The effect of object density in fluorescence microscopy is of course well studied 14 , 15 , 16 , 17 and here we provide further insight into its role. Therefore, in a highly dense object pattern, not surprisingly, the probability that an object is not affected by algorithmic resolution is severely reduced. For example, consider cellular membrane receptor clusters distributed in a completely spatially random fashion with a density of 1 cluster per square micrometer as in Fig. Thus, the difference in algorithmic resolution between the two algorithms can have drastic effects on the analysis of the data.
Further, it is only for cluster densities of 0. However, this probability decreases significantly to Other spatial distributions can be assessed through similar analysis of their nearest-neighbor distribution function. This will be demonstrated by applying this probabilistic approach to localization microscopy. Localization-based superresolution methods use repeat stochastic excitation of small subsets of the fluorophores in a sample 4.
To quantify this, let us denote q to be the probability of an object appearing in any given frame of the dataset. Here, G q is the nearest-neighbor distribution function for the subset of objects that appear in an arbitrary frame. This shows how the probabilistic resolution increases by separating the objects among frames.
We will illustrate these concepts with two examples: clustered objects and tubulin data. Suppose the single molecules to be localized exist in clusters. A simple model one may apply in this setting is that the cluster centers are completely spatially random and single molecules are distributed about the cluster centers according to a 2D spherically symmetric Gaussian distribution. An example realization of this process is shown in Fig.
This demonstrates the extra demands that clustered data present. Probabilistic resolution. The blue crosses show the observed proportion of correctly localized molecules. A demonstration of this calculation is shown in Fig. Verification of this analysis is provided in Fig. An experimental tubulin dataset is also considered to further verify the results presented.
The original dataset, Dataset 1, is comprised of 50, frames and contains approximately 3. Averaging pairs of images, we are able to create Dataset 2 that consists of 25, frames, each with double the object density. Dataset 3, formed by averaging triplets of frames, consists of 16, frames with triple the object density. This is repeated up to Dataset 10, which consists of frames with ten times the object density. The number of localizations generated by each algorithm for each dataset is shown in Fig. These results demonstrate two key points. The first is it can be seen that Algorithm 2, the algorithm with the smallest resolution limit, has the largest number of localizations.
Furthermore, Algorithm 1, which has a similar algorithmic resolution limit, has a very similar number of localizations. However, Algorithm 3, which has an algorithmic resolution limit almost twice as large as Algorithms 1 and 2, produces far fewer localizations.
This is consistent with the presented theory; a larger algorithmic resolution results in a smaller probabilistic resolution under the same molecule density, and as such we would expect fewer localizations. The second point is that the number of localizations decreases as the density increases. Again, this is predicted under our theoretical framework since we show probabilistic resolution decreases with density. This demonstrates that experimental observations on the performance of different algorithms are consistent with the findings of the paper, which have been reached from simulation methods.
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If the clathrin-coated pit data of Fig. This indicates that at distances above twice the algorithmic resolution limit for the individual algorithms, the clathrin-coated pit locations do not show any deviation from complete spatial randomness. Resolution has been analyzed in microscopy going back to the classical criteria by Rayleigh and Abbe. Those criteria address the performance of the imaging optics. A resolution measure based on the Fourier ring coefficient was introduced that can be computed directly from an acquired image and takes into account the standard deviation with which a single molecule can be localized Common to these recent approaches is they do not take into account that different object-based image analysis algorithms can have very different algorithmic resolution limits.
The evaluation of algorithms for single-molecule image analysis is complex and a number of approaches have been used in the past, many of those comparing the estimated locations with the ground truth of simulated data The introduction of the concept of algorithmic resolution in this paper provides an additional tool by determining the minimum distance beyond which the algorithm can reliably distinguish different objects. The analysis presented here also provides important insights into why algorithms perform differently on more classical evaluation approaches see e.
