Impact Statement

Cryogenic serial plasma focused ion beam milling scanning electron microscopy is an emerging microscopy technique that is used to visualize 3D structures of biological features at mesoscale resolutions. This technique requires common postprocessing of data such as alignment and charge mitigation to enable robust segmentation and analysis. In addition, approaches are needed to quantify data quality to enable an assessment of features and tune data acquisition parameters to enable optimal image acquisition. This article presents Okapi-EM, a combination of software tools designed to facilitate these important initial steps in assessing and processing images from these experiments. These tools have been assembled as a plugin for a popular 3D biological image visualizer called napari, making their usage user-friendly and readily accessible.

2. Implementation

Napari is an open-source, user-friendly, data visualization application that runs in Python programming language⁽Reference Sofroniew, Lambert and Evans¹⁷⁾. It supports three-dimensional visualization of different data types (i.e., image and labels) interactively. Its annotation features are useful in biological image analysis and processing. Furthermore, its functionality can be extended with support for plugins, allowing continuous development of image processing methods alongside of experimental microscopy advances. Developing within the Python ecosystem also facilitates deep learning approaches with easy access to modern machine learning packages such as scikit-learn⁽Reference Pedregosa, Varoquaux and Gramfort¹⁹⁾ and PyTorch⁽Reference Paszke, Gross, Massa, Lerer, Bradbury, Chanan, Killeen, Lin, Gimelshein, Antiga, Desmaison, Kopf, Yang, DeVito, Raison, Tejani, Chilamkurthy, Steiner, Fang, Bai, Chintala, Wallach, Larochelle, Beygelzimer, Alché-Buc, Fox and Garnett²⁰⁾. The plugin installation uses the well-established package management system (PyPI), which also enables plugin installation chains, where a single plugin can install many other plugins or packages as needed to create a bundle. Bundling several plugins in this way means that a whole toolbox can be created for specific data processing workflows or research themes (e.g., devbio-napari⁽Reference Haase²¹⁾, or napari-assistant⁽Reference Haase and Savill²²⁾, both of which were tailored for biological image processing). Napari supports and opens a range of image formats through the skimage library, including multi-page tiff. It also hosts a variety of other image processing and file format plugins so Okapi-EM tools can be integrated into and combined with other workflows. Okapi-EM can be installed through napari’s plugin search and installer engine. Otherwise, it can be downloaded freely at https://github.com/rosalindfranklininstitute/okapi-em, or installed from Python package index (PyPI). Okapi-EM was developed for minimum python version 3.7 and installations of other dependency packages using PyPI or napari’s plugin installation engine are automatic.

Currently, Okapi-EM incorporates three major image processing tools which are organized in the user interface as separate tabs: stack alignment, charge suppression, and resolution measurement (Figure 1). Each tab contains user interface elements to adjust settings and execute data analysis.

Figure 1. View of the napari application highlighting the Okapi-EM plugin (green rectangle at right), and the currently available tools (pink rectangle at right). Each Okapi-EM plugin has its own tab with appropriate options for its use displayed. See Supplementary Figure S1 for a detailed view and description of the options available in each plugin.

3. Stack Alignment

In FIB-SEM, samples are milled in preparation for imaging. During this process, a number of issues can cause drift or misalignment in the image stack, such as the movement of the sample due to mechanical stress, small temperature variations and slight, iterative, stage movements going from milling to imaging positions that accumulate over time⁽Reference Stephensen, Darkner and Sporring²³⁾. Inaccurate placement of the milling area by the operator or software may also lead to the observation of shearing during subsequent imaging⁽Reference Yuan, de Moortèle and Epicier²⁴⁾. These misalignments may further be amplified by factors including charging effects and instabilities caused by external disturbance⁽Reference Yuan, de Moortèle and Epicier²⁴⁾. As a result, an important element in the alignment of the resultant SEM images is the shearing or skewing along the direction perpendicular to the ion beam (or other layer removal option) trajectory⁽Reference Stephensen, Darkner and Sporring²³⁾, which is particularly significant along the slow scan direction. Before any subsequent visualization or analysis tasks, aligning the image stack is a crucial step.

