Fig. 1. Concept of upsampled PSF modeling. First, the z-stack data of beads are collected (top left), from which ROIs are extracted to construct an experimental PSF library at different positions (top center). Next, the initial global parameters (upsampled PSF represented by a 3D matrix) and local parameters (3D position, photon count, and background) are used to generate a fine PSF library. Then, the fine PSF library is pixel-integrated to create a coarse PSF library (top right). The loss function, based on MLE, is calculated between the coarse PSF library and the experimental PSF library (top center). Finally, through backpropagation (L-BFGS-B optimization algorithm), both global and local parameters are updated iteratively, ultimately yielding the final upsampled PSF. We also localize the experimental bead data with both the upsampled PSF (bottom center) and averaged coarse PSF, revealing that only the upsampled PSF could achieve unbiased and optimal localization results (bottom left).

Fig. 2. Comparison of the CRLB and RMSE for PSFs with different pixel sizes. (a)–(c) Comparison of the theoretical CRLB for x, y, and z for PSFs with different pixel sizes. (d)–(f) Comparison of the RMSE in the x, y, and z directions localized with PSFs of different pixel sizes. Rup represents the percentage degradation in CRLB or RMSE averaged over the range from −600 to 600 nm, relative to the values obtained with a pixel size of 110 nm. Each molecule contributes a total of 2000 photons, with a background level of 2×10−3  photons/nm2. Localization accuracy at each z position is computed from RMSE of the localization results from 1000 single molecules.

Fig. 3. Validation of the upsampled PSF modeling using simulated data from astigmatic PSF. (a) The 330 nm pixel size PSF (top) and 110 nm pixel size astigmatic PSF (bottom) estimated from 300 simulated data sets with a pixel size of 330 nm. (b)–(d) Comparison of the impact of the estimated 330 and 110 nm pixel size astigmatic PSF on the localization accuracy of the 330 nm pixel size simulated data along the x, y, and z directions, respectively. Each molecule contributes a total of 10,000 photons, with a background level of 2×10−3 photons/nm2. The x/y/z bias as a function of z is defined as in Appendix F Eqs. (F1) and (F2). Solid lines and shaded areas in localization plots indicate mean and standard deviation of bias, respectively.

Fig. 4. Nup96-AF647 in U2OS cells reconstructed using the upsampled PSF model. (a) Single-molecule data of Nup96-AF647 with a pixel size of 127 nm were 2×2 binned to produce undersampled data with a pixel size of 254 nm. Localization of the undersampled data using the upsampled PSF yielded a superresolved image. The red boxes indicate the single-molecule image with different pixel sizes. In the superresolution image, the white dashed box in the lower left corner corresponds to a magnified view of the region marked by the solid box. (b) XZ view of the selected region (white dashed lines) in (a) reconstructed from single-molecule data using PSFs of different pixel sizes. (c) Superresolution image of 321 nm pixel size single-molecule data of Nup96-AF647 analyzed with 321 and 107 nm pixel size PSFs, respectively. The XZ views of the selected region [white dashed lines in the left image of (c)] were compared.

Fig. 5. Comparison of the shape and intensity distribution of 330 nm pixelated PSFs at different positions. (a) PSFs with a pixel size of 330 nm at different x positions showing the shape of PSF is spatially variant. xc is the x position of the PSF, with 0 nm locating at the center of the middle pixel. (b) Relative intensity distribution at the white dashed line in (a) for PSFs at different x positions.

Fig. 6. Validation of spline fitting for different sampling rates. (a) To match the pixel size of the upsampled PSF with the simulated or experimental data, a convolution operation is performed on the estimated raw upsampled PSF (Raw) to generate the pixel integrated upsampled PSF (Convolved), using a convolution kernel size of bin×bin with all values set to 1 [Eq. (B15)]. The convolved PSF is used for localizing experimental data. The bin factor represents the ratio between the pixel size of the simulated or experimental data and the pixel size of the upsampled PSF, with its value constrained to integer values. (b) The spline fitter returns the z-position results by fitting the same simulated data using upsampled PSFs at different sampling rates. (c) Localization results of 110, 55, and 27.5 nm pixel size PSF for 110 nm pixel size single-molecule data. The simulated data consist of 51 axial positions evenly distributed along −600 to 600 nm with 1000 single molecules at each axial position. Each molecule contributes a total of 2000 photons, with a background level of 2×10−3 photons/nm2.

