1. Introduction

Since its introduction in 2006, single molecule localization microscopy (SMLM) [1–3] has progressively advanced, becoming a key tool for imaging nanoscale structures in thin biological specimens, such as cultured cells [4,5] and tissue sections under 10 µm thick [6,7]. Yet, the challenge of imaging thicker specimens with SMLM persists. The advent of lightsheet-based SMLM [8] has marked a turning point by confining illumination to the focal plane of the detection objective lens through side illumination from a separate lens. This method significantly reduces background fluorescence and minimizes optical aberrations when the refractive indices of the sample and immersion oil are matched, both crucial for achieving high-quality deep tissue imaging. More recent developments include the use of reflected lightsheet geometry [9–13], which can ease the need for orthogonal placement of the illumination objective or even enable a single objective configuration. However, these advancements necessitate specific sample holders or geometric arrangements, potentially complicating sample preparation or microscope use.

Oblique plane SMLM (obSTORM [14]) is an alternative approach to lightsheet SMLM that achieves oblique lightsheet imaging with a single objective configuration in sample space. This layout is fully compatible with standard microscopy specimens prepared on glass slides, making it convenient for accommodating laterally wide tissue sections and small animal samples, just like conventional microscope platforms. However, even with its capability to conduct SMLM imaging of thick biological specimens up to ∼60 µm or thicker, obSTORM’s imaging performance, encompassing localization precision, imaging resolution, and volumetric imaging speed, demands enhancements. This performance constraint primarily stems from the irregular behavior of the point spread function (PSF) in both its shape and size through focus [14], the cause of which remains not fully understood. Specifically, the PSF shapes in obSTORM deviate from the conventionally-used Gaussian models, and the sensitivity of PSF size variations is constrained across the positive defocus range. Consequently, the previous use of the Gaussian PSF model for three-dimensional (3D) localization compromised localization precision, which in turn negatively affected imaging resolution. Moreover, the exclusion of single molecule data from the positive defocus range, due to suboptimal localization precision, effectively halved the volumetric imaging speed [14].

In this study, our goal is to enhance the 3D localization precision of single molecules in obSTORM by employing a refined PSF model based on cubic splines [15–18]. We construct this model from experimental PSF calibration images and evaluate its modeling accuracy and localization performance in comparison with the conventional Gaussian PSF model. Our approach aims to improve localization precision and extend the range for 3D localization to both defocus directions – an enhancement previously unattainable with obSTORM. We validate our method by conducting a comparative analysis of STORM images of a scanned fluorescent bead and mouse retina tissue sections, each reconstructed using the cubic spline and Gaussian PSF models, respectively.

2. Materials and methods

2.1. obSTORM system and PSF calibration

Figure 1 shows the simplified layout of obSTORM [14]. A water-immersion objective (O1; UPLSAPO 60XW/1.2, Olympus) is used to illuminate the specimen with an oblique sheet of light, and single molecule images along the xo oblique plane are refocused and reoriented to the focal plane of the remote objective (O2; MPlanApo 50×/0.95, Olympus) to be directly captured by the camera (Cascade II 512, Photometrics). The O1 objective has a numerical aperture (NA) of 1.2, and the angle (α) of oblique plane is primarily set at 45°. The setup uses a polarizing beam splitter and a quarter-wave plate to improve the isotropy of the vectorial PSF shape and double the light collection efficiency, compared to setups that use only a nonpolarizing beam splitter [14]. The total imaging magnification is ∼110.7 and the image pixel size in object space is 144.5 nm. For additional experimental details on the obSTORM system, refer to Ref. [14].

 

Fig. 1. Conceptual schematic of obSTORM. O, objective; DM, dichroic mirror; L, lens; PBS, polarizing beam splitter; PM, periscope mirror; QWP, quarter-wave plate; M, mirror; EF, emission filter; CL, cylindrical lens. The lenses L1, L2, and L3 have focal lengths of 200 mm, 180 mm, and 300 mm, respectively. Experimental PSF images (normalized to maximum intensity) at different defocuses in the z_o direction are shown. Scale bar: 500 nm.