While there have been efforts at improving image acquisition hardware and developing real-time correction⁽Reference Dahmen, Engstler and Pauly^25–Reference Schaffer, Wagner and Schaffer²⁷⁾, these methods are limited by the need to ensure that the sample is not overexposed during data collection. Therefore, they do not provide fine alignment correction and require the use of subsequent alignment software⁽Reference Yuan, de Moortèle and Epicier²⁴⁾. Several software packages/plugins are available for this task, such as the closed-source and pay-for-service Amira by ThermoFisher⁽²⁸⁾ and open-source options such as ImageJ plugins Linear Stack Alignment with SIFT⁽²⁹⁾ and TrakEM2⁽Reference Cardona, Saalfeld and Schindelin³⁰⁾. Although these software can in general align the images so that there is smooth transition between the slices, a major drawback is that they do not consider the physical process by which the images were acquired, thus they can introduce distortions that are not present in the sample. During alignment, if incorrect transformation parameters are chosen, because slices are aligned to their neighbors, this can cause a cascade of transformations that can ultimately distort the shape of the stack⁽Reference Hennies, Lleti and Schieber³¹⁾. Notably, the ImageJ plugins Linear Stack Alignment with SIFT and TrakEM2 do not offer options to perform alignment with shearing instead of rotation in particular directions, which is what images obtained by scanning line-by-line need to be compensated for.

3.1 Alignment method

We have developed an improved image alignment method that aims to provide users with a variety of transformation options to appropriately describe image acquisition processes and to reduce the likelihood of introducing nonphysical distortions during alignment of 3D stacks. It employs scale-invariant feature transform, or SIFT, a widely used algorithm for the detection of local landmarks⁽Reference Vourvoulakis, Kalomiros and Lygouras³²⁾. Since SIFT tends to include some false matches that could affect the alignment outcome, filtering is necessary to remove these outliers. Users can choose between percentile and RANSAC filters to achieve this. The former option is faster but less robust, while the latter is more computationally costly, but more reliable in cases where matches are scarce, or exhibiting misalignment. Users will then define a transformation that would best describe the physical distortion of the images depending on the image acquisition technique (Table 1) by adding or removing components of the transformation matrix, including translation, rotation, shearing, uniform scaling and stretching or shrinking, or selecting affine transformation which allows all six degrees of freedom. A description of each option is shown in Supplementary Figure S2. For instance, if the option shearing along the $ y $ -axis is enabled, and no scaling or shearing in $ x $ -axis are chosen, then the transformation matrix and translation vector to be obtained is:

(1) $$ {\overrightarrow{p}}^{\prime }={M}_{\mathrm{she},x}\bullet \overrightarrow{p}+\overrightarrow{t} $$

Table 1. Types of transformations that are commonly used to correct distortions from different volumetric electron microscopy techniques.

^a Deformations between subsequent serial section images.

^b Deformation between projection images of the same thin section taken at different tilt angles.

or:

(2) $$ \left[\begin{array}{c}x^{\prime}\\ {}y^{\prime}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ {}k& 1\end{array}\right]\bullet \left[\begin{array}{c}x\\ {}y\end{array}\right]+\left[\begin{array}{c}{x}_t\\ {}{y}_t\end{array}\right], $$

where $ \overrightarrow{p}=\left(x,y\right) $ is a feature point on the source image and $ {\overrightarrow{p}}^{\prime }=\left(x^{\prime },y^{\prime}\right) $ is its location after shearing with $ {M}_{she,x} $ being the linear transform for shearing along $ x $ by the parameter of $ k $ , and translation of $ \overrightarrow{t}=\left({x}_t,{y}_t\right) $ .

The alignment algorithm starts from the first image slice and calculates the optimized transformation parameters between consecutive slices along the whole stack. Similar features between images are identified and matched. The distance between a pair of feature points is $ d=\left|{\overrightarrow{p}}^{\prime }-\overrightarrow{p}\right| $ and, considering all the feature points pairs between images, the sum of the squared distances is minimized by adjusting the parameters $ k,{x}_t,{y}_t $ .

3.2 Rotational distortion introduced with existing software

As discussed previously, limiting the modes of transformation according to the physical process of image acquisition is crucial. Failing to do so when aligning FIB-SEM image stacks could result in a large angle rotation in the image slices that is not actually present in the sample. To demonstrate this, an artificial shearing transformation was applied to the original image (Figure 2a) acquired using cryogenic serial pFIB-SEM to mimic a shear force distortion (Figure 2b). For this image pair, using SIFT, many matches can be found on the left side of the image, especially in the triangular and circular region (Figure 2c). Using least square or other optimization methods without restraining the transformation mode results in a large angle rotation or an affine transformation that aligns those feature-rich regions well, but fails to align other parts of the image and introduces a nonphysical distortion (Figure 2d).

