Localization uncertainty in nta measurements

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Tracking error is defined[@catipovic2013Improving the quantification of Brownian motion] as the uncertainty in time and position for each particle. The localization of the particle is limited by the number of photons, shot noise, etc.

It must be noted, also that brownian motion itself generates localization uncertainties[@ernst2013Measuring a diffusion coefficient by single-particle tracking: statistical analysis of experimental mean squared displacement curves]. Although longer exposure times allow the collection of more light, they induce a larger uncertainty if the particles can diffuse a lot during that time window. This is quadratically problematic for small particles, since they scatter less light AND move faster.

The reduced localization error can be expressed as:

$$x=\frac{\sigma^2}{D\Delta t}$$


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