Object density has been well recognized as causing significant problems for the object-based image analysis algorithms 14 , 15 , 16 , Here, we have derived analytical approaches that investigate how algorithmic resolution and object density impact the probability of an object being affected by resolution when the data are analyzed with an image analysis algorithm with specific algorithmic resolution limit.
Understanding this probabilistic resolution is key to determining if post-processing methods, for example clustering algorithms 20 that estimate cluster sizes and the number of objects per cluster, can be used with confidence.
The approach to defining algorithmic resolution that is presented here depends on the availability of test data with objects that are distributed in a completely spatially random fashion. Such data are completely uncorrelated and consequently gives rise to a pair-correlation function that is identically equal to 1. The estimation algorithm that is investigated is then used to estimate locations of these objects based on simulated data with the given object locations.
Subsequently, an estimated pair-correlation function is computed based on the estimated locations. The algorithmic resolution limit is defined through the distance below which the estimated pair-correlation function deviates from that based on the true locations, i. This approach could be modified to use test data that gives rise to a different pair-correlation function. The point at which the estimated pair-correlation function deviates from the theoretical one could be used as a method of comparing two or more different algorithms.
For the approach to be relevant to experimental settings, it is important that the simulated data used to determine the algorithmic resolution limit reflects the data that will be acquired when the algorithm is applied in an experimental setting. As we have seen, the determination of the algorithmic resolution limit does depend on the photon count of the simulated single-molecule data in certain circumstances, such as extremely low photon counts.
We have also seen a dependence on the density of the sampled objects, again in extreme cases. It is equally clear that other model parameters that determine the appearance of the images of the objects under study need to be matched to the experimental settings to be able to expect reliable results. If experimental data were available that can be guaranteed to be made up of objects that are located in a completely spatial random manner, then such data could also be used.
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However, guaranteeing that objects are located in a completely spatially random fashion would be very difficult to achieve. Unless such a guarantee is available, simulated data are to be preferred as complete spatial randomness of these test data is critical for the approach. We have introduced a methodology to systematically assess the algorithmic resolution limit of object-based image analysis algorithms and to evaluate the impact of the limitations on the analysis of microscopy data.
We hope that the approaches presented will contribute to a systematic evaluation of such algorithms that are of relevance not only to microscopy applications but to other object-based imaging scenarios such as those arising, for example, in astronomy. Spatial statistics has played an important role in many areas of cell biology. We hope that the results presented here will contribute to an improved understanding of the methodology and lead to the avoidance of misinterpretation of the acquired data.
HMEC-1 cells were fixed using 1. Cells were washed twice with PBS between each incubation and finally immersed in 1. Signal from the sample was also filtered through this filterset before being acquired by the camera. The cells were fixed using 3. The fixed cells were then permeabilized using 0.
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All devices including lasers, shutters, and cameras were controlled and synchronized using custom-written software in the C programming language. Buffer solutions in which the BS-C-1 cell samples were prepared were replaced with an imaging buffer immediately prior to the imaging of each sample.
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All buffer ingredients were purchased from Thermo Fisher Scientific. The imaging buffer was freshly prepared for each acquisition. A coverslip was placed over the sample and sealed using vacuum grease. The activation laser power was kept low throughout the acquisition.
For the simulated image in Fig. Deterministic structures consist of D molecules located at evenly spaced points on the circumference of a circle of radius r. For the simulated images analyzed to obtain the results in Fig. The total photons detected at the k th pixel from all clathrin-coated pits within the region represented by the image is denoted by S k.
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For the simulated image of clathrin-coated pits in Fig. For each clathrin-coated pit, f d is modeled as a Gaussian profile given by. The photon count detected at the k th pixel is modeled as a Poisson random variable with mean given by,. For each molecule, f d is modeled as an Airy profile given by. When simulating images of fluorescently labeled tubulin molecules that are stochastically photoactivated and detected, each image is taken as an image of single molecules and is simulated as described above.