Single-molecule localization microscopy (SMLM) [1–3] has emerged as a crucial tool in breaking the diffraction limit, enabling the exploration of subcellular structures, including the observation of proteins, lipids, DNA, and RNA at nanoscale. Despite advancements in achieving three-dimensional (3D) resolution [4–12] in SMLM, imaging beyond a few micrometers into the tissue remains challenging [13,14]. One prominent impediment lies in sample-induced aberrations distort and blur single-molecule emission patterns, known as point spread functions (PSFs), which are key to accurate localization of molecules, thereby introducing inaccuracies in the inference of 3D molecular positions. Several methods have been proposed for extending the imaging depth of single molecules. Techniques such as light-sheet illumination [15] and adaptive optics [14,16–18] have been employed to reduce background fluorescence and eliminate positioning errors and image artifacts caused by aberrations. At the same time, advanced sample-preparation techniques [19,20] have been proposed to minimize background and scattering in the fluorescence microscopy of tissues, thereby enhancing the penetration depth of probes during immunolabeling. Despite these advancements, the true PSF affected by the system and sample environment remains unknown and cannot be accurately obtained, causing localization inaccuracy and resolution degradation during deep region imaging. Therefore, accurately obtaining the PSF considering both system- and sample-induced aberrations has become the key for deep tissue SMLM. In addition, acquiring an accurate PSF is of significant importance for multiple applications, including the correction and analysis of aberrations introduced by the instrument and sample environment [14,21], quantification and enhancement of microscopy system performance [22,23], PSF engineering (e.g., double helix [9], phase ramp [24], tetrapod [25], or biplane configuration [7]), and enhancement of resolution in SMLM.

To obtain the PSF model, current approaches rely on calibrations generated from fiducial markers such as beads or gold nanoparticles [14,26–31], which take images of bead stacks moving at certain intervals in the axial direction and subsequently utilize phase retrieval (PR) algorithms for PSF retrieval. However, photons emitted by these fiducial markers never traverse the cellular or tissue sample, resulting in the ignorable aberrations induced by the sample environment, such as refraction mismatch caused by structural distribution unevenness and anisotropy of the sample. In addition, the bead size ranges from 40 to 200 nm, significantly exceeding the size of a fluorescent molecule (approximately 2 nm). Therefore, the PSF model retrieved by beads has limitations when localizing in SMLM, such as unsatisfactory uncertainty in localization precision and resulting localization deviations in 3D.

Recently, Xu et al. introduced INSPR to generate an in situ PSF model directly derived from raw data in 3D SMLM using a modified Gerchberg–Saxton (GS) algorithm [32], which takes into account both the system-induced and sample-induced aberrations and is well-suited to the SMLM experimental conditions. Subsequently, Liu et al. introduced uiPSF, a universal and robust toolbox designed to extract accurate PSF models for most SMLM imaging modalities, either from bead stacks or in situ from blinking emitters [33]. This approach leverages the automatic differentiation functionality of TensorFlow and uses inverse modeling to extract accurate PSF models which accounts for various realistic experimental conditions.

Drawing inspiration from Xu’s approach [32], we introduce vectorial in situ PSF retrieval (VISPR), founded on a vectorial PSF model and maximum likelihood estimation (MLE) phase retrieval algorithm. In contrast to the traditional scalar PSF model, the vectorial PSF model, which accounts for the vectorial nature of light, is better suited for high numerical aperture (NA) microscope systems [34]. Additionally, compared to the traditional GS algorithm, which is susceptible to background noise and neglects high-frequency information in the PSF spatial domain because of the incorporation of an empirical optical transfer function (OTF) scaling function in data processing [35], the MLE phase retrieval algorithm achieves more accurate fitting of the pupil and PSF. The MLE algorithm effectively considers high-frequency information within the PSF spatial domain and exhibits insensitivity to background noise, given its consideration of single-molecule patterns being Poisson-distributed. Consequently, VISPR achieves accurate in situ PSF fitting under SMLM conditions, due to the consideration of model deviation, acquisition of PSF high-frequency information, and insensitivity to background noise.

