Fig. 1. Concept of VISPR. After the single-molecule dataset (left) is acquired in SMLM experiments, a PSF library is obtained by segmentation. VISPR uses a starting pupil function with constant Zernike coefficients to obtain the reference vector PSF stacks. Then the reference PSF stacks assign each emitter pattern to different z-axis positions according to its similarity with the template. These axially assigned PSFs are subsequently grouped, aligned, and averaged to form a 3D PSF stack, which is then used to estimate the PSF model using MLE phase retrieval. The new pupil generates an updated reference vectorial PSF model for the next iteration. This process iterates about 6–8 times to converge and get the final PSF model.

Fig. 2. Performance quantification of VISPR on simulations. (a) Simulated single-molecule dataset located randomly over an axial range from −800 to +800  nm with a known PSF model. (b) Phase of the VISPR pupil (left), the ground-truth pupil (middle), and the residual error (right). The RMSE is 11.33 nm. (c) x−y and x−z views of the ground-truth 3D PSF (top row), VISPR-retrieved 3D PSF (middle row), and INSPR-retrieved PSF (bottom raw). In the −400 to 400 nm region, VISPR and INSPR are very similar to the real PSF. However, in the region beyond the z-axis 400 nm, INSPR shows a severe deviation from the real PSF due to the lack of high-frequency information, while VISPR still maintains high similarity with the real PSF. (d) Localization precision, accuracy, and mean value in the z direction at different axial positions with real PSF, VISPR, and INSPR. Scale bars, 100 nm in (c).

Fig. 3. Reconstruction results on simulated microtubule. (a) x−y overviews of the simulated microtubules resolved by VISPR from the 3D SMLM data. (b), (c) x−z overviews of the simulated microtubules resolved by VISPR and INSPR PSF. (b1), (b2) Enlarged x−z views of the areas indicated by the blue and orange boxed regions in (b), respectively. (d1), (d2) Intensity profiles along the z direction within the white lines in (b1), (b2), comparing the VISPR resolved profiles (blue solid lines) with the ground truth (red dashed lines). (c1), (c2) Enlarged x−z views of the areas indicated by the blue and orange boxed regions in (c), respectively. (e1), (e2) Intensity profiles along the z direction within the white lines in (c1), (c2), comparing the INSPR resolved profiles (blue solid lines) with the ground truth (red dashed lines). Scale bars, 3 μm in (a)–(c), and 0.5 μm in (b1), (b2), (c1), (c2).

Fig. 4. Experimental beads validation of VISPR on oblique astigmatism-dominant aberration. (a) The original data of beads, the PSF obtained by PR, the PSF obtained by VISPR, and the PSF obtained by INSPR are presented at different z-axis depths. Scale bar: 100 nm. (b) The localizing results of the verification beads. For each depth from −600 to 600 nm with a 50 nm increment, 50 consecutive bead images are captured and reconstructed by PR, VISPR, and INSPR. Both PR and VISPR localization results exhibit a step-like upward trend, reflecting the movement of the piezo positioner. Linear fitting lines are applied to the positioning results, with all points falling within the 95% prediction area. (c) Localization precision, accuracy, and mean value in x,y,z-positions at different axial positions with PR, VISPR, and INSPR. This comprehensive analysis demonstrates the effectiveness and reliability of VISPR.

Fig. 5. 3D super-resolution reconstruction of immunofluorescence-labeled Nup98 on the nuclear envelope in U2OS cells using VISPR PSF, INSPR PSF, and PR PSF. (a), (d) x−y overview of Nup98 on the bottom and top surfaces of the nucleus. (b), (e) The PSFs of the bottom and top surfaces obtained by VISPR, INSPR, and PR are presented at different z-axis depths. (c), (f) x−z cross-section of the selected region in (a), (d). (g), (l) Intensity profiles along the white dashed lines in (c) and (f), which demonstrate the higher quality of VISPR PSF. Scale bars, 5 μm in (a), (d); 1 μm in (c), (f); 100 nm in (b), (e).

Fig. 6. Supplement to Fig. 2. (a) Coefficients of 21 Zernike modes retrieved by VISPR compared with the ground truth (red circles). The RMSE is 10 nm. (b) Localization precision, accuracy, and mean value in x, y directions at different axial positions with real PSF, VISPR, and INSPR.

Fig. 7. Performance of VISPR estimated from simulated single-molecule blinking data. (a) Comparison of the ground-truth PSF and VISPR estimated PSF from single molecules located in the axial range from −600 to +600  nm. The simulated data are randomly distributed from −600 to 600 nm on the z-axis with a total of 4000 photons and a background level of 100. Scale bar, 100 nm. (b) Comparison of the ground-truth coefficients of the 21 Zernike modes (blue line) and the amplitudes of the 21 Zernike modes retrieved from VISPR (orange line). (c) Localization precision of 3D positions. 1000 repeated fitting calculations were performed to determine the localization precision. CRLB is the Cramér-Rao lower bound.

Fig. 8. Performance of VISPR under varying signal-to-background ratios. (a) Display of the ground-truth PSF in different photon (I) and background (bg) conditions. In each condition, the intensity profile along the white dashed lines at the 0 nm position PSF is analyzed to calculate the ratio of the peak signal to background. Notably, under the condition of 200 photons and 100 background, the peak signal-to-background ratio (SBR) plummets to as low as 2:1. Scale bar, 100 nm. (b) Comparison of the ground-truth pupil and VISPR estimated pupil from single molecules located in the axial range from −600 to +600  nm under diverse signal-to-background ratios. (c) Root mean square error (RMSE) between the VISPR estimated pupil and the ground-truth pupil in different photon (I) and background (bg) conditions. In each condition, the ground-truth pupil is randomly sampled from different combinations of 21 Zernike amplitudes for each trial, amounting to 11 trials in total. This comprehensive analysis sheds light on the robustness and accuracy of VISPR under conditions reflective of real-world variations in signal-to-background ratios.

Fig. 9. Experimental validation of VISPR on vertical astigmatism-dominant aberration. (a) The original data of beads, the PSF obtained by PR, the PSF obtained by VISPR, and the PSF obtained by INSPR are presented at different z-axis depths. (b) Partial FOV showing the sample with sparse single beads. The localization precision, accuracy, and mean value of the beads pointed by the yellow arrows are measured. (c) A sketch of the sample used to measure localization precision at different depths. The 100 nm particles are on the upper surface of the coverslip and moved by a piezo stage to different axial depths. (d) Localization precision, accuracy, and mean value curve across the depth of field of the two candidate molecules. Scale bars: 2 μm for (b) and 100 nm for others.

Fig. 10. 3D super-resolution reconstruction of immunofluorescence-labeled Nup98 on the nuclear envelope in U2OS cells using VISPR. (a), (d) x−y overview of Nup98 on the bottom and top surfaces of the nucleus. (b), (e) Subregions, as indicated by the white boxed regions in (a) and (d), showing the hollow structure of Nup98, with distinctive ring-like structures. (c), (f) Intensity profile along the white dashed line in (b) and (e), featuring subtle invaginations and undulations. Scale bars: 3 μm for (a) and (d), 0.5 μm for (b) and (e).