![]() Clathrin-coated pits play a major role in endocytosis. However, importantly for our considerations, all possible spatial configurations of events are sampled.Īn important question in cell biology is whether or not structures are organized in a regular way or do not have particular relations among them. These occur purely by chance and are not due to some underlying correlation structure within the data. So at first sight some spatial configurations of events might be seen that resemble clusters, whereas in other areas large “empty” patches might be seen (see Supplementary Figure 1). Here, it is also instructive to recall that completely spatially random data are those in which the events occur completely at random and independently of each other. Clustering, in which objects are typically closer to each other than one would expect under complete spatial randomness, is characterized through positive values of this function, whereas deviations from 0 to negative values indicates inhibition or regularity, meaning that the spacing of points is somewhat larger than that in completely spatially random data. The L( r) − r function is non-zero if the point pattern is not completely spatially random. ![]() This implies that the related L( r) − r function, where \(L(r): = \sqrt \) for r > 0, is constant and equal to zero. Ripley’s K-function, which describes the expected number of objects within a distance r of an arbitrary object, is given by K( r): = πr 2, r > 0, for a completely spatially random pattern. For a spatial pattern that is uniformly distributed (in the probabilistic sense), also termed completely spatially random, the pair-correlation function g, which describes the relationship between pairs of objects that are a distance r apart, is given by the identity function g( r) = 1, r > 0. Central to this analysis is the notion of algorithmic resolution which we introduce to characterize an algorithm’s ability to resolve objects.ĭetecting the effect of algorithmic resolutionįirst, we show that insufficient “algorithmic resolution” of an image analysis algorithm can have a significant impact on the outcome of the analysis of spatial patterns which is typically carried out using the pair-correlation function or Ripley’s K-function 11 (see Supplementary Note 2). This theoretical background is introduced in Supplementary Note 1 and rigorously developed in Supplementary Notes 2– 7. Methods of spatial statistics, which have been extensively used in different scientific disciplines 11, form the theoretical background for the development of this manuscript and underpin the presented methods for evaluating location-based image analysis algorithms. A specific example that we will consider in detail relates to the question of whether the distribution of clathrin-coated pits is purely random or exhibits other spatial characteristics such as clustering. Here, we use methods of spatial statistics to quantitatively evaluate the resolution capabilities of location-based image analysis algorithms and to demonstrate the impact of resolution limitations on the analysis of object-based imaging data. The assessment of such algorithms in terms of their resolution capabilities is, however, largely unexplored. The success of such imaging experiments is, therefore, to a large extent dependent on how well these algorithms can resolve the imaged objects 10. 2, 3, 4), experiments to investigate the arrangement of molecular complexes on the cellular membrane such as clathrin-coated pits 5, 6, experiments tracking single particles 7, 8 or subcellular organelles 9, etc.Ĭommon to the analysis of experimental data produced by such “object-based” imaging experiments is the central role that image analysis algorithms play in the identification and localization of the underlying objects, be they single molecules, clathrin-coated pits, etc. Examples are localization-based superresolution experiments (PALM, STORM, etc. This is particularly relevant for the many modern imaging experiments and corresponding image processing algorithms for which the detection of objects (e.g., molecules, molecular complexes, subcellular organelles) form an integral aspect. However, no systematic approach is yet available for the evaluation of the often complex image processing algorithms that have become central to the analysis of the imaging data that today is acquired by highly sensitive cameras. Rayleigh’s and Abbe’s resolution criteria 1 were developed for observations with the human eye and had a major influence on the development of optical instruments. ![]() Resolution is one of the most important properties of an imaging system, yet it remains difficult to define and apply.
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