In this paper, we first introduce the theoretical foundations of VISPR, including the vector PSF model and MLE algorithm. Subsequently, we employed both simulations and practical experiments to validate the efficacy of VISPR. VISPR effectively fits the deformation of the pupil surface by taking into account the system- and sample-induced aberrations, and accurately retrieves the 3D PSF model, achieving the theoretical Cramér-Rao lower bound (CRLB) and faithfully reflecting three-dimensional information of single molecules. This enhancement led to improved accuracy in the reconstruction of 3D SMLM. Furthermore, our methodology exhibits robustness under low signal-to-noise ratio (SNR) conditions and proves adaptable to a diverse range of systems capable of SMLM experiments.

VISPR overcomes the limitations inherent in bead-based phase retrieval methods [14,26–31] by directly retrieving the 3D PSF model from the raw single-molecule blinking dataset obtained from 3D SMLM experiments. The fundamental concept of 3D-SMLM is illustrated in Fig. 1. It consists of three significant steps: (1) PSF library construction, wherein the SMLM dataset is segmented into a single-spot PSF library representing a random axial distribution of the 3D PSF to be retrieved; (2) PSF library assignment, involving the utilization of an ideal vector PSF (generated by a constant pupil and using a vectorial PSF forward model) as a reference PSF to categorize the PSF library into classes based on axial information, employing mathematical frameworks of expectation-maximization [36] and k-means [37] to calculate the similarity between images; and (3) 3D PSF model estimation, wherein temporarily axially assigned single-molecule spots are assembled, aligned, and averaged to form a 3D PSF stack. Subsequently, MLE phase retrieval is employed to estimate the new pupil and Zernike coefficients of the PSF stack. This updated pupil generates a refined vectorial reference template, and steps 2–3 are iteratively repeated. For data with a low SNR, 6–10 iterations are required to converge. To avoid the PSF degeneracies that occurred during the process mentioned in Ref. [32], we use prior knowledge of the astigmatism orientation to generate an asymmetric reference PSF, which helps us to build the unique in situ vectorial PSF model.

For optical systems with a low NA, the intensity profile in the focal plane and the complex signal in the pupil plane are related via Fourier transform, as described by the Fresnel approximation [34]. A necessary condition for the validity of Fresnel approximation is that the effect of light polarization on the diffracted images is negligible. The scalar PSF model can be related to the wavefront aberration ϕ(x) as PSFs∝|F2D{A(ρ,φ)·ej(ϕpos(ρ,φ,x0,y0,z0)+ϕab(ρ,φ,x0,y0,z0))}|2,(1)where F2D is the 2D Fourier transform, and (ρ,φ)∈R2 is the normalized polar coordinate in the back focal plane (BFP). (x0,y0,z0) is the 3D position of the single molecule in the sample plane. PSFs is the intensity of the optical field in the focal plane. A(ρ,φ) is the amplitude of the pupil plane. ϕpos(ρ,φ,x0,y0,z0) indicates the phase shift correlated with the emitter position and objective position, which is defined as ϕpos(x)=2πλ(x0kx+y0ky+z0kz),(2)with kz=1−kx2−ky2,(3)where λ is the wavelength, and (kx,ky) are the x and y components of the unit wave vector. ϕab(x) represents field-dependent aberrations and is expressed as a linear sum of variance normalized Zernike polynomials: ϕab(x)=∑n+|m|≤8Cnm(x)·Znm(u),(4)where x=(x0,y0,z0), u=(ρ,φ), and Cnm are the aberration coefficients; Znm is the normalized Zernike polynomial; n is the radial order and m is the angular frequency.

