Fig. 1. The workflow of JSFR-NL-SIM can be divided into two parts: preparation procedure and computational routine. During the preparation procedure, optimization functions and coefficient functions are required to calculate in advance. In the computational routines, parameters estimation is first performed and then 25 raw images are filtered with the pre-calculated attenuation function. The intermediate images are obtained by summing the results of multiplying the coefficient functions with the filtered images. The final SR image is then reconstructed by filtering the optimization functions.

Fig. 2. Workflow of the parameter estimation of the proposed JSFR-NL-SIM. (a) Raw image stack. (b) Separated spectrum components. (c) Flowchart of the wave vector estimation using the proposed method. (d) Initial phase estimation. (e) SR image reconstructed by the full-process workflow.

Fig. 3. Simulation results to demonstrate the quality of JSFR-NL-SIM. (a) Ground truth. (b) Widefield image. (c) SR image reconstructed by JSFR-SIM. (d) SR image reconstructed by JSFR-NL-SIM. (e) SR image reconstructed by JSFR-NL-SIM with two additional higher-order harmonics (2HOH). (f) Zoom-in views of the blue boxes and yellow boxes in (b)–(e). (g) Spectra of (b)–(e). (h) Intensity profiles of green lines in (b)–(e). (i) Execution time of JSFR-based workflow and HiFi-based workflow in the same GPU environment; unit, millisecond. (j) Ideal spectrum.

Fig. 4. Performance analysis of JSFR-NL-SIM reconstructed on raw data with high out-of-focus signal and noise. (a) Widefield image of F-actin filaments. (b)–(d) SR images reconstructed using Wiener-NL-SIM provided by BioSR, HiFi-NL-SIM, and JSFR-NL-SIM. (e) Magnified images of (a)–(d), corresponding to the green box of (a), where out-of-focus signal and filament artifacts are produced by GT-SIM but eliminated in HiFi-NL-SIM and JSFR-NL-SIM. In (f) the yellow arrows indicate the sidelobe artifact while in (g), the red arrows represent the out-of-focus signal. The intensity profiles of both yellow lines are shown in (h). The scale bars are 5 μm in (a)–(d), 1 μm in (e), and 500 nm in (f) and (g).

Fig. 5. Discussion of reconstruction parameters on final SR image. (a)–(d) The effect of attenuation parameter, Wiener-type filter’s parameter, optimization filter1 parameter, and optimization filter2 parameter. The intensity of (a)–(d) is the white line along plane (e). (f)–(h) Intensity of the images filtered with Wiener-type filter, optimization filter1, and optimization filter2. The scale bars are 5 μm in (a)–(c) and 1 μm in (d).

Fig. 6. Reconstruction results from the BioSR dataset. (a) displays the reconstruction results of F-actin filaments, including widefield image and the SR images reconstructed by traditional Wiener-NL-SIM (termed GT-SIM, provided by BioSR dataset), HiFi-NL-SIM, and the proposed JSFR-NL-SIM. (b) provides zoomed-in views of (a), corresponding to the white box in (a). (c) Intensity profiles of the white lines in (b). (d), (e) Location resolution estimation and corresponding histogram of blue box in (a). (f) Decorrelation analysis to evaluate resolution. The scale bars are 5 μm in (a) and 1 μm in (b).

Fig. 7. Structure similarity index measure difference among three algorithms. (a)–(c) SR images reconstructed by HiFi-NL-SIM, JSFR-NL-SIM, and GT-SIM-a (Wiener-NL-SIM). (d) Enlarged images of different colored boxes from (a). The numbers at the upper-left corner in the insets of (b)–(d) indicate the structural similarity of the current image, compared with the original HiFi-NL-SIM image.

Fig. 8. Comparison of workflow of HiFi-NL-SIM and JSFR-NL-SIM. The preliminary preprocessing of both methods is similar. (a) The workflow of HiFi-NL-SIM and (b) the workflow of JSFR-NL-SIM.

Fig. 9. Comparison of the workflow of the Wiener-NL-SIM, HiFi-NL-SIM, and JSFR-NL-SIM. (a) Workflow of the Wiener-NL-SIM. (b) Workflow of the HiFi-NL-SIM. (c) Workflow of the JSFR-NL-SIM.

Fig. 10. Execution time of the JSFR-NL-SIM, Wiener-NL-SIM, and HiFi-NL-SIM in both CPU and GPU environments. (a) Execution time of the three algorithms in the CPU environment. (b) Execution time of the three algorithms in the GPU environment. (c) Exact execution time of the three algorithms.

Fig. 11. Wave vector errors of classical COR algorithm and DFT algorithm.