Fig. 1. Two-dimensional localization of a single molecule ^[21]. (a) In the acquisition step, sparsely distributed single molecule images are recorded; (b) in the analysis step, the two-dimensional coordinates of the single molecules are precisely localized in each frame and then accumulated to reconstruct the super-resolution image

Fig. 2. Changes of each PSF at different axial positions and optical path layout for SLM modulation. (a) The standard PSF; (b) astigmatism PSF; (c) double helix PSF^[27-28]; (d) phase ramp PSF^[29]; (e) spiral PSF^[30]; (f) self-bending PSF^[33]; (g) optical path layout for SLM modulation

Fig. 3. Tetrapod PSF optimized at different depths of field^[37]. (a) The pupil function, (b) theoretical PSF, (c) experimental PSF, (d) localizing accuracy of Tetrapod PSF optimized for 6 µm depth of field. (e)～(h) The same as (a)～(d), but for Tetrapod PSF optimized for 10 µm depth of field.

Fig. 4. Simultaneous measurement of emission wavelength and 3D position of single molecules^[48]. (a) Optical path design with an SLM placed on the back focal plane; (b) pupil function of curved grating; (c) PSF distributions of three different wavelengths at different localizations. The longer the wavelength, the farther the distance between the two side lobes

Fig. 5. Information extraction method based on artificial neural network ^[52]

Fig. 6. Localization deviation caused by the dipole’s direction^[53]. (a) PSF xz section (left) and xy section (right) of single molecule with a rotation angle, polar angle and azimuth angle of 15°, 45° and 0° respectively, and corresponding localization deviations; (b) PSF with the same polar angle and azimuth angle as (a) and rotation angle of 60°. (c) and (d) are the lateral offsets generated by different rotation angles and polar angles, respectively; (e) physical meaning of rotation angle, polar angle and azimuth angle of the dipole

Fig. 7. Dipole orientation localization method based on double helix PSF^[58] . (a) Optical path layout; (b) and (c) are imaging channels in two polarization directions, respectively, and their pupil functions are (i) and (ii) respectively; (d) the upper and lower figures are PSF of horizontal and vertical channels, respectively; (e) and (f) are LA and LD indicators respectively. There are four possible orientations when only the LA indicator is considered. Red and blue represent LD indexes of transmission channel and reflection channel respectively

Fig. 8. Simultaneous localization of the single-molecule orientation and three-dimensional location based on vortex PSF^[60] . (a) Vortex PSF’s optical path; CRLB of azimuth angle (b) and polar angle (c) in single molecule imaging with 4000 photons and 10 background photons; (d) 2D position and azimuthal angle of the λ -DNA. The false color represents the azimuthal angle, as shown in the lower left corner

Fig. 9. 4Pi-PSF decomposed by IAB model^[65]. (a) The optical path layout of 4-channel 4Pi-SMLM and the PSFs of each channel; (b) localization accuracy of each dimension obtained by photometric method and IAB model fitting for single molecule of 2,000 photons collected by each objective lens; (c) ideal 4Pi-PSF and decomposed IAB matrix

Fig. 10. Flowchart of single molecule localization based on convolutional neural network. (a) Network training steps; (b) localizing steps