The scalar PSF model only focuses on the Fourier transform relationship between the pupil and PSF according to the Fresnel approximation while ignoring the vector nature of light, which is feasible in low-NA situations. However, for high-NA optical systems [38] (e.g., NA≥0.6), the vector nature of light cannot be neglected because the bending of the rays created by a lens introduces a significant z component of the electromagnetic field in the region behind the lens. The PSF can be modeled according to the vector theory of diffraction [39] by considering the x,y,z components of the field right after the lens separately for the polarization vector with x and y components (for a collimated beam, the z component is approximately zero). Each volume element of the sample represents an ensemble of independently emitting fluorophore molecules, which we describe as uncorrelated dipole radiators (with random orientations) that can rotate and wobble freely. The PSF is treated as summed image of three orthogonal dipoles: x,y,z directions. Thus, our calculation of the vectorial PSF simply consists of calculating and adding six intensity PSFs incoherently, which are modulated by six field vectorial coefficients: PSFv∝∑p=x,y∑d=x,y,z|F2D{A′(ρ,φ)·ej(ϕpos(ρ,φ,x0,y0,z0)+ϕab(ρ,φ,x0,y0,z0))·Ep,d}|2,(5)with A′(ρ,φ)=A(ρ,φ)/cosθ,(6)where A′(ρ,φ) indicates the amplitude function with the obliquity factor to account for the angle θ between the Poynting vector and the normal to the imaging plane. The six coefficients Ep,d represent the polarization vectors of the electrical field with components p=x,y in the image plane with orthogonal dipole components d=x,y,z contributing to components p: Ex,d=TpPd→cosφ−TsSd→sinφ,(7a)Ey,d=TpPd→sinφ+TsSd→cosφ,(7b)with P→=[cosθmedcosφcosθmedsinφ−sinθmed],S→=[−sinφcosφ0].(7c)Here, P→ and S→ are light-field polarization vectors. Tp and Ts are the Fresnel transmission coefficients for the p and s polarizations, respectively. The six field components can be used to calculate the total electrical field energy at any point after the lens. In particular, they determine the intensity collected by an imaging plane in any orientation.

MLE is an effective method for fitting Poisson distribution models. We know that the measurement noise of the camera among pixels is independent and identically distributed with a Poisson function. Therefore, the PSF intensity profile on the camera is also Poisson distributed, which can be accurately fitted by MLE especially for low signal-to-noise conditions.

The proper procedure for MLE phase retrieval is to adjust the fitting parameters to minimize the MLE for the Poisson distribution. We used the Levenberg–Marquardt (L-M) algorithm [29] to minimize the maximum-likelihood-based loss function given by χMLE2=2(∑k(fk−xk)−∑k,xk>0xkln(fk/xk)),(8)where fk is the expected number of photons in the pixel k from the model PSF function with parameters, and xk is the measured number of photons in the raw data. This function is minimized to determine the maximum likelihood of the Poisson process.

We first validated the feasibility of our VISPR method extensively with single-molecule simulations. We primarily focus on the performance comparison between VISPR and INSPR under in situ biological sample conditions, where both sample-induced and system-induced aberrations are considered. PR, being an in vitro PSF retrieval method that relies on calibrations generated from fiducial markers such as beads or gold nanoparticles, does not account for sample-induced aberrations and therefore was not included in our simulation comparisons. Specifically, we generated a randomly blinking dataset of single-molecules [Fig. 2(a)] based on the cubic-spline representation of a ground-truth oblique astigmatic PSF model with a set of Zernike aberrations consisting of 21 Zernike modes (Wyant order) [40] and a random axial range of ±800nm (4000 total photons/localization, 100 background photons per pixel, z-range from −600 to 600 nm). Each molecule was added to a constant background and degraded by Poisson noise. In the simulated experiments, VISPR demonstrated its proficiency by successfully estimating the ground-truth pupil [Fig. 2(b)] with a residual error of 11.3 nm [root mean square (RMS) error of the wavefront] and accurately determining the Zernike amplitude coefficients with an error of 10 nm for all 21 modes [Fig. 6(a)]. As shown in Fig. 7, the VISPR PSF is close to the CRLB under different aberration conditions.

