Resolution limit of image analysis algorithms.

The resolution of an imaging system is a key property that, despite many advances in optical imaging methods, remains difficult to define and apply. Rayleigh's and Abbe's resolution criteria were developed for observations with the human eye. However, modern imaging data is typically acquired on highly sensitive cameras and often requires complex image processing algorithms to analyze. Currently, no approaches are available for evaluating the resolving capability of such image processing algorithms that are now central to the analysis of imaging data, particularly location-based imaging data. Using methods of spatial statistics, we develop a novel algorithmic resolution limit to evaluate the resolving capabilities of location-based image processing algorithms. We show how insufficient algorithmic resolution can impact the outcome of location-based image analysis and present an approach to account for algorithmic resolution in the analysis of spatial location patterns.

An information-theoretic approach to designing the plane spacing for multifocal plane microscopy.

Multifocal plane microscopy (MUM) is a 3D imaging modality which enables the localization and tracking of single molecules at high spatial and temporal resolution by simultaneously imaging distinct focal planes within the sample. MUM overcomes the depth discrimination problem of conventional microscopy and allows high accuracy localization of a single molecule in 3D along the z-axis. An important question in the design of MUM experiments concerns the appropriate number of focal planes and their spacings to achieve the best possible 3D localization accuracy along the z-axis. Ideally, it is desired to obtain a 3D localization accuracy that is uniform over a large depth and has small numerical values, which guarantee that the single molecule is continuously detectable. Here, we address this concern by developing a plane spacing design strategy based on the Fisher information. In particular, we analyze the Fisher information matrix for the 3D localization problem along the z-axis and propose spacing scenarios termed the strong coupling and the weak coupling spacings, which provide appropriate 3D localization accuracies. Using these spacing scenarios, we investigate the detectability of the single molecule along the z-axis and study the effect of changing the number of focal planes on the 3D localization accuracy. We further review a software module we recently introduced, the MUMDesignTool, that helps to design the plane spacings for a MUM setup.

Designing the focal plane spacing for multifocal plane microscopy.

Multifocal plane microscopy (MUM) has made it possible to study subcellular dynamics in 3D at high temporal and spatial resolution by simultaneously imaging distinct planes within the specimen. MUM allows high accuracy localization of a point source along the z-axis since it overcomes the depth discrimination problem of conventional single plane microscopy. An important question in MUM experiments is how the number of focal planes and their spacings should be chosen to achieve the best possible localization accuracy along the z-axis. Here, we propose approaches based on the Fisher information matrix and report spacing scenarios called strong coupling and weak coupling which yield an appropriate 3D localization accuracy. We examine the effect of numerical aperture, magnification, photon count, emission wavelength and extraneous noise on the spacing scenarios. In addition, we investigate the effect of changing the number of focal planes on the 3D localization accuracy. We also introduce a new software package that provides a user-friendly framework to find appropriate plane spacings for a MUM setup. These developments should assist in optimizing MUM experiments.

Comparison of estimation algorithms in single-molecule localization.

Different techniques have been advocated for estimating single molecule locations from microscopy images. The question arises as to which technique produces the most accurate results. Various factors, e.g. the stochastic nature of the photon emission/detection process, extraneous additive noise, pixelation, etc., result in the estimated single molecule location deviating from its true location. Here, we review the results presented by [Abraham et. al, Optics Express, 2009, 23352-23373], where the performance of the maximum likelihood and nonlinear least squares estimators for estimating single molecule locations are compared. Our results show that on average both estimators recover the true single molecule location in all scenarios. Comparing the standard deviations of the estimates, we find that in the absence of noise and modeling inaccuracies, the maximum likelihood estimator is more accurate than the non-linear least squares estimator, and attains the best achievable accuracy for the sets of experimental and imaging conditions tested. In the presence of noise and modeling inaccuracies, the maximum likelihood estimator produces results with consistent accuracy across various model mismatches and misspecifications. At high noise levels, neither estimator has an accuracy advantage over the other. We also present new results regarding the performance of the maximum likelihood estimator with respect to the objective function used to fit data containing both additive Gaussian and Poisson noise. Comparisons were also carried out between two localization accuracy measures derived previously. User-friendly software packages were developed for single molecule location estimation (EstimationTool) and localization accuracy calculations (FandPLimitTool).

A resolution measure for three-dimensional microscopy.

A three-dimensional (3D) resolution measure for the conventional optical microscope is introduced which overcomes the drawbacks of the classical 3D (axial) resolution limit. Formulated within the context of a parameter estimation problem and based on the Cramer-Rao lower bound, this 3D resolution measure indicates the accuracy with which a given distance between two objects in 3D space can be determined from the acquired image. It predicts that, given enough photons from the objects of interest, arbitrarily small distances of separation can be estimated with prespecified accuracy. Using simulated images of point source pairs, we show that the maximum likelihood estimator is capable of attaining the accuracy predicted by the resolution measure. We also demonstrate how different factors, such as extraneous noise sources and the spatial orientation of the imaged object pair, can affect the accuracy with which a given distance of separation can be determined.

Quantitative study of single molecule location estimation techniques.

Estimating the location of single molecules from microscopy images is a key step in many quantitative single molecule data analysis techniques. Different algorithms have been advocated for the fitting of single molecule data, particularly the nonlinear least squares and maximum likelihood estimators. Comparisons were carried out to assess the performance of these two algorithms in different scenarios. Our results show that both estimators, on average, are able to recover the true location of the single molecule in all scenarios we examined. However, in the absence of modeling inaccuracies and low noise levels, the maximum likelihood estimator is more accurate than the nonlinear least squares estimator, as measured by the standard deviations of its estimates, and attains the best possible accuracy achievable for the sets of imaging and experimental conditions that were tested. Although neither algorithm is consistently superior to the other in the presence of modeling inaccuracies or misspecifications, the maximum likelihood algorithm emerges as a robust estimator producing results with consistent accuracy across various model mismatches and misspecifications. At high noise levels, relative to the signal from the point source, neither algorithm has a clear accuracy advantage over the other. Comparisons were also carried out for two localization accuracy measures derived previously. Software packages with user-friendly graphical interfaces developed for single molecule location estimation (EstimationTool) and limit of the localization accuracy calculations (FandPLimitTool) are also discussed.

Localizing single molecules in three dimensions - a brief review.

Single molecule tracking in three dimensions (3D) in a live cell environment holds the promise of revealing important new biological insights. However, conventional microscopy based imaging techniques are not well suited for fast 3D tracking of single molecules in cells. Previously we developed an imaging modality multifocal plane microscopy (MUM) to image fast intracellular dynamics in 3D in live cells. Recently, we have reported an algorithm, the MUM localization algorithm (MUMLA), for the 3D localization of point sources that are imaged using MUM. Here, we present a review of our results on MUM and MUMLA. We have validated MUMLA through simulated and experimental data and have shown that the 3D-position of quantum dots (QDs) can be determined with high spatial accuracy over a wide spatial range. We have calculated the Cramer-Rao lower bound for the problem of determining the 3D location of point sources from MUM and from conventional microscopes. Our analyses shows that MUM overcomes the poor depth discrimination of the conventional microscope, and thereby paves the way for high accuracy tracking of nanoparticles in a live cell environment. We have also shown that the performance of MUMLA comes consistently close to the Cramer-Rao lower bound.