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obSTORM employs a cylindrical lens positioned in front of the camera to introduce astigmatism into PSFs [4], enabling 3D localization with the third localization direction (z_o) perpendicular to the oblique plane (xo). For PSF calibration, a z_o-stack of PSF images of deep-red fluorescent beads (P7220, Invitrogen) is captured by moving the sample objective (O1) in the z direction with a 30 nm step size, which corresponds to a z_o interval of 21.2 nm calculated by ${z_o} = z\cos \alpha $ [14]. As shown in Fig. 1, the experimental PSF shapes show asymmetry along the x and o axes, particularly at larger defocus values. The cause of this behavior has not been fully identified. We also note that, while only the beads on the coverslip were assessed, the PSF shapes were presumed consistent across the sample depth, given the refractive index match between the objective lens and the specimen in the aqueous STORM imaging buffer.

2.2. PSF modeling: Gaussian function vs. cubic spline

In previous obSTORM demonstrations, the PSF shape was approximated as a 2D Gaussian function:

Similarly, the lateral (xo) shift of the PSF’s central position through focus, influenced by the asymmetric pupil and PSF shapes about the x and o axes, was also extracted and modeled using 5^th order polynomials for correction.

In contrast, cubic spline (or cspline) modeling offers the flexibility to model complex-shaped PSFs that may vary considerably across defocus. This adaptability arises from the approach of modeling each voxel of the 3D PSF using distinct piecewise 3^rd order polynomials [15]. As introduced in Ref. [15], the PSF centered at (${\mu _x},{\mu _o},{\mu _{{z_o}}})$ can be modeled using cubic spline as

2.3. STORM imaging data

To evaluate the relative performance of cubic spline and Gaussian PSF models in obSTORM imaging (α = 45°), we referenced raw image datasets of 3D PSF stacks of dark-red beads and volumetric single molecule images of mouse retina tissues from Ref. [14]. The 60 µm-thick section of mouse retina tissue was immunostained with Alexa Fluor 647 to target the protein kinase C alpha (PKCα) of rod bipolar cells. The volumetric STORM data comprises 10 optical sections, each spaced at a ∼504 nm z_o-step interval and containing 7,000-10,000 image frames recorded at a rate of 28 frames/s. For more comprehensive details concerning sample preparation and STORM image acquisition, refer to Ref. [14].

In addition, we utilized a STORM image dataset of a “straight line” specimen. This dataset was generated by slow scanning of a deep-red bead (Ø180 nm) over a ∼3.6 µm range in the z direction, while capturing images of the moving bead over ∼18,000 frames at 50 frames/s. The illumination was adjusted to detect ∼960 photons from the bead per camera frame, a count which aligns with the average photon count of Alexa Fluor 647 molecules in obSTORM, as the light collection efficiency is ∼25% of that in conventional 1.4-NA STORM systems. In this instance, the average constant background was found to be ∼17 photons per pixel.

2.4. Localization analysis and image reconstruction

We used SMAP software for the localization analysis based on the cspline PSF model, which utilizes maximum-likelihood estimation (MLE) for the estimator $\hat{\theta } = [{{\mu_x},{\mu_o},\; N,\; b,{\mu_{{z_o}}}} ]$. One of the four 3D PSF stack datasets used for cspline PSF modeling was selected, and its PSF images were localized to calculate Cramér-Rao lower bound (CRLB) precision in SMAP based on the fitting result of $\hat{\theta }$. For comparison, we fitted the same bright bead data with the Gaussian PSF model and evaluated the CRLB precision by calculating the inverse of the Fisher information matrix using custom MATLAB codes for obSTORM [14]. The estimated CRLB precision values from both PSF models were scaled to correct for the dimensional distortion in the o direction caused by the cylindrical lens (∼1.19) and an excess noise factor resulting from the EMCCD camera (∼1.41).

To localize the single molecule data of the “straight line” bead and mouse retina sections, we used cspline and Gaussian fitting from SMAP and custom MATLAB codes, respectively. Once localization was achieved, we adjusted the data to correct for the cylindrical distortion present in the o direction. For mouse retina tissues, we filtered the localization data to exclude single molecules having CRLB precisions greater than 60 nm in xo or 130 nm in z_o. Lastly, we rendered the data for visualization through SMAP software.