Fig. 7. CRLB and RMSE improved by using upsampled PSF model and novel spline PSF fitter. (a) Comparison of RMSE for 330 nm pixel data, with and without pixel integration [convolution in Fig. 6(a)]. RMSE_con shows results after convolution, while RMSE_raw shows results without convolution. Only with convolution (RMSE_con) does the localization achieve the theoretical CRLB limit. (b) Comparison of RMSE for localizing 330 nm data with 110 nm upsampled PSF and 330 nm PSF using our novel fitting algorithm. The results show that only the upsampled PSF achieves the theoretical CRLB limit. Simulation parameters match those in Fig. 6.

Fig. 8. Comparison of localization accuracy at different lateral positions. Localization accuracy of 110 nm pixel size PSFs on 330 nm single-molecule data at different lateral positions, (0 nm, 0 nm), (0 nm, 115 nm), (115 nm, 0 nm), and (115 nm, 115 nm) for (a), (b), (c), and (d), respectively. Here, the center of the central pixel is set as coordinate (0 nm, 0 nm). We used a vector PSF model (Appendix D) to generate 1000 simulated single molecules at different x and y positions at each axial position. Twenty-five axial positions uniformly distributed along −600 to 600 nm were evaluated. Each molecule contributes a total of 2000 photons, with a background level of 2×10−3 photons/nm2. Localization accuracy at each z position is computed from the RMSE of the localization results from 1000 single molecules.

Fig. 9. Validation of the upsampled PSF modeling using simulated data from tetrapod PSF. (a) The 330 nm pixel size PSF (top) and 110 nm pixel size PSF (bottom) estimated from 300 simulated tetrapod PSF data sets with a pixel size of 330 nm. (b)–(d) Comparison of the impact of the estimated 330 and 110 nm pixel size tetrapod PSF on the localization accuracy of the 330 nm pixel size simulated data along the x, y, and z directions, respectively. Each molecule contributes a total of 10,000 photons, with a background level of 2×10−3 photons/nm2. The x/y/z bias as a function of z is defined as in Appendix F, Eqs. (F1) and (F2). Solid lines and shaded areas in localization plots indicate mean and standard deviation of bias, respectively.

Fig. 10. Validation of the upsampled PSF modeling using experimental data from astigmatic PSF. (a) 321 nm pixel size experimental bead data (top) and the estimated 321 and 107 nm pixel size astigmatic PSF (bottom). (b)–(d) Comparison of the impact of the estimated 321 and 107 nm pixel size astigmatic PSF on the localization accuracy of the 321 nm pixel size experimental data along the x, y, and z directions, respectively. The x/y/z bias as a function of z is defined as in Appendix F, Eqs. (F1) and (F2). Solid lines and shaded areas in localization plots indicate mean and standard deviation of bias over 50 beads, respectively.

Fig. 11. Detailed layout of the optical setup. M, mirror; DM, dichroic mirror; L, lens; TS, translation stage; FC, fiber coupler; fiber, single-mode fiber; BFP, back focal plane; FW, filter wheel; TBL, tube lens; AP, aperture; QPD, quadrant photodiode. The excitation lasers are first reflected by dichroic mirror DM1 and coupled into a single-mode fiber through the fiber coupler FC. Before being reflected by the main dichroic mirror DM2 to enter the objective for sample illumination, the beam is collimated and reshaped by a pair of lenses (L1 and L2) and a slit at AP1. In the imaging path, the fluorescence collected by the objective passes through the dichroic mirror DM2 and is filtered by the filter wheel FW. It is then focused using a tube lens TBL. Subsequently, the fluorescence passes through a 4f system composed of lenses L3 (focal length=150  nm) and L4 (focal length=30  nm), before being detected by the camera. Additionally, a beam excited by a 785 nm laser, reflected off the coverslip, is detected by the quadrant photodiode (QPD), providing feedback control to the Z-stage for focus locking.

Fig. 12. Comparison of the impact of PSF Gaussian blurring on the theoretical CRLB. The σ corresponds to the σx and σy terms in Appendix D, Eq. (D8). When σ=0, Gaussian blurring is not applied. The pixel size of the data used is 330 nm. The simulation parameters are the same as those in Fig. 5.