= Solution
<PALM> separates emitters in time. A sparse subset of photoactivatable or photoconvertible labels is switched on so that its diffraction-broadened images can be fitted individually; repeated activation, imaging and localization builds a map from many isolated detections. The positional uncertainty can be much smaller than the <point spread function> width because many <photons> estimate one image centre: ideally $\sigma_{\rm loc}\sim s/\sqrt{N_\gamma}$ for image width $s$ and detected <photon> number $N_\gamma$.
That is <fluorescence localization precision>, not automatically the structural resolution of the reconstructed cell. Pixelation, background, imperfect optics, drift, label linkage, blinking and labelling density matter. Repeated localizations of one molecule cannot fill unlabelled gaps; a crowded frame with overlapping images cannot simply be treated as many isolated emitters. For a static structural map, the structure must remain sufficiently stable while the many frames are accumulated, or its time dependence must be reconstructed explicitly.
For mobility in living cells, keep activated <proteins> observable for successive frames and link isolated positions into trajectories before bleaching. Repeating this for successive sparse subsets yields many trajectories without making the entire <protein> population simultaneously fluorescent. This is the <sptPALM> strategy.<High-density mapping of single-molecule trajectories with photoactivated localization microscopy by Suliana Manley and others>[Combination of sparse photoactivation and live-cell single-particle tracking]{ref}
For free isotropic two-dimensional <diffusion> with per-coordinate localization error $\sigma_{\rm loc}$ and instantaneous exposures,
$$
\boxed{\langle|\mathbf r(t+\Delta t)-\mathbf r(t)|^2\rangle
=4D\Delta t+4\sigma_{\rm loc}^2.}
$$
Fit several lags or a full trajectory <likelihood> to infer D and its uncertainty. Finite exposure causes <motion blur>; for a uniform shutter of duration $t_E\le\Delta t$, a simple correction is $4D(\Delta t-t_E/3)+4\sigma_{\rm loc}^2$. Confinement, directed motion and switching binding states change the lag dependence. Short tracks, missed detections and false linking in crowded regions can mimic apparent heterogeneity, so mobility classes require explicit measurement-error and tracking controls. A trajectory map recovers dynamics; it is not identical to a static <PALM> density map.
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