Next, we conducted a comparative analysis of VISPR and INSPR [32]. As shown in Fig. 2(c), within the −400 to +400nm range, both INSPR- and VISPR-retrieved PSFs demonstrated significant similarity with the ground-truth PSF. As the depth along the z-axis extended beyond ±400nm, the VISPR-retrieved PSF maintained a notably high degree of similarity to the real PSF by considering the high-frequency information of the PSF. However, the INSPR PSF model introduces noticeable distortions with the ground-truth PSF, because of the inappropriate PSF model and failure to consider the high-frequency information of the data.

Moreover, we conducted a three-dimensional reconstruction of the simulated single-molecule datasets using separate ground-truth 3D PSF, INSPR PSF, and VISPR PSF models. The evaluation encompasses localization precision, accuracy, and mean values in x,y,z at various axial positions, utilizing 100 simulated molecules at each axial position (ranging from −800 to 800 nm at 20 nm intervals) to comprehensively illustrate the performance of VISPR and INSPR. The localization precision is calculated as the standard deviation of the fitted positions. The localization accuracy is calculated as the root mean squared error between the fitted and ground-truth positions. Figure 2(d) shows that the evaluated precision, accuracy, and mean value in the z direction, localized by the VISPR-retrieved 3D PSF model, exhibited excellent similarity with the ground-truth PSF, even in the large z-axis. In contrast, substantial deviations were observed between the results located by INSPR and the ground truth. Notably, the mean z-position of the INSPR PSF within the −400 to 400 nm range closely aligns with the ground-truth PSF; however, a considerable deviation becomes evident beyond the 400 nm range. Additionally, the localization results in the x, y directions also demonstrate that VISPR exhibits a stronger ability to accurately fit the ground-truth PSF [Fig. 6(b)].

Subsequently, we further verified the performance of VISPR under conditions of low signal-to-background ratios, considering the conditions of conventional single-molecule experiments characterized by pronounced background noise. We tested the RMSE between the VISPR fitted pupil and the ground-truth pupil under varying signal-to-background ratio conditions [different photon numbers (I) and backgrounds (bg) conditions]. As shown in Fig. 8, even when confronted with a peak signal-to-background ratio (SBR) as low as 2:1 (2000 total photons, 100 background photons per pixel, z-range −600 to 600 nm), with a molecular number of 1000, the RMSE of the estimated pupil was also approximately 33 nm. Furthermore, with an increase in the number of molecules, the estimated pupil achieves a higher accuracy, reaching approximately 20 nm. This analysis sheds light on the robustness and accuracy of VISPR under conditions reflective of real-world variations in signal-to-background ratios.

Furthermore, we used VISPR and INSPR to reconstruct the simulated microtubule structure [Fig. 3(a)], respectively. The ground truth of the microtubule is obtained from the EPFL 2016 SMLM Challenge training dataset [41]. Initially, a ground-truth oblique astigmatic PSF model was employed to generate a simulated single-molecule dataset. Next, the single-molecule dataset of microtubules was input into VISPR and INSPR, and the 3D PSF model retrieved separately was employed to reconstruct the microtubule super-resolution image [Figs. 3(b) and 3(c)]. The results reveal that within the −300 to 300 nm region of the z-axis, the x–z reconstruction outcomes of VISPR [Figs. 3(b1) and 3(d1)] and INSPR [Figs. 3(c1) and 3(e1)] are closely aligned with the ground-truth values. However, as the depth extended beyond 500 nm, the z localization of INSPR deviated significantly from the ground truth by approximately 200 nm [Figs. 3(c2) and 3(e2)]. In contrast, the results obtained with VISPR exhibited a good fit, closely aligning with the ground truth [Figs. 3(b2) and 3(d2)].

