Unified Resolution Bounds for Conventional and Stochastic Localization Fluorescence Microscopy

Eran A. Mukamel and Mark J. Schnitzer

Phys. Rev. Lett. 109, 168102 – Published 17 October 2012

Abstract

Superresolution microscopy enables imaging in the optical far field with $\sim 20 nm$ -scale resolution. However, classical concepts of resolution using point-spread and modulation-transfer functions fail to describe the physical limits of superresolution techniques based on stochastic localization of single emitters. Prior treatments of stochastic localization microscopy have defined how accurately a single emitter’s position can be determined, but these bounds are restricted to sparse emitters, do not describe conventional microscopy, and fail to provide unified concepts of resolution for all optical methods. Here we introduce a measure of resolution, the information transfer function (ITF), that gives physical limits for conventional and stochastic localization techniques. The ITF bounds the accuracy of image determination as a function of spatial frequency and for conventional microscopy is proportional to the square of the modulation-transfer function. We use the ITF to describe how emitter density and photon counts affect imaging performance across the continuum from conventional to superresolution microscopy, without assuming emitters are sparse. This unified physical description of resolution facilitates experimental choices and designs of image reconstruction algorithms.

Received 3 September 2010

DOI:https://doi.org/10.1103/PhysRevLett.109.168102

Authors & Affiliations

Eran A. Mukamel^1,3,* and Mark J. Schnitzer^2,4,†

¹Department of Physics, Stanford University, Stanford, California 94305, USA
²Departments of Applied Physics and Biology, Stanford University, Stanford, California 94305, USA
³Center for Brain Science, Harvard University, Cambridge, Massachusetts 02138, USA; Center for Theoretical Biological Physics, University of California, San Diego, California 92093, USA
⁴Howard Hughes Medical Institute, Stanford University, Stanford, California 94305, USA

^*eran@post.harvard.edu
^†mschnitz@stanford.edu

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 109, Iss. 16 — 19 October 2012

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Real-space analysis of localization accuracy depends on the configuration of emitters. (a) A scene with two emitters has 4 degrees of freedom (inset). The Cramér-Rao lower bound (CRLB) for estimation of each of the 4 coordinates (solid lines; red, $x_{0}$ ; cyan, $y_{0}$ ; green, $d_{x}$ ; blue, $d_{y}$ ) matches the variance of a maximum-likelihood estimator (□). (b)–(d) Real-space analysis for a scene containing 6 emitters. (b) Observed intensity at each pixel. (c) Fisher information matrix for the emitters’ 12 $X$ , $Y$ coordinates. (d) Localization accuracy, determined by simulating 100 independent estimates of the emitters’ locations, with $n = 17$ photons per emitter, chosen for illustrative purposes. Contours denote 2 s.d. from the true position of each emitter, as determined by $[J_{i j}]^{- 1}$ .Reuse & Permissions
Figure 2
The information transfer function, $F (k)$ , bounds image estimation accuracy. (a) $F_{0} (k)$ for conventional imaging ( $n = 1$ photon/emitter; blue and red curves) and superresolution ( $n = 100$ , green and cyan). (b) Mean squared localization error for a truncated Airy disk PSF (width parameter $σ$ , defined in 16) analyzed by maximum-likelihood fitting (red) approaches the Fisher limit for $n ≳ 100$ photons. The effects of parameters on image quality are shown in simulations of a synthetic image (c), and of microtubules in a mammalian cell (d). The top row shows the true sample, lower rows show estimates using (from the top) $ρ_{e} σ^{2} = {10, 10, 10^{3}, 10^{3}}$ and $n = {1, 100, 1, 100}$ . Scale bars are $1 μ m$ , $ρ_{e}$ is the total emitter density and $n$ is the number of photons per emitter. Simulations used an Airy disk PSF (width $σ = 200 nm$ ), and the true image in (d) is based on experimental data from 22.Reuse & Permissions
Figure 3
ITF for images of discrete emitters reveals the optimal density. (a) ITF for a fixed number of photons, $N$ , with an Airy disk PSF. The Fisher information about emitter location for an isolated emitter is $J^{(1)} = σ^{- 2}$ per photon, and $N = n A ρ_{e}$ is the total number of observed photons. The ITF declines quadratically for superresolution techniques with sparse labeling ( $ρ_{a} ≪ σ^{- 2}$ , dot-dashed black line). For conventional microscopy with dense labeling ( $ρ_{a} ≫ σ^{- 2}$ ) the ITF declines more sharply (solid black curve) and goes to zero at the diffraction limit (black dashed line). Colored curves show the median ITF taken over an ensemble of random simulated scenes with $ρ_{a} σ^{2} = 0.001$ (blue), 0.025 (green), 0.1 (red) and 0.2 (cyan). Error bars are $\pm 1 s . d .$ (b) ITF for a fixed acquisition time. (c) Contour plot showing the ITF as a function of emitter density and spatial frequency for fixed imaging time. Vertical colored lines correspond to curves in (a) and (b). Black horizontal line denotes the conventional diffraction limit. Gray line indicates the emitter density that maximizes the ITF at each spatial frequency. (d) The optimal emitter density as a function of $k$ . (e) The ITF per unit of image acquisition time, summed over all spatial frequencies greater than the conventional diffraction limit.Reuse & Permissions

Physical Review Letters