Figure 2. Nonphysical distortion in the alignment process if the modes of transformation are not appropriately limited. (a) Original SEM image of yeast cells. (b) Artificially distorted image after a shearing transformation was applied to the original image. A shear factor of 0.15 was selected for visibility. (c) Feature points found in (b) using SIFT, with two feature-dense regions highlighted with circle and triangle markers. (d) Possible alignment result when the modes of transformation are not restrained, where even though the feature-rich regions are well-aligned, a nonphysical rotation is introduced. Data is FIB-SEM of yeast, available at EMPIAR with ID 11416.

3.3 Results

Okapi-EM stack alignment was applied to a cryogenic serial pFIB/SEM image stack of yeast cells (109 slices). As described in Table 1, translation, shear x–axis, and stretch along y-axis were chosen to compute an alignment based on the landmarks found using SIFT with RANSAC outlier filtering. Cross-sectional views are used to visualize the alignment result (Figure 3a). To compare outcomes, the same image stack was aligned in Fiji plugins TrakEM2 and Linear Stack Alignment with each transformation option offered as well. Substantial drift and distortion can be observed in the unaligned stack (Figure 3b,j and Supplementary Movie S1). After alignment, cell shapes are restored with mostly smooth outlines (Figure 3c,k and Supplementary Movie S2). While all alignment approaches improved the cross-sectional alignment to some degree, distortion can still be observed in the outline of the cells when using either TrackEM2 or Linear Stack Alignment (Figure 3d–i,l–q), especially at the y = 2 μm when using TrakEM2 and when affine transformation is selected in Fiji Linear Stack Alignment.

Figure 3. Alignment results and comparison between Okapi-EM, TrakEM2, and Fiji Linear Stack Alignment with SIFT (a) Cross-sectional views of SEM image stack of yeast cells (109 slices along z direction, slice size 20.7 × 13.8 μm²) are obtained. (b–i) Cross-sectional view of a 11.2 × 3.7 μm² area at y = 2 μm of the unaligned stack and the aligned stacks using Okapi-EM alignment with RANSAC and {translate, shear x, stretch y} selected, Fiji plugin TrakEM2 with rigid transformation, similarity transformation, or affine transformation selected, respectively, and Fiji plugin Linear Stack Alignment with rigid transformation, similarity transformation, or affine transformation selected respectively. (j–q), Cross-sectional view of a 15.0 × 3.7 μm² area at y = 11 μm of the aligned stacks using the three aforementioned alignment methods and settings respectively. Data is EMPIAR-11416. In both Fiji plugins, rigid transformation allows translation and rotation. Similarity transformation allows translation, rotation, and scaling. Affine transformation is defined as shown in Supplementary Figure S2. Details about the rendering of the cross-section images is detailed in Supplementary Figure S8.

4. Charge Artifact Mitigation

Charging artifacts often appear when insulating materials interact with the electron beam⁽Reference Dumoux, Glen and Ho^1, Reference Spehner, Steyer and Bertinetti³⁾. This effect is normally minimized by adjusting beam energy or scanning parameters, and its severity depends on the target substance being imaged. Biological samples can present a challenge, with cells often containing different compartments with distinct electrical conduction properties that cannot be completely balanced through adjustments in acquisition parameters. Inevitably, images will show evidence of charging, causing artifacts which manifest in the SEM images collected in the form of elongated dark regions in the scanning direction that extend asymmetrically beyond the charging feature itself (Figure 4a,g). The practical outcome of this artifact is the partial obscuring of biological features nearby to insulating materials. This makes both manual and automated downstream data processing and analysis (e.g., segmentation or visualization) more difficult.

Figure 4. (a–f) SEM images and plots illustrating the charge artifact removal algorithm on a lipid droplet from a yeast dataset. Data is EMPIAR-11416. (a) SEM image of a lipid droplet within a yeast cell (image size 6.75 × 1.35 μm²). Arrows indicate a scanning line of interest, with its signal profile in (c–e) as blue line. Red rectangle represents data region where signal was averaged (width set by the nlinesavaerage parameter) resulting in signal profile in (c) as orange line. (b) Annotation image of the charge center, corresponding to the charging artifact in Figure (a). Green line in plot (c) is the line profile of the annotation along the row of interest (arbitrary units). (d) Line profiles describing the background corrected signal (blue line) and the optimized functions $ {f}_{\mathrm{left}} $ and $ {f}_{\mathrm{right}} $ , (red and purple) respectively. (e) Line profiles of the uncorrected signal (blue line) and corrected signal (brown line). (g–i) SEM images illustrating charge artifact suppression on myelinated sheaths found in mouse brain (image size 20.7 × 10.3 μm²). Data is EMPIAR-11415. (g) Original SEM image displaying copious amounts of charging. (h) SEM image with overlaid segmentation of charging centers and extending to complete rings of myelin sheaths. (i) Result after applying filter with default parameters.