To evaluate the performance of VISPR, experiments on 100 nm fluorescent nanospheres, which were settled on the upper surface of the coverslip, were conducted as a proof of concept. We added an oblique astigmatism-dominant aberration with other random but non-primary Zernike coefficients to the imaging system by deformable mirror (DM). The validation experiment was divided into three parts. (1) We staged and captured the beads from −600 to 600 nm with a 50 nm step. The average of the stacks of bead candidates with high photon intensity within the field of view (FOV) was utilized for in vitro PSF/pupil retrieval using the vectorial model and MLE fitting. This PSF (PR PSF) and pupil were regarded as the ground truth of the system. (2) We staged and captured the beads from −600 to 600 nm with a random step interval, i.e., the step sizes range from 2 to 7 nm randomly, to mimic in situ SMLM raw data [Fig. 9(c)]. VISPR and INSPR methods were applied to get the in situ PSF model and pupil function, respectively. It can be seen that in Fig. 4(a), both PR and VISPR can accurately retrieve the shape of the beads. In contrast, the shape of the INSPR PSF model at large depths (>300nm) deviates significantly from the PR PSF and real beads, because INSPR cannot retrieve the high-frequency information of the beads image. (3) We captured 50 repeated bead images for each depth from −600 to 600 nm with a 50 nm step. For each bead candidate in the FOV, PR PSF, VISPR PSF, and INSPR PSF models were utilized to fit the 3D coordinates at each depth, and the average values of x/y/z coordinates of the PR PSF result at each depth were regarded as the ground-truth coordinates for this specific bead candidate. Figure 4(b) shows that both PR and VISPR localization results exhibit a step-like upward trend, indicating the movement of the piezo positioner. Then, the localization precision (Δx/Δy/Δz) and corresponding accuracy (mean square error relative to the ground-truth coordinates) of PR/VISPR/INSPR fitting at each depth were calculated. For the VISPR/INSPR localization results, we added a global 3D displacement to the original fitting coordinates to compensate for the offset between the PR PSF and VISPR/INSPR PSF models. This is reasonable because the translation of the 3D reconstruction does not affect the final structural information. In Fig. 4(c), we present a comparative analysis of the x,y,z axis localization precision and accuracy for beads using PR/VISPR/INSPR, respectively. Due to the sample stage’s random jitter, the z-position of the beads in the same z-axis area will also deviate, which will cause fluctuations in precision and accuracy. However, by comparing precision, accuracy, and mean value at different z-axis positions, we still find that the average positions of VISPR and PR on the z-axis positioning can correctly reflect the movement of the stage. In contrast, the position of INSPR produces obvious deviations from the PR PSF, which illustrates the limitations of INSPR.

To further explore the robustness of VISPR, we conducted bead experiments using other types of aberrations, i.e., vertical astigmatism-dominant aberration with other random but non-primary Zernike coefficients, as shown in Fig. 9(a). The PSF estimated by VISPR demonstrated a clear linear relationship between the fitted z-position and the actual objective z-position. In contrast, INSPR exhibited distortion, indicating a potential misalignment of the PSF model with the actual data.

VISPR enables the measurement and analysis of both system- and sample-induced aberrations, facilitating the extraction of the authentic 3D PSF model in SMLM. This capability enables the precise three-dimensional localization of single molecules. To further validate the robustness and performance of VISPR, we conducted experiments on imaging Nup98-AF647 on the nuclear envelope in U2OS cells. Since the aberrations in the nuclear pore sample are relatively mild compared to deep tissues, we artificially introduced the randomly unknown high-order aberrations in addition to the aberrations of the nuclear pore sample itself to the DM, to mimic the deep tissue imaging situation.