3. Results and discussions

We evaluated the accuracy of the cubic spline PSF model in representing the structural features of obSTORM PSF shapes. As shown in Fig. 2, the cspline PSFs closely resembled the experimental PSF images of a deep-red bead across the entire z_o range of 2.4 µm. The error images, which depict the differences between the experimental and model PSFs, revealed only small deviations at all z_o values, with a maximum pixel-wise intensity difference of 4.0% (relative to the peak intensity of one). This is a notable improvement over the previous Gaussian PSF models. Although the experimental PSFs had a Gaussian shape for -0.6 µm ≤ z_o ≤ 0.3 µm, the pixel-wise error remained as high as 10.8% (including 7.3% error even at focus, z_o = 0 µm). The Gaussian PSF model could reasonably capture the change in PSF ellipticity at -1.2 µm ≤ z_o ≤ -0.9 µm, but the maximum error exceeded 21.8%. Moreover, for 0.9 µm ≤ z_o ≤ 1.2 µm, the Gaussian model failed to represent the two upper tails of the PSF and had a larger PSF size than the experimental PSFs, resulting in a modeling error as large as 27.8%.

 

Fig. 2. Experimental PSF and its numerical models based on cubic spline and elliptical Gaussian function. The maximum intensity of the images was normalized to one. Errors represent pixel-wise difference between the measured and model PSFs. Scale bar: 500 nm.

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To further assess the effectiveness of the cubic spline PSF model over the Gaussian PSF model, we compared the CRLBs on localization precision calculated from the localization results of the bright bead images used for PSF calibration. The bright bead had a photon count of ∼53,000 photons and a constant background of ∼182 photons/pixel, when averaged over all z_o positions. As depicted in Fig. 3, the cspline PSF model demonstrated superior performance over the entire z_o range compared to the Gaussian model. In the x and o directions, the cubic spline PSF model improved the CRLB precision by approximately 1.1-1.6 times, depending on the z_o position (on average, 1.25× and 1.16× for all z_o in x and o directions, respectively), with a gradual improvement observed with increasing defocus to ±1.2 µm. This trend was quite consistent with the larger modeling errors observed with increasing defocus in Fig. 2.

 

Fig. 3. Comparison of CRLBs on localization precision in (a) x, o, and (b) z_­o directions for a bright deep-red bead when localized using cubic spline and Gaussian PSF models.

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The improvement in CRLB precision in the z_o direction varied significantly depending on the value of z_o, with an average improvement factor of 2.0× for all z_o values. For -0.9 µm < z_o < -0.1 µm, the cubic spline PSF model provided minimal improvement in CRLB precision, with an improvement factor of about 1-1.5 times. This was due to the dramatic change in ellipticity in PSF shapes in this range, where PSF size was a reasonably good figure of merit for defining the axial position of z_o. Therefore, the previous obSTORM method restricted the working range for 3D localization to the negative defocus side (-1.0 µm ≤ z_o ≤ 0 µm), given the poor localization precision from the Gaussian PSF model on the positive defocus side due to insensitive PSF size changes. In contrast, the cubic spline PSF model substantially improved z_o precision by about 2-4 times over positive z_o values by adeptly capturing minor PSF shape variations. Moreover, the cubic spline PSF model maintained relatively uniform CRLB precision in the z_o direction across the entire z_o range, including both ends (-1.2 µm < z_o < -0.9 µm and 0.9 µm < z_o < 1.2 µm), without any performance degradation seen with the Gaussian PSF model. Consequently, by modeling the obSTORM PSF with a cubic spline, the full 2.4 µm z_o range is expected to be utilizable.

As CRLB precision alone is not sufficient to explain SMLM imaging resolution, we also evaluated the imaging resolution by obtaining STORM images of a z-scanned bead. Ideally, the scanned bead should appear as a narrow straight line, with its width representing the imaging resolution. The bead was adjusted to a brightness level of approximately 960 photons, which is similar to the brightness of Alexa fluor 647 during obSTORM imaging. As shown in Figs. 4(a) and 4(b), the reconstructed STORM images using the cspline PSF model showed narrower linewidths over the entire z_o range, with particularly noticeable improvements in the positive defocus range. In xo plane view, the full width at half maximum (FWHM) linewidths of cross-sectional intensity profiles were found to be improved by 1.6×, 1.3×, and 1.3× for three different z_o positions (-1.0, 0.3, and 1.0 µm), respectively, as shown in Figs. 4(c-e). These improvements were consistent with the CRLB improvements in the x direction by 1.4×, 1.2×, and 1.4× for each z_o position in Fig. 3(a). Therefore, it can be concluded that the improved PSF model using cubic spline has led to a ∼1.3× improvement in imaging resolution in the x direction. Based on the CRLB precision data presented in Fig. 3(a), a similar improvement factor in the o direction (on average, ∼1.2×) is expected.