A previous attempt to mitigate charging in SEM images presented by Spehner et al. ⁽Reference Spehner, Steyer and Bertinetti³⁾ utilizes the python scikit-image dilation/morphology function⁽Reference Robinson and Whelan³³⁾. This method separates the charging tail signals from the SEM images and then partially subtracts them from the original image. In our hands, this method did not work with the charging artifact tails in our datasets (Supplementary Figure S6), possibly indicating it is specific to the datasets it was developed for or to the instrument used to collect them.

Okapi-EM contains a charging artifact suppression tool⁽Reference Perdigão³⁴⁾ which is designed to restore the image contrast within and around charging artifacts, while retaining the charge centers themselves. As the artifact appears elongated along the direction of scanning (laterally) and is therefore influenced by the rastering nature of the scanning, it immediately suggests that to reverse this effect in the images, a row-by-row filter that uses information of the surrounding areas, in particular along the direction perpendicular to scanning, should be used to subtract the charging effect. With Okapi-EM’s chafer, the restoration method operates sequentially row-by-row in down and up passes, and the estimation of the charging artifact tails is done by fitting with a smoothing function, which is later subtracted.

For this filter to work a semantic segmentation of the charging centers must be provided as a separate layer in napari. Within the plugin user interface, there is a field where user sets this layer as being the annotation of the charging artifacts (Supplementary Figure S1B). This segmentation can be done manually or by using shallow or deep-learning⁽Reference Ronneberger, Fischer and Brox³⁵⁾ predictors using either tools within napari (as demonstrated here) or tools available elsewhere^(28, Reference Kremer, Mastronarde and McIntosh³⁶⁾ and then loaded in napari. In our experience, simple thresholding methods to label either charging centers or full artifacts do not work well due to the presence of other features of similar intensity. Correct annotation of the charging centers is crucial as it provides both an inverted mask of charging locations for correction and an indication of where the charging artifact is relative to the charging center, hence being useful for choosing initial values during optimization of the functions used (see below).

The filtering scheme works in the following way. First, it takes the previous rows (Figure 4a, red box) and averages perpendicular to the scanning direction (vertical here). This is used as an approximation of the signal without the charging effect. The difference between the current row (Figure 4a, blue arrows) and the previous-row average is assumed to be an estimation of the effect of the charging (Figure 4c–e, blue line). Simply subtracting this estimation of charging effects from the current row of data gives noisy results from row-to-row. Instead, the tails of the charging effect signal (artifact that is outside of the charging center) can smoothed or modeled by curves, and when subtracting this from the original signal, it gives results that are more consistent from scanning row to row. Throughout this data processing tasks, the data within the charging artifact region is not considered in the calculations; the charging artifact label in Figure 4b is used as a negative mask and also to direct the direction of the function to optimize in either side (see below).

We have trialed several functions that may fit and optimize best to the charging artifact tail signals. Exponential ( $ {Ae}^{- kx} $ ) and Gaussian functions ( $ A{e}^{-{x}^2/{\unicode{x03C3}}^2} $ ) can fit reasonably well to these tails but overall give poor results when reconstructing images, in particular at the function optimization step the Gaussian filter often fails to converge to suitable parameters (see Supplementary Figure S5). Some of the reasons why it fails to converge is that the signal is often flat in the rows above the charging artifacts, while other times, there are other biological features nearby that interfere with the function fitting, leading to unrealistic curve fitting parameters. Instead, we found that a shifted logistic sigmoid function (or Fermi-Dirac distribution type) works very well for this task. This function appears like a smoothed step function and is widely used in machine learning algorithms as an activation function, having the advantage that it “saturates” on either side of the curve, which is more characteristic of the charging artifact tails observed. Because the tails themselves were asymmetrical, the functions used to mitigate them were different for the left side and right side of the charging artifacts, and given by:

(3) $$ {f}_{\mathrm{left}}=A\left(\frac{1}{e^{\left(\frac{x-{x}_0}{\sigma}\right)}+1}-1\right),\hskip0.5em {f}_{\mathrm{right}}=A\left(-\frac{1}{e^{\left(\frac{x-{x}_0}{\sigma}\right)}+1}\right) $$

with $ A $ , $ {x}_0, $ and $ \sigma $ being parameters to be fitted, and functions $ {f}_{\mathrm{left}} $ fitted on the left of the labeled artifact, and $ {f}_{\mathrm{right}} $ fitted on the right side.

