same dataset and to achieve the temporal resolution necessary to identify the single emitter antibunching and estimate the number of emitters present, we suggest to use hardware based on the fiber-coupled single-photon avalanche detector array architecture, which has been introduced first for calorimetric purposes in particle physics Recently, the detector array technology has been improved significantly in terms of array size, readout rates and on-chip data processing 36 , In turn these technological advances have already led to first demonstrations of optimized super-resolution imaging by means of utilizing antibunching and postselection via a fiber bundle camera It offers low noise and negligible cross talk among pixels along with a large fill factor in conjunction with independently working detectors that can operate at a variable synchronized frame rate.

Single photon avalanche diode arrays 12 without fiber coupling are also a viable option, but typically suffer from rather large pixel sizes and small fill factors. We envision that by combining novel approaches in terms of detector hardware development, on-chip data processing and statistical analysis and optimized algorithms and by combining QUEST with other techniques, real-time super-resolution imaging may become feasible. To model the noise background, noise photons are added using an exponential distribution with a decay time of 5.

For the noise series, we vary this decay time, but keep the total simulated duration constant. Hell, S.

Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy. Klar, T. Subdiffraction resolution in far-field fluorescence microscopy. Three kingdoms have been overtaken by three evil lords and only Tyor, a teenage boy with magical powers, can restore peace to the land with the help of a bumbling elder, wizard and a hero A mighty hero battles the son of Satan and his evil witch ally to save a kingdom from being taken over by the duo. When three of the inhabitants of the primitive, all-female Kingdom of Kimbia are sentenced to death, a stranger from a neighboring Kingdom must overthrow the corrupt rules in order to prevent their doom and bring peace and harmony to both kingdoms.

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