Optimizing Nonrigid Registration for Scanning Transmission Electron Microscopy Image Series

Abstract Achieving sub-picometer precision measurements of atomic column positions in high-resolution scanning transmission electron microscope images using nonrigid registration (NRR) and averaging of image series requires careful optimization of experimental conditions and the parameters of the registration algorithm. On experimental data from SrTiO3 [100], sub-pm precision requires alignment of the sample to the zone axis to within 1 mrad tilt and sample drift of less than 1 nm/min. At fixed total electron dose for the series, precision in the fast scan direction improves with shorter pixel dwell time to the limit of our microscope hardware, but the best precision along the slow scan direction occurs at 6 μs/px dwell time. Within the NRR algorithm, the “smoothness factor” that penalizes large estimated shifts is the most important parameter for sub-pm precision, but in general, the precision of NRR images is robust over a wide range of parameters.


Introduction
Scanning transmission electron microscopy (STEM) can routinely obtain images with sub-Ångstrom spatial resolution. With appropriate processing, the positions of atomic columns in these images can be measured much more precisely than the resolution, with picometer-and even sub-picometer precision reported using various approaches (Jones & Nellist, 2013;Sang & LeBeau, 2014;Yankovich et al., 2014;Jones et al., 2015;Ophus et al., 2016). The precision here is referred to as the random variation in measurements, not the systematic difference between measurement and the truth, which is referred to as accuracy. Although the measured values could be off from the true values, we can still use the relative values to study small strains in crystalline materials one atomic column at a time. This has the potential to impact materials research on topics, including ferroic distortions (Lee et al., 2015;Rojac et al., 2016), microstructural strains (Oni et al., 2015;Tang et al., 2015), octahedral rotation angles (Borisevich et al., 2010;Wang et al., 2016), catalytic activities (Nilsson Pingel et al., 2018), and characterization of point defects (Hwang et al., 2013;Feng et al., 2017).
The precision of atom positions determination is influenced by two factors: image signal-to-noise ratio (SNR) (Van Aert et al., 2002;De Backer et al., 2013 and image distortions (Jones & Nellist, 2013;Sang & LeBeau, 2014). With a low-noise detector like most modern STEM detectors, SNR is determined by the electron dose, which is proportional to the exposure time at fixed beam current. Distortions in STEM refer to the actual probe positions on the sample deviating from the expected perfect regular grid, and they arise from two sources: sample drift during acquisition and scan distortion. As STEM images are acquired in a pixel-by-pixel serial scan, sample drift means the sample is in different positions during the acquisition of different pixels in the frame. As a result, the final image can be sheared and sometimes uninterpretable, especially when the frame exposure time is long. Scan distortions include the random offset deviations at the beginning of each row in the row-wise scan pattern, which displace the beam positions of a whole scan row and result in flagging and skipping artifacts in the collected image (Jones et al., 2017). More random distortions effecting small groups of pixels can also result from floor vibration, acoustic noise in the microscope lab, and higher-frequency instability in the scan electronics (Muller et al., 2006).
Both high electron dose and minimum scan distortions are necessary to obtain a high-precision image. A common way to achieve both is to acquire a series of fast acquisition frames in which every single frame has small distortion, register the frames to one another to computationally compensate for sample drift, then average the registered frames to get a single high-quality image. Kimoto et al. (2010) reported several pm precision in atom positions detected in annular dark-field (ADF) images using sample drift detection and compensation. Sample drift was detected using cross-correlation and compensated using rigid shifts that move all the pixels in a single frame by the same vector. Rigid registration effectively captures overall sample drift but does not capture the local distortions arising from pixelwise STEM acquisition or scan distortions. developed a nonrigid registration (NRR) to align STEM images which can compensate for the effects of sample drift and scan distortions, which Yankovich et al. (2014) used to achieve sub-pm precision imaging from series of high-angle annular dark-field (HAADF) images. In NRR, every pixel in each frame has its own shift vector, which better fits the serial, pixel-by-pixel acquisition scheme of STEM. NRR can, in principle, undo all of the forms of drift and distortion, if the appropriate pixel-by-pixel distortions can be found. However, estimating the distortions if all possible shifts of every pixel are allowed is quite challenging, so some information about what kinds of distortions are present must be built into the algorithm. The details of prior information and the algorithmic implementation are discussed elsewhere Berkels et al. (2014) , but they include general assumptions about the imaging process such as that scan distortions have zero mean over many frames, and that large-magnitude shifts must effect many pixels because they arise from slow processes like sample drift, while small-magnitude shifts may affect smaller numbers of pixels because they arise from fast processes like electronic instability. The NRR algorithm was later improved by Berkels & Liebscher (2019) to reduce the bias toward preserving distortions in the keyframe to which all the other images are registered.
Other related methods have achieved similar sub-pm precision to NRR by acquiring, registering, and averaging a series of images (Braidy et al., 2012;Jones & Nellist, 2013;Sang & LeBeau, 2014;Jones et al., 2015;Ophus et al., 2016;Ning et al., 2018), but they vary in the prior knowledge applied to the registration. Some methods focus on correcting flagging and skipping artifacts by using prior knowledge of the features in atomic resolution input images. By assuming a periodic structure and using the strong peaks in Fourier space from the periodic structure as a reference, Braidy et al. (2012) detect and correct the artifacts using phase maps generated from the streaks in the Fourier space. By assuming a smooth atomic resolution image with round-shaped atoms, Jones & Nellist (2013) correct flagging by minimizing the difference between consecutive rows, and correct skipping by reordering rows to generate round atoms. By assuming the distortion patterns are similar within each fast scan line and can be fitted by a linear function, Jones et al. (2015) improved upon NRR methods by applying a row-fitting constraint to better fits and correct the random offset deviations at the beginning of each row scan. These methods are better at correcting flagging and skipping than NRR, but they can only be applied to images that contain features matching their assumptions.
Another path to adding prior information is to rotate the scan