 

Fig. 4. 3D STORM imaging of a dark-red fuorescent bead moving in the z direction. Line trajectories of the bead reconstructed using (a) cubic spline and (b) Gaussian PSF models. Scale bar: 500 nm. (c-h) Cross-sectional intensity profiles for the magenta and green lines shown in panels (a) and (b). The intensity data and their Gaussian fits are displayed as dots and lines, respectively. FWHM linewidths obtained from the fits are displayed as numbers.

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As shown in Figs. 4(f-h), the cspline PSF model resulted in significant improvements in the FWHM widths of the intensity cross-sections along the diagonal direction at z_o = -1.0, 0.3, and 1.0 µm in the oz_­o plane, by factors of 2.1×, 2.5×, and 1.9×, respectively. The corresponding CRLB precision improvements (cspline vs. Gaussian PSF models) were estimated as 2.0×, 4.1×, and 2.5×, respectively, by root mean square of the CRLB data in o and z_o directions in Fig. 3. While the improvements in FWHM width and CRLB precision were consistent at z_o = -1.0 µm, the FWHM improvements at z_o = 0.3 and 1.0 µm were not as large as the CRLB improvements. One possible explanation is that at these z_o positions, some of the localization data from the Gaussian PSF model were overly scattered, and such excessively dispersed molecule data points were not included in the FWHM calculation. The ∼2-fold resolution improvement shown in Figs. 4(f-h) was primarily due to the improved localization precision along the z_o direction. Therefore, it can be said that the cspline PSF model has improved the z_o resolution approximately two-fold at these z_o positions, and the entire 2.4 µm range of z_o could be utilized for 3D localization.

We subsequently evaluated the performance improvements on real tissue samples. As shown in Fig. 5, we reconstructed volumetric STORM images (α = 45°) of Alexa Fluor 647-labeled PKCα from a mouse retina slice (using single molecule data from [14]) based on each of the PSF models. The STORM images contained 10 optical sections, captured with a z_o-step interval of ∼504 nm. We used only single molecules identified within z_o = [-1, 0] µm range in each section, with estimated CRLB precisions under 60 nm in xo and 130 nm in z_o, adhering to the same filtering condition as the previous reconstruction [14]. The cspline PSF model provided a more accurate structural reconstruction with increased densities of single molecules at several field locations, as indicated by white arrows in Figs. 5(a) and 5(b). Enhanced image reconstruction was also evident in cross-sectional views at different field locations, as showcased in Figs. 5(c-h). In contrast to the inferior stitching quality between adjacent optical sections (highlighted by white arrows in Figs. 5(d) and 5(f)) achieved using the Gaussian PSF model, the cspline PSF model produced seamless stitching with continuous intensity outlines of membrane-bound PKCα. Further, as Fig. 5(i) shows, the intensity cross-sections were narrower by factors of 1.37×, 1.17×, and 1.22× in the x, o, and z_o directions, respectively, when localized using the cspline PSF model. These ratios aligned closely with the improvement ratios of average localization precisions shown in Fig. 5(j), which were (x) 1.24×, (o) 1.14×, and (z_o) 1.24×. These values also corresponded with the average CRLB precision ratios of the calibration bead (cspline vs Gaussian PSF models) in Fig. 3 when averaged over z_o = [-1, 0] µm, i.e., (x) 1.19×, (o) 1.13×, and (z_o) 1.28×. Thus, it can be concluded that the cspline PSF model enhances image resolution by approximately 1.15-1.25× depending on axis directions, when using 1-µm-thick optical section (z_o = [-1, 0] µm) with CRLB precision filtering.

 

Fig. 5. A comparative study of volumetric obSTORM images of PKCα within a mouse retina section, reconstructed using (a) cubic spline and (b) Gaussian PSF models. (c-h) Cross-sectional views along the white dashed lines marked in panels (a) and (b). (i) The cross-sectional intensity profiles at the white boxed regions, projected along the longer direction of the boxes, in panels (c-h). FWHM values are displayed. (j) The histograms of CRLB precision for single molecule data in (a) and (b). The mean precisions in each axis direction are shown. Scale bars: 5 µm (a, b), 500 nm (c-f), and 2 µm (g, h).

