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. 2016 Jul 1;33(7):B36-57.
doi: 10.1364/JOSAA.33.000B36.

Fisher Information Theory for Parameter Estimation in Single Molecule Microscopy: Tutorial

Free PMC article

Fisher Information Theory for Parameter Estimation in Single Molecule Microscopy: Tutorial

Jerry Chao et al. J Opt Soc Am A Opt Image Sci Vis. .
Free PMC article

Abstract

Estimation of a parameter of interest from image data represents a task that is commonly carried out in single molecule microscopy data analysis. The determination of the positional coordinates of a molecule from its image, for example, forms the basis of standard applications such as single molecule tracking and localization-based super-resolution image reconstruction. Assuming that the estimator used recovers, on average, the true value of the parameter, its accuracy, or standard deviation, is then at best equal to the square root of the Cramér-Rao lower bound. The Cramér-Rao lower bound can therefore be used as a benchmark in the evaluation of the accuracy of an estimator. Additionally, as its value can be computed and assessed for different experimental settings, it is useful as an experimental design tool. This tutorial demonstrates a mathematical framework that has been specifically developed to calculate the Cramér-Rao lower bound for estimation problems in single molecule microscopy and, more broadly, fluorescence microscopy. The material includes a presentation of the photon detection process that underlies all image data, various image data models that describe images acquired with different detector types, and Fisher information expressions that are necessary for the calculation of the lower bound. Throughout the tutorial, examples involving concrete estimation problems are used to illustrate the effects of various factors on the accuracy of parameter estimation and, more generally, to demonstrate the flexibility of the mathematical framework.