After optimizing $ {f}_{\mathrm{left}} $ and $ {f}_{\mathrm{right}} $ (Figure 4d, red and purple curves, respectively) these functions are subtracted from the row signal (excluding masked regions), resulting in a charge-mitigated signal that still represents the charging object (Figure 4e, brown). Running this process row-by-row in up-down and down-up passes results in a filtered image (Figure 4f), which is a significant improvement compared to Figure 4a, suppressing the main artifacts, while recovering some of the washed-out data previously hidden below.

In addition to SEM images of lipid droplets found in yeast cells, a second dataset featuring myelinated sheaths from mouse brains was used to test this process (Figures 4g–i). The manual segmentation of charging centers (Figure 4h) included both the charging centers and whole organelles, despite many regions not displaying charging artifact tails (including these noncharging regions has no negative impact, see Supplementary Figure S7). The filtered image (Figure 4i) demonstrates a substantial visual improvement to the image quality. We note that in regions near significant charging effects, although the contrast can be matched to the remaining image, there are no recoverable biological features present. This is particularly noticeable around the elongated diagonal biological feature on the right side of the image.

The main advantage of mitigating charging artifacts is the improved contrast of organelles near the charge centers. As such, both manual and automatic segmentation tools are expected to perform better when charging artifacts are absent, as these are notoriously sensitive to contrast changes which could lead to poor visualization and missing or misidentified organelles.

To better understand the effects of our charge mitigation scheme, we have compared the background in the vicinity of charging centers in corrected images (e.g., Figure 4f or i) to their uncorrected counterparts (e.g., Figure 4a or g), taking care to exclude regions that have been labeled as charging centers from the calculation. Through filtering, we expect to “uncover” biological features that were previously obscured by the charging tails and as such would expect a decrease in the standard deviation of the signal intensity around the charging center. The standard deviation of the signal intensity of the results shown in Figure 4f is 60% lower than the signal intensity in Figure 4a. Similarly, the standard deviation signal in Figure 4i (excluding labeled regions) is 58% less than in Figure 4g, suggesting the regions previously obscured by the charging artifacts are now in a more comparable range of signal intensity to the regions throughout the rest of the dataset. Through visual inspection, it is clear that while some information is recovered, at close proximity to the charging centers information is lost due to the presence of the charging artifact.

5. Resolution Estimation Using One-Image FRC

Measuring image quality enables users to evaluate the best imaging protocols for the requirements of their research question. Image resolution is a helpful measure of image quality as it can be directly related to physical dimensions, so it is intuitive to interpret. Resolution can be defined as “a maximum spatial frequency at which the information content can be considered reliable”⁽Reference Penczek³⁷⁾.

One method of determining image resolution is via a two-image Fourier Ring Correlation (FRC)⁽Reference Saxton and Baumeister³⁸⁾, which has been implemented in fluorescence microscopy. It is both a measure of resolution and consistency between images within a dataset. In this method, two independently acquired images of the same field-of-view are compared in the frequency domain to find the highest frequency where the images can be said to be similar. This is performed by determining the frequency where cross-correlation drops below a given threshold, often 1/7 = 0.143, as commonly employed in cryo-electron microscopy⁽Reference Rosenthal and Henderson^39, Reference van Heel and Schatz⁴⁰⁾. The requirement for two independently acquired images is challenging to fulfill as it cannot be done retrospectively, and repeated acquisition could introduce image artifacts (i.e., beam damage), affecting the resolution.

Koho et al. proposed a one-image FRC calculation based on subsampling a single image to produce pairs of images⁽Reference Koho, Tortarolo and Castello⁴¹⁾. This method arranges alternate pixels of an image in a checkerboard pattern to create subimages from a subset of pixels and then calculates the FRC between the subimages. However, in reality, the calculated FRC from these subimages is not equal to the value determined from two separate experimental images, but it can be calibrated (see below). A calibration function is then applied to match the one-image FRC from subimages to the gold-standard two-image FRC for the specific microscope and imaging conditions used.