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Lastly, we evaluated the potential of increasing the z_o sectioning interval with the cspline PSF model by reconstructing STORM images from only five alternating optical sections (sections 1, 3, 5, 7, and 9) of the mouse retina data in Fig. 5, but now using the full range of each optical section, z_o = [-1.2, 1.2] µm. As demonstrated in Figs. 6(a) and 6(b), the z_o CRLB precision of single molecules from these 5 sections mirrored closely the trend of the CRLB precision distribution for the calibration bead shown in Fig. 3(b), showing the worst precision around z_o = 300 nm. By using the cspline PSF model, the average CRLB precision improved significantly by approximately 1.7-fold, decreasing from 154.8 nm (Fig. 6(b)) with the Gaussian model to 91.3 nm (Fig. 6(a)). This considerable improvement led to clearer membrane-like outlines of PKCα in the cross-sectional view (Fig. 6(c)), as opposed to the distorted structure (indicated by the white arrow in Fig. 6(d)) observed when the full z_o range was applied in the Gaussian PSF model. Upon applying single molecule filtering based on a threshold precision of 130 nm in z_o, only a small fraction (∼15% of single molecules) was excluded in the cspline PSF model. Importantly, even when only 5 sections were used, the resulting STORM images exhibited clear and distant outlines of PKCα without discernible stitching artifacts (Fig. 6(e)), maintaining a quality comparable to those reconstructed from the full 10 sections (Fig. 6 g). Conversely, using the Gaussian PSF model led to a significant data loss, as nearly half (∼48%) of single molecules were discarded. This resulted in a fragmented outline of PKCα, indicated by the white arrows in Fig. 6(f), along with incomplete removal of the distorted structure. This result underscores the benefits of using the cspline PSF model, which utilizes the full z_o range of 2.4 µm and supports at least a two-fold increase in the z_o interval between adjacent optical sections. Such an enhancement could reduce the image acquisition time in volumetric obSTORM by half.

 

Fig. 6. Distributions of CRLB precision in z_o for data from 5 alternating optical sections of PKCα within a mouse retina slice, as determined by (a) cspline and (b) Gaussian PSF models. (c-f) Volumetric STORM images (corresponding to the yellow-boxed area in Fig. 5) and cross-sectional views along white dashed lines. These images show the effects of single molecule filtering (threshold: 130 nm in z_o CRLB precision) under each PSF model. (g) The magnified view of the yellow-boxed area in Fig. 5(a) (derived from the complete set of 10 sections) and the cross-sectional view for comparison.

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4. Conclusion

In this study, we have demonstrated the superior accuracy of cubic splines in modeling the through-focus PSF shapes in obSTORM, achieving less than 4% deviation. This refinement offers a more sophisticated PSF model compared to the conventional Gaussian model. Remarkably, this improved model enhances the 3D localization precision, resulting in a substantial improvement in estimated CRLB precisions by 1.1-1.6× in xo and 1-4× in z_o across the full z_o range of [-1.2, 1.2] µm. These enhancements in localization precision have been substantiated by FWHM improvements in scanned bead STORM images, with a factor of 1.3-1.6× in xo and 1.9-2.5× in z_o at three different z_o positions.

When applied to single molecule data of real mouse retina tissues, the cubic spline PSF model has improved the obSTORM imaging resolution by an average of 1.15-1.25× in all axis directions. Notably, this improvement factor is achieved even when compared against the optimally trimmed resolution in the previous obSTORM reconstruction by limiting the section range to z_o = [-1, 0] µm (an interval where the Gaussian PSF model performs relatively well) and by aggressively filtering out single molecules with subpar localization precision. Moreover, the cubic spline model enhances the stitching quality of neighboring optical sections in volumetric STORM images of PKCα and allows for a relaxation of the z_o interval for volumetric obSTORM to at least double when using the full z_o range of 2.4 µm.

To conclude, our incorporation of the cubic spline PSF model into obSTORM significantly propels the deep-tissue STORM technology forward, enhancing imaging fidelity and expediting volumetric imaging time. This advancement could significantly enhance the capability of super-resolution obSTORM imaging in tissue and small animal studies.