First, we reconstructed a high-resolution 3D volume of Nup98 at the bottom surface of the nucleus, ∼3μm above the coverslip [Fig. 5(a)]. We used VISPR PSF, INSPR PSF, and PR PSF to reconstruct the SMLM raw data obtained from the experiments. Our observations revealed distinctive ring-like structures that span the lower surface of the nuclear envelope, featuring subtle invaginations and undulations (Fig. 10). Notably, as shown in Fig. 5(c), due to the proximity of the nuclear pore complex (NPC) to the coverslip, the reconstructed distributions within the region of approximately −200 to 200 nm at the bottom of the concave structure along the axial direction were relatively similar when comparing VISPR, INSPR, and in vitro bead PR methods. However, when the depth extended beyond 400 nm, both INSPR and PR exhibited inaccurate, distorted, and diffuse reconstruction at the edge of the concave structure, as observed in the intensity profile [Figs. 5(g)–5(i)]. It is clear that the NPC reconstructed by INSPR dispersed completely at depths greater than 400 nm, leading to reconstruction failure. In contrast, the reconstruction of VISPR exhibited consistent and converging axial distributions at the edges of the concave structure.

Subsequently, we proceeded to image Nup98 at the top region of the nucleus, ∼5μm above the coverslip [Fig. 5(d)]. This specific imaging condition, employing an oil immersion objective lens (NA=1.45) and a standard imaging buffer (n=1.346), inherently introduced substantial sample-induced aberrations. We employed VISPR, INSPR, and the in vitro PR methods based on fluorescent beads affixed to the coverslip to reconstruct the same field of view. Through a comparative analysis, we found that within the localization region of −200 to 200 nm, the reconstruction results using VISPR and INSPR exhibited a consistent distribution along the z-axis [Fig. 5(f)]. In contrast, reconstructions using the in vitro PR approach showed gradual spread and diffusion over a large depth range, accompanied by noticeable distortions and a reduction in resolution, as observed in the intensity profile [Figs. 5(j)–5(l)].

It is worth noting that although the reconstruction quality of VISPR is superior to that of INSPR and PR, the two layers of the NPC are nearly unresolved in the z direction [Figs. 5(c) and 5(f)]. After analyzing the experimental conditions and sample situation, we found this to be reasonable for the following reasons. (1) We deliberately added randomly unknown high-order aberrations by using a DM in addition to the aberrations of the nuclear pore sample itself to better match the situation of deep tissue imaging. Due to the artificially introduced high-order aberrations and the inherent complex aberrations of the NPC biological sample, the reconstructed image quality is inevitably affected. (2) We used a commercially available sample in which Nup98 was labeled with primary and secondary antibodies. The antibody complexes always introduce quite influential linkage errors due to their big sizes (20–30 nm) in imaging ultrafine structures [42]. Therefore, it is challenging to resolve the two axially spaced layers of Nup98, which are approximately 50–60 nm apart, using traditional astigmatic SMLM.

By evaluating the reconstruction results on both the lower and upper surfaces of the nuclear membrane, VISPR demonstrates its capability to not only resolve the central structure of the nuclear pore within the radial axis, but also maintain a consistent distribution along the axial axis. Despite the increasing influence of the refractive-index mismatch as the depth increases, VISPR’s reconstructed structure exhibits remarkable consistency. INSPR’s localization results within the −200 to 200 nm region are consistent with VISPR, but as the depth along the z-axis increases, INSPR’s reconstructed results exhibit distortion and dispersion. Additionally, the in vitro PR method neglects sample-induced aberrations such as refraction mismatch, leading to distortion and significant spreading of reconstructions along the extensive z direction. These results underscore VISPR’s capacity of obtaining the accurate PSF model even at considerable depths.