Figures

Fig. 1
Fig. 1
Fundamental data model example. (a) The photon distribution profile fθ for an in-focus and stationary molecule whose image is modeled with the Airy pattern. This profile assumes that the molecule is located at (x0, y0) = (0 nm, 0 nm) in the object space, and that photons of wavelength λ = 520 nm are collected using an objective lens with a numerical aperture of na = 1.4 and a magnification of M = 100. While the profile is defined over the detector plane ℝ2, the plot shows its values over the 150 μm × 150 μm region over which it is centered. (b) A fundamental image simulated according to the profile in (a), with each dot representing the location at which a photon is detected. The simulation assumes a constant photon detection rate of Λ0 = 10000 photons/s and an acquisition time of tt0 = 0.05 s, such that the mean photon count in an image is Λ0 · (tt0) = 500. In this particular realization, N0 = 516 photons are detected, of which 492 fall within the 150 μm × 150 μm region shown, over which the image is centered.
Fig. 2
Fig. 2
Poisson data model example. (a) The mean pixelated image of the in-focus and stationary molecule of Fig. 1, computed over an 11×11-pixel region with a pixel size of 13 μm × 13 μm and a mean background photoelectron count of βθ,0 = 10 per pixel (obtained as the product of the constant background photon detection rate Λ0b=24200photons/s and the uniform spatial distribution fθb(x,y)=1/20449μm-2, integrated over each pixel and the acquisition time of tt0 = 0.05 s). The molecule is located at (x0, y0) = (656.5 nm, 682.5 nm) in the object space, which corresponds to (5.05 pixels, 5.25 pixels) in the image space, assuming (0, 0) in both spaces coincides with the upper left corner of the 11×11-pixel region. In terms of the photoelectron count due to the molecule, the mean image contains 476.15 out of the total of Λ0 · (tt0) = 500 mean number of photoelectrons distributed over the detector plane. All other relevant parameters are as specified in Fig. 1. (b) A Poisson realization of the mean image in (a). The photoelectron count in each pixel is drawn from the Poisson distribution with mean given by the photoelectron count in the corresponding pixel in (a).
Fig. 3
Fig. 3
CCD data model and EMCCD data model examples. (a) A CCD realization of the mean image in Fig. 2(a). The electron count in each pixel is the sum of a random number drawn from the Poisson distribution with mean given by the photoelectron count in the corresponding pixel in Fig. 2(a), and a random number drawn from the Gaussian distribution with mean η0 = 0 electrons and standard deviation σ0 = 8 electrons. In this particular realization, the Poisson random numbers are taken directly from the Poisson realization of Fig. 2(b). (b) An EMCCD realization of the mean image in Fig. 2(a). The electron count in each pixel is the sum of a random number drawn from the probability distribution of Eq. (14) with electron multiplication gain g = 950 and νθ,k given by the photoelectron count in the corresponding pixel in Fig. 2(a), and a random number drawn from the Gaussian distribution with mean η0 = 0 electrons and standard deviation σ0 = 24 electrons.
Fig. 4
Fig. 4
Limits of accuracy under the fundamental data model. (a) The limit of accuracy δx0 for the estimation of the x0 coordinate of an in-focus and stationary molecule from a fundamental image is plotted as a function of the expected number of photons detected in the image. Curves are shown that correspond to the detection of photons of wavelengths λ = 620 nm (dotted), 520 nm (solid), and 440 nm (dashed), which are collected by an objective lens with numerical aperture na = 1.4. The expected photon counts range from 100 to 10000, and are obtained as the product of the constant photon detection rate Λ0 = 10000 photons/s and acquisition times tt0 ranging from 0.01 s to 1 s. The image of the molecule is described by the Airy image function. (b) The limit of accuracy δΛ0 for the estimation of the photon detection rate Λ0 = 10000 photons/s from a fundamental image is shown as a function of the expected number of photons detected in the image.
Fig. 5
Fig. 5
Limit of the accuracy for separation distance estimation under the fundamental data model. The limit of accuracy δd for the estimation, from a fundamental image, of the distance d separating two in-focus and stationary molecules is plotted as a function of the distance d. Curves are shown that correspond to the detection of photons of wavelengths λ = 620 nm (◇), 520 nm (*), and 440 nm (○), which are collected by an objective lens with numerical aperture na = 1.4. Photons are detected from each molecule at a rate of Λ0 = 10000 photons/s over an acquisition time of 0.3 s, such that Λ0 · (tt0) = 3000 photons are on average detected from each molecule in a given image. The image of each molecule is described by the Airy image function. The inset shows the portion of the curves from d = 1 nm to d = 50 nm.
Fig. 6
Fig. 6
Comparing limits of accuracy corresponding to different data models and variations thereof. (a) The limit of accuracy δx0 for the estimation of the x0 coordinate of an in-focus and stationary molecule from an image is plotted as a function of the effective pixel size for the Poisson data model (*) and the Poisson data model assuming the absence of background noise (+). Also shown are the limits of accuracy corresponding to the finite detection area variation of the fundamental data model (dashed line), the finite detection area variation of the fundamental data model assuming the absence of background noise (dotted line), and the fundamental data model (solid line), which are plotted as horizontal lines as they do not depend on the effective pixel size. For the two cases involving the Poisson data model, the physical pixel size is 13 μm × 13 μm, and the different effective pixel sizes are obtained by varying the lateral magnification from M = 20 to M = 966.67. The image consists of a 15×15-pixel