A variety of recent imaging techniques are able to beat the diffraction limit in fluorescence microcopy by activating and localizing subsets of the fluorescent molecules in the specimen, and repeating this process until all of the molecules have been imaged. In these techniques there is a tradeoff between speed (activating more molecules per imaging cycle) and error rates (activating more molecules risks producing overlapping images that hide information on molecular positions), and so intelligent image-processing approaches are needed to identify and reject overlapping images. We introduce here a formalism for defining error rates, derive a general relationship between error rates, image acquisition rates, and the performance characteristics of the image processing algorithms, and show that there is a minimum acquisition time irrespective of algorithm performance. We also consider algorithms that can infer molecular positions from images of overlapping blurs, and derive the dependence of the minimimum acquisition time on algorithm performance.
Deep Dive into Theoretical Limits on Errors and Acquisition Rates in Localizing Switchable Fluorophores.
A variety of recent imaging techniques are able to beat the diffraction limit in fluorescence microcopy by activating and localizing subsets of the fluorescent molecules in the specimen, and repeating this process until all of the molecules have been imaged. In these techniques there is a tradeoff between speed (activating more molecules per imaging cycle) and error rates (activating more molecules risks producing overlapping images that hide information on molecular positions), and so intelligent image-processing approaches are needed to identify and reject overlapping images. We introduce here a formalism for defining error rates, derive a general relationship between error rates, image acquisition rates, and the performance characteristics of the image processing algorithms, and show that there is a minimum acquisition time irrespective of algorithm performance. We also consider algorithms that can infer molecular positions from images of overlapping blurs, and derive the dependence
Recent work has shown that the diffraction limit to the resolution of an optical microscope, believed to be ~l/2 since the work of Abbe (1), can be overcome by a variety of techniques (2). One promising road to single-molecule imaging in the far-field is to work with fluorophores that can switch between a fluorescent (activated) state and a nonfluorescent (dark) state, e.g., quantum dots (3,4), photoactivatable fluorescent proteins (5)(6)(7), or pairs of cyanine dyes (8,9). If the average distance between activated fluorophores is significantly larger than l then nearly all of the bright spots in the image are the result of single molecules. The centers of these bright spots can, in principle, be identified with subpixel accuracy (10,11), enabling accurate localization of individual molecules. A different subset of the fluorophores is then selected at random (via the stochastic nature of light absorption) and activated, and the process is repeated to localize the new set of activated fluorophores. Depending on the implementation, the minimum resolvable feature size ranges from 20 to 40 nm in the lateral direction and ~50 nm in the axial direction (12,13).
A common issue in all of these techniques is the number of molecules that can be activated at any one time. If too many molecules are simultaneously activated, then there is a high probability that a single bright spot will include light from multiple activated molecules and the center identified will not be an actual molecular position, introducing artifacts between two closely spaced molecules. However, decreasing the fraction of activated molecules increases the number of activation cycles needed to reliably image every molecule. Most imaging approaches with switchable molecules therefore use an algorithm to identify whether a bright spot contains one or more molecules. Typical algorithms include tests of a spot’s shape (ellipticity) (9). intensity (3,14,15), or fitting to the imaging system’s point spread function (6) via the CLEAN algorithm (16). If the algorithm finds that a spot contains multiple molecules, it is rejected and its center is not determined.
The questions that we address here are: Given the performance characteristics of some algorithm for rejecting bright spots with multiple molecules, what are i), the maximum possible error rate and ii), the minimum number of activation cycles needed when operating at a given error rate? Also, if instead of rejecting bright spots with small numbers of molecules we are able to solve the inverse problem and identify molecular positions, what is iii), the minimum number of activation cycles needed and how does it depend on the number of molecules the algorithm can simultaneously localize in a given bright spot?
The concept of a rejection algorithm is illustrated in Fig. 1: a spot of width l contains n molecules. At reported resolutions of l/10 or better, n could exceed 100 in 2D or 1000 in 3D imaging. If the spot contains one activated molecule, the algorithm should accept the spot for analysis, and if the spot contains m R 2 molecules the algorithm should reject the spot. In practice, the algorithm will not work perfectly and will sometimes reject a single-molecule image or accept an m-molecule image. We will denote by f m (m R 1) the probability that a m-molecule image is accepted by the algorithm. We assume that f m is calculated by averaging over all possible positions and orientations of m molecules, the distribution of light emission rates under the imaging conditions, and noise in the imaging system. The performance of the algorithm for a given imaging system and fluorescent species is completely determined by {f m }, and an ideal algorithm has
It is straightforward to set a lower bound on the number of activation cycles needed. To ensure that almost all molecules are imaged at least once and that the variance of the number of times imaged is small (for precision in brightness measurements) we must image each molecule an average of T » 1 times; because position measurements are noise-limited, T depends in part on the number of photons captured per activation cycle (10). We denote by p the probability that any individual molecule is activated in a given cycle, and p 1 ¼ np(1 À p) nÀ1 is the probability that a given bright spot has exactly one activated molecule. Note that the value of p is determined by the dosage of the activation pulse. Over N activation cycles, the expected number of times that our algorithm identifies a bright spot containing exactly one molecule will be f 1 p 1 N ¼ f 1 np(1 À p) nÀ1 N, which must equal nT to image each probe the desired number of times. Minimizing N with respect to p gives p min time ¼ 1=n:
Setting p > 1/n increases the acquisition time while also increasing the probability of multiple activated molecules per bright spot, so the minimum value of N is N min ¼ nT/f 1 (1 À 1/n) nÀ1 . In the limit n » 1 (valid for high resolution), (1 À 1/n) n z e À1 (wher
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