To further explain this difference, we compared VISPR PSF, INSPR PSF, and PR PSF, as illustrated in Figs. 5(b) and 5(e). Notably, to mimic the complex biological environment of deep tissue, we artificially introduced randomly unknown higher-order aberrations via the DM. As a result, Fig. 5(b) shows that the PR PSF is not a perfectly tilted astigmatism but rather exhibits distortions caused by these higher-order aberrations. Analyzing the VISPR PSF from both the bottom and top surfaces of the nucleus reveals that, with an increase in imaging depth, the extent of sample-induced aberrations, such as spherical and coma aberrations from optical sections, intensifies. The VISPR-retrieved pupils effectively captured this depth-dependent variation in sample-induced aberrations, their decomposed Zernike amplitudes, and the corresponding axially stretched vectorial PSFs. Additionally, the VISPR PSF accurately retrieved the artificially introduced higher-order aberrations.

We also found that within the −400 to +400nm range, both INSPR- and VISPR-retrieved PSFs demonstrated significant similarity. However, because INSPR uses a scalar PSF model and the modified GS algorithm, the INSPR-retrieved PSF exhibits noticeable distortion and fails to recover high-frequency details beyond ±400nm on the z-axis compared to the VISPR PSF. This leads to INSPR’s failure to accurately reconstruct the NPC region beyond ±400nm on the z-axis, which is consistent with observations from our simulation experiments. As for the PR method, the PSF retrieved from fluorescent beads only measured instrument imperfections and failed to consider sample-induced aberrations and their depth-dependent variations, such as refractive-index mismatch. Consequently, the reconstruction quality of the PR PSF is significantly lower compared to VISPR and INSPR. Given that sample-induced aberrations can vary significantly from specimen to specimen, resulting in the degradation of axial reconstruction for in vitro PR algorithms, our results underscore the consistent ability of VISPR to achieve accurate and high-resolution 3D reconstructions.

In this work, we propose VISPR, a robust in situ phase retrieval framework that can directly retrieve 3D PSF models considering both system- and sample-induced aberrations from raw single-molecule datasets. VISPR achieves precise in situ 3D PSF modeling, enabling the accurate localization of single molecules in three dimensions and facilitating accurate 3D SMLM. Our approach was validated using simulation datasets and practical experiments. In the simulation experiment, we confirm that VISPR exhibits superior fitting ability compared to INSPR [32], which is attributed to enhancements in both the PSF model and phase retrieval algorithms. With the help of the vectorial PSF model and MLE phase retrieval algorithm, VISPR effectively addresses issues related to model deviation and the absence of high-frequency information at substantial depths along the z-axis. Furthermore, even in scenarios with low signal-to-noise ratios, VISPR demonstrates robust performance. It is essential to note that VISPR’s efficacy relies on single-molecule data within a specific z-axis range, and potential limitations may arise if the majority of single-molecule data are clustered around the same z-axis position. Since VISPR relies on averaged single-molecule images to construct a PSF stack at a few axial positions, coupled with the impact of noise, some details of the real PSF model may become blurred and neglected. This means that VISPR’s precision still cannot truly match the accuracy of a real PSF. Additionally, one of the potential future development directions for VISPR is to utilize GPU acceleration, vectorization, or parallel computing to enhance its computational speed, due to the increased complexity of vectorial PSF modeling and its current CPU implementation.

Traditional phase retrieval methods relying on beads are inadequate for SMLM data at large depths, such as intracellular and extracellular cellular targets deep inside tissues. This inadequacy stems from the fact that many biological samples exhibit significant variations in the refractive index, a feature not captured by beads. VISPR, in contrast, emerges as a robust solution capable of accurately retrieving the in situ 3D PSF model. Its strength lies in the simultaneous consideration of system- and sample-induced aberrations. Additionally, VISPR improves the accuracy of the PSF model by solving the problem of PSF model distortion at large depths and by considering the high-frequency information of the dataset. Noteworthy is VISPR’s versatility, demonstrating applicability in low signal-to-noise ratio SMLM situations and compatibility with various microscope modalities. This versatility provides invaluable insights for optimizing optical system performance, correcting and analyzing aberrations induced by the system and samples, estimating single-molecule properties, and enhancing imaging contrast and resolution.

Acknowledgment. We would like to thank Xin Liu (Zhejiang University) for the support and kind assistance.