region at M = 100, and a proportionately scaled pixel region at each of the other magnifications. At each magnification, the molecule is positioned such that the center of its image, given by the Airy image function, is located at 0.05 pixels in the x direction and 0.25 pixels in the y direction with respect to the upper left corner of the center pixel of the image. For the two cases involving the finite detection area variation of the fundamental data model, δx0 is computed using the unpixelated equivalent of the M = 100 scenario, though any other scenario can be used to obtain the same result. For all cases, photons are detected from the molecule at a rate of Λ0 = 6000 photons/s over an acquisition time of tt0 = 0.05 s, such that Λ0 · (tt0) = 300 photons are on average detected from the molecule over the detector plane (and 289.5 photons are on average detected from the molecule over a given finite-sized image). For the two cases involving a background component, a background detection rate of Λ0b=22500photons/s is assumed, along with a uniform background spatial distribution of fb(x, y) = 1/A μm−2, (x, y) ∈ C, where C is the region in the detector plane corresponding to the image and A is the area of C. For example, for the M = 100 scenario, fb(x, y) = 1/38025 μm−2, such that there is a background level of β0 = 5 photoelectrons per pixel over the 15×15-pixel region. For all cases, it is assumed that photons of wavelength λ = 520 nm are collected by an objective lens with numerical aperture na = 1.4.
Fig. 7
Fig. 7
Comparing limits of accuracy corresponding to the CCD and EMCCD data models. (a) For the CCD data model, the limit of accuracy δx0 for the estimation of the x0 coordinate of an in-focus and stationary molecule from an image is plotted as a function of the effective pixel size. Limits of accuracy are shown for three different levels of readout noise in each pixel, characterized by standard deviations of σ0 = 6 electrons (●), 1 electron (◇), and 0.5 electrons (○). In each case, the mean of the readout noise in each pixel is η0 = 0 electrons. The limit of accuracy for the Poisson data model (with background noise) from Fig. 6 (*) is shown as a reference for comparison, since the CCD limits of accuracy are computed by adding readout noise to the same assumed conditions. All details not mentioned are therefore exactly as given in Fig. 6 for the Poisson data model. The inset provides a clearer view of some data points for five relatively large effective pixel sizes. (b) Zoomed-in version of (a) with the addition of δx0 corresponding to the EMCCD data model with readout noise of mean η0 = 0 electrons and standard deviation σ0 = 48 electrons in each pixel, and an electron multiplication gain of g = 1000.
Fig. 8
Fig. 8
Comparing noise coefficients corresponding to the CCD and EMCCD data models. The noise coefficient αk for the kth pixel of an image is plotted as a function of the mean photoelectron count νθ,k in the pixel. Noise coefficients are shown for two EMCCD detectors operated at an electron multiplication gain of g = 1000, with readout noise of standard deviations σ0 = 24 electrons (□) and 48 electrons (+) in each pixel, and for two CCD detectors, with readout noise of standard deviations σ0 = 0.5 electrons (○) and 1 electron (◇) in each pixel. The readout noise mean is η0 = 0 electrons in all cases.
Fig. 9
Fig. 9
Limits of the axial coordinate estimation accuracy for a conventional microscopy setup and a two-plane MUM setup. The limit of accuracy δz0 for the estimation of the z0 coordinate of an out-of-focus and stationary molecule from a conventional image (*) or a pair of two-plane MUM images (○) is plotted as a function of the coordinate z0. Focal planes 1 and 2 of the MUM setup are denoted by vertical dashed lines, and are located at z0 = 0 nm and z0 = 450 nm, respectively. The value of z0 is specified with respect to focal plane 1, which coincides with the focal plane of the conventional setup. In both setups, an image consists of an 11×11-pixel region, acquired using a CCD detector with a pixel size of 13 μm × 13 μm and readout noise with mean η0 = 0 electrons and standard deviation σ0 = 6 electrons at each pixel. The image of the molecule is described by the Born and Wolf image function. For the conventional setup, the molecule is laterally positioned such that the center of its image is located at 5.05 pixels in the x direction and 5.25 pixels in the y direction. Photons are detected from the molecule at a rate of Λ0 = 10000 photons/s over an acquisition time of tt0 = 0.1 s, such that Λ0 · (tt0) = 1000 photons are on average detected from the molecule over the detector plane (and 952.3 photons are on average detected from the molecule over a given image when the molecule is in focus). A background detection rate of Λ0b=24200photons/s is assumed, along with a uniform background spatial distribution of fb(x, y) = 1/20449 μm−2, such that there is a background level of β0 = 20 photoelectrons per pixel over the 11×11-pixel image. Photons of wavelength λ = 520 nm are collected by an objective lens with magnification M = 100 and numerical aperture na = 1.4. The refractive index of the immersion medium is n = 1.518. For the MUM setup, the collected molecule and background photons are split equally between the images corresponding to the two focal planes. Therefore, the photon detection rate is Λ0 = 5000 photons/s per image, and the background detection rate is Λ0b=12100photons/s per image. For the image corresponding to focal plane 1, all other details are as given for the conventional setup. For the image corresponding to focal plane 2, the magnification is M = 98.18, calculated using a formula from [76] for a plane spacing of Δzf = 450 nm and a tube length of 160 mm. The lateral position of the image of the molecule and the background spatial distribution are accordingly adjusted based on this slightly smaller magnification.

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