The method of Koho et al. has been adapted for serial FIB/SEM imaging in a software tool called Quoll⁽⁴²⁾. Quoll is an open-source, user-friendly tool, and library to calculate the local resolution of single images with the one-image FRC. This method can be applied to any imaging modality once the calibration has been performed. The 2D image is first split into tiles, and the one-image FRC is calculated on each, returning a map of the spatial variation of resolution across the image, and a plot of its distribution. This process can be done on multiple 2D images within a 3D stack to understand resolution throughout the dataset. If large regions of featureless background or artifacts are present, it could skew the output as measurements would be taken on areas without appropriate levels of information present; these can be excluded through masking prior to assessment.

5.1 Calibration

The FRC curve obtained from the subsampling method of the one-image FRC is shifted from the gold-standard two-image FRC, where the same details in the one-image FRC curve are shifted to lower frequencies than they should be. As a result, resolution values reported by the one-image FRC are higher than the gold-standard two-image FRC, so calibration of the one-image FRC is required to match the resolution obtained by both methods. The rationale for this calibration is explained in further detail by Koho et al. in their original implementation of the one-image FRC resolution measurement⁽Reference Koho, Tortarolo and Castello⁴¹⁾. This is an instrument-specific calibration as it is affected by the noise generation of the instrument. The calibration dataset requires two repeated images of the same field of view taken at several pixel sizes/magnifications. Ideally, there should be no significant, artificial changes between the images such as artifacts or image shifts, as these could affect the two-image FRC that is calculated between them. If necessary, image registration can be used to correct shifts and choosing specimens which do not suffer from specimen degradation through repeated imaging is helpful (i.e., inorganic options such as gold or polystyrene beads).

In our recent cryogenic pFIB/SEM study⁽Reference Dumoux, Glen and Ho¹⁾, calibration was performed using images of polystyrene beads of 1 μm diameter (Abvigen, Newark, NJ) in cryogenic conditions. The images show the same field-of-view at 1.12, 2.25, and 4.5 nm pixel size. The final dataset consisted of six images, with a pair of images taken at each pixel size and SEM angle (angle between the SEM and the specimen face). The linear stack alignment with SIFT plugin in Fiji 2.3.0/1.53q was used to register each pair of images to each other, to ensure the exact same field-of-view was considered for both images⁽Reference Schindelin, Arganda-Carreras and Frise⁴³⁾. The images were cropped to cover the same physical field-of-view for all pixel sizes, 2000 × 2000 nm² for the 52° images and 1600 × 1600 nm² for the 90° images (Figure 5). The number of pixels in each pair of images was different due to the varying pixel size.

Figure 5. Calibration dataset of polystyrene beads used to calculate the gold-standard two-image FRC. These images were taken at 90° SEM angle. A region-of-interest (blue box) covering 2 × 2 μm² was used for the calibration, where the average one-image FRC and two-image FRC were calculated for these regions at all pixel sizes. Scale bars represent 2 μm.

The FRC curve (cross-correlation vs. frequency) was calculated from each pair of images, and normalized to a scale of 0–1, where 1 was the maximum frequency calculated (Figure 6a). This normalization enabled direct comparison of FRC curves between images. The normalized frequency at which the cross-correlation fell below 0.143⁽Reference Rosenthal and Henderson³⁹⁾ was taken as the reference resolution ( $ {r}_{\mathrm{ref}} $ ). The one-image FRC resolution was also calculated following the checkerboard sampling method of Koho et al., this was the $ {r}_{\mathrm{co}1} $ .

Figure 6. Calibration of the one-image FRC measurement to the gold-standard two-image FRC. (a) Two-image FRC curve and one-image FRC curves before and after calibration for the image pair at 4.5 nm pixel size at 52° SEM angle. (b) $ {r}_{\mathrm{co}1}/{r}_{\mathrm{ref}} $ versus $ {r}_{\mathrm{co}1} $ scatter plot (blue dots) and calibration curve $ {r}_{\mathrm{co}1}/{r}_{\mathrm{ref}}=2.067+0.099\log \left(0.085{r}_{\mathrm{co}1}\right) $ , where $ {r}_{\mathrm{co}1} $ is the average resolution measured from the one-image FRC for the image pair (Figure a, normalized frequency value at the location the curve crosses the correlation cutoff line of 1/7), and $ {r}_{\mathrm{ref}} $ is the resolution from the gold-standard two-image FRC measured from the image pair. Calibration shifted the uncalibrated one-image FRC curves along the frequency axis to match the two-image FRC curves, ensuring that the resolution measurement for the one-image FRC matches the gold standard.

