State space approach to single molecule localization in fluorescence microscopy

Vahid, Milad R.; Chao, Jerry; Kim, Dong‐Young; Ober, Raimund J.

doi:10.1364/boe.8.001332

Cited by 7 publications

(7 citation statements)

References 23 publications

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“…(2.4) and (2.5), respectively. We next in Corollary 5.5 to Theorem 5.2 calculate the Fisher information matrix for these special cases, and show that the obtained results are consistent with the results presented in [22,37,35,36].…”

Section: )supporting

confidence: 83%

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Vahid¹,

Hanzon²,

Ober³

2020

SIAM J. Imaging Sci.

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Tracking of objects in cellular environments has become a vital tool in molecular cell biology. A particularly important example is single molecule tracking which enables the study of the motion of a molecule in cellular environments by locating the molecule over time and provides quantitative information on the behavior of individual molecules in cellular environments, which were not available before through bulk studies. Here, we consider a dynamical system where the motion of an object is modeled by stochastic differential equations (SDEs), and measurements are the detected photons emitted by the moving fluorescently labeled object, which occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to uniform time points which have been commonly used in similar dynamical systems. The measurements are distributed according to optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., an Airy profile for an in-focus and a three dimensional profile, such as Born and Wolf, for an out-offocus molecule with respect to the detector. For some special circumstances, Gaussian image models have been proposed. In this paper, we introduce a stochastic framework in which we calculate the maximum likelihood estimates of the biophysical parameters of the molecular interactions, e.g., diffusion and drift coefficients. More importantly, we develop a general framework to calculate the Cramér-Rao lower bound (CRLB), given by the inverse of the Fisher information matrix, for the estimation of unknown parameters and use it as a benchmark in the evaluation of the standard deviation of the estimates. There exists no established method, even for Gaussian measurements, to systematically calculate the CRLB for the general motion model that we consider in this paper. We apply the developed methodology to simulated data of a molecule with linear trajectories and show that the standard deviation of the estimates matches well with the square root of the CRLB. We also show that equally sampled and Poisson distributed time points lead to significantly different Fisher information matrices.

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Section: )supporting

confidence: 83%

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Vahid¹,

Hanzon²,

Ober³

2020

SIAM J. Imaging Sci.

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“…We also calculated the standard deviations of the estimates. These standard deviations, which are computed as 57.4 nm and 59.6 nm for the x 0 -and y 0 -locations of the molecule, respectively, are close to the localization accuracy, i.e., the positive definite square root of the CRLB, which is given as 58.37 nm for both xand y-directions, reported in [18], [28], [29], [30].…”

Section: Example 53supporting

confidence: 64%

Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

Vahid

Hanzon

Ober

2021

IEEE Trans. Comput. Imaging

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The advent of single molecule microscopy has revolutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which was not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel, during an exposure time interval. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories, from pixelated images, a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Fisher information matrix for this parameter estimation problem. The analytical results are complicated through the fact that the observation process in a microscope prohibits the use of standard Kalman filter type approaches. The analytical framework presented here is illustrated with examples of low photon count scenarios for which we rely on Monte Carlo methods to compute the associated probability distributions.

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“…13 Here, we provide a similar presentation of the study, but apply the algorithm on different simulated and experimental data. In particular, the simulated image data sets considered here have a different background level of 30 photons per pixel, and we find the results to be similar to the results reported previously.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the simulated image data sets considered here have a different background level of 30 photons per pixel, and we find the results to be similar to the results reported previously. 13 …”

Section: Introductionmentioning

confidence: 99%

A state space based approach to localizing single molecules from multi-emitter images

Vahid

Chao

Ober

2017

Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIV

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Single molecule super-resolution microscopy is a powerful tool that enables imaging at sub-diffraction-limit resolution. In this technique, subsets of stochastically photoactivated fluorophores are imaged over a sequence of frames and accurately localized, and the estimated locations are used to construct a high-resolution image of the cellular structures labeled by the fluorophores. Available localization methods typically first determine the regions of the image that contain emitting fluorophores through a process referred to as detection. Then, the locations of the fluorophores are estimated accurately in an estimation step. We propose a novel localization method which combines the detection and estimation steps. The method models the given image as the frequency response of a multi-order system obtained with a balanced state space realization algorithm based on the singular value decomposition of a Hankel matrix, and determines the locations of intensity peaks in the image as the pole locations of the resulting system. The locations of the most significant peaks correspond to the locations of single molecules in the original image. Although the accuracy of the location estimates is reasonably good, we demonstrate that, by using the estimates as the initial conditions for a maximum likelihood estimator, refined estimates can be obtained that have a standard deviation close to the Cramér-Rao lower bound-based limit of accuracy. We validate our method using both simulated and experimental multi-emitter images.

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State space approach to single molecule localization in fluorescence microscopy

Cited by 7 publications

References 23 publications

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

A state space based approach to localizing single molecules from multi-emitter images

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