Effect of unknown emitter intensities on localization accuracy in stochastic optical localization nanoscopy using single frames

Abstract

The Cramér–Rao lower bound (CRLB) indicates the inherent power of stochastic optical localization nanoscopy in spatially resolving emitters. The CRLB for a known and identical intensity of emitters was obtained previously. In practice, the intensities of emitters at times are unknown and must be estimated along with emitter localization. To study effect of unknown emitter intensities on localization accuracy, we analyze Fisher information and CRLB and obtain their formulas in three extended cases: emitter intensities are (i) known and arbitrary, (ii) known to be identical with an unknown value, and (iii) all unknown. The effect of unknown emitter intensities on CRLB is then numerically investigated in three scenarios: 2D emitters on a line with an Airy point spread function (PSF), 2D randomly distributed emitters with a Gaussian PSF, and 3D randomly distributed emitters with an astigmatic PSF. In all three scenarios it is shown that in comparison with the case of a known and identical emitter intensity, cases (i) and (ii) slightly increase CRLB; however, case (iii) significantly increases CRLB no matter whether the emitter intensities are identical or not. These results imply that, in practice, the emitter intensities, which are known a priori to have an identical value, incur little penalty on localization accuracy. In contrast, the emitter intensities, which are all unknown, significantly lower localization accuracy unless the emitter density is so low as to have little chance of overlapped PSFs. The analytical and numerical results can be extensively applied to set up an experiment, develop localization algorithms, and benchmark localization algorithms.