A calibration function was fitted to the plot of $ {r}_{\mathrm{co}1}/{r}_{\mathrm{ref}} $ against $ {r}_{\mathrm{co}1} $ (Figure 6b). This calibration function shifted the one-image FRC curve to match the two-image FRC, so that the resolution from both methods were comparable. This calibration function is instrument-specific, so it can be reused for any images taken from that instrument, provided that the image acquisition parameters are within the range of the calibration dataset.

5.2 Application to cryogenic serial pFIB/SEM data

Quoll was used in a recent publication to develop cryogenic serial pFIB/SEM imaging of biological specimens⁽Reference Dumoux, Glen and Ho¹⁾. Here, Quoll resolution measurement was demonstrated on five different biological specimens (R. rubrum, HeLa cells, Vero cells, mouse brain, Saccharomyces cerevisiae). The measurements were used to show that imaging at 90° SEM angle produced better results than 52°, to determine the depth of field of the instrument, and to show that there was no degradation of image quality through serial sectioning of the specimen, even at depths of ≈25 μm. These performance evaluations would not have been as accurate and exhaustive without the quantitative measurements from image resolution estimation with Quoll.

The image resolution reported by Quoll was validated against biological structures with known physical sizes. The local image resolution was calculated for a cryogenic serial pFIB/SEM image of HeLa cells on tiles measuring 256 × 256 pixels, which corresponded to a physical field-of-view of 1.73 × 1.73 μm². These images contained nuclear pore complexes and centrosomes, which are approximately 120 and 200 nm in diameter, respectively⁽Reference Lin and Hoelz^44, Reference Alieva and Uzbekov⁴⁵⁾. These structures were clearly resolvable in the images, and resolution in the tiles containing these structures surpassed the known diameters of these structures, validating the resolution measurements (Figure 7). FRC measurements are carried out on tiles within image slices and the reported measures provide a plot of the mean and standard deviation of resolution across the slice, as well as a heatmap for visualization. It is important to emphasize that these values are not the highest resolution visible within the slice, but instead a representation of the mean resolution of each tile. The goal of this method is to provide a quality metric for the raw data as a whole.

Figure 7. Validation of Quoll resolution measurements compared to physical features of known sizes in HeLa cells. Images were taken with 6.745 nm pixel size at 52° SEM angle. Nuclear pore complexes (a,b) of 120 nm diameter were clearly resolved in the images, as indicated by the arrows in (b), and the resolution in this region was estimated as better than the resolvable features. Similarly, in (c) and (d), centrosomes of 200 nm diameter could be resolved from the images (indicated by the arrow in d), and resolution was better than the size of the centrosomes. (a,c) A resolution heatmap is overlaid onto regions of the image showing nuclear pore complexes and centrosomes respectively, where the colors represent the resolution values of that local region and numbers are the local resolution values in nm calculated on the respective rectangular regions. (b,d) are the zoomed-in regions indicated in (a) and (c) with a red square, with arrows indicating the position of the nuclear pore complexes and centrosomes, respectively. Data is EMPIAR-11419.

We found that tile size does not affect the overall resolution distribution of the image. The local image resolution was calculated on cryogenic pFIB/SEM images of S. cerevisiae and R. rubrum, at 3.37 and 1.94 nm pixel sizes, respectively. The images were sampled with tile sizes of 128 × 128, 256 × 256, and 512 × 512 pixels. The difference in minimum and maximum median resolutions for all tile sizes was within 0.38–0.48 pixels. The Kruskal–Wallis H-test was applied with the null hypothesis that the population median of all groups was equal⁽Reference Kruskal and Wallis^{46, 47)}. The null hypothesis could not be rejected (p > 0.05) so the median resolution for all three tile sizes was considered equal (Table 2).

Table 2. Summary statistics of resolution distribution measured from different tile sizes.

Note. Tile size was not found to affect the overall resolution distribution of the image. The resolution distribution median was found to be equal for all tile sizes (p > .05) by the Kruskal–Wallis H-test. All measurements are rounded to three significant figures.