Download We analyze singe emitter localization at high background levels

THE EFFECT OF BACKGROUND ON LOCALIZATION UNCERTAINTY IN
SINGLE EMITTER IMAGING
Bernd Rieger and Sjoerd Stallinga
Quantitative Imaging Group, Department of Imaging Science and Technology
Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands
Email: [email protected]
KEY WORDS: point spread function, maximum likelihood estimation, Cramer-Rao lower
bound, free dipole emitter
We analyze singe emitter localization at high background levels. Thompson, Larson and
Webb have published a widely quoted formula for the localization accuracy as a function of
signal and background photon count [1] for Least Mean Squares (LMS) fitting. In the
presence of shot noise LMS fitting is not
appropriated and should be replaced by Maximum
Likelihood Estimation (MLE). The latter estimation
closely follows the Cramer-Rao lower bound
(CRLB) for a wide range of signal photon counts and
up to very large background levels and can be
computed in real time using graphics cards [2]. The
use of an idealized Gaussian to fit measured spots
that originate from freely and rapidly rotating dipole
emitters appears to work well for all parameters
considered [3]. Mortensen and co-workers presented
an exact but not closed form expression for the
localization uncertainty [4] that applies to MLE and
thus corrects the widely used formula of [1]. We
propose the following analytical approximation for
the localization uncertainty for MLE fitting:
σ2 ⎛
2τ ⎞
(Δx) 2 = a ⎜⎜1 + 4τ +
⎟,
N ⎝
1 + 4τ ⎟⎠
τ = 2πσ a2b / ( Na 2 ), σ a2 = σ 2 + a 2 /12
Here a is the camera pixel size, N is the total signal
photon count, b the background photon count per
pixel and σ is the width of the Gaussian that is used
to fit the PSF. This relatively simple closed-form
expression approximates the minimum achievable
localization uncertainty excellently for all signal and background levels. We present
simulation results that prove the validity and usefulness of our formula for an idealized
Gaussian ground truth PSF model or a realistic free dipole ground truth PSF model [3].
[1] R.E. Thompson, D.R. Larson, and W.W. Webb, Biophysical Journal, 82:2775–2783,
2002.
[2] C.S. Smith, N. Joseph, B. Rieger, and K.A. Lidke, Nature Methods, 7(5):373–375, 2010.
[3] S. Stallinga, and B. Rieger, Optics Express, 18(24):24461-24476, 2010.
[4] K.I. Mortensen, L.S. Churchman, J.A. Spudich, and Flyvbjerg, Nature Methods, 7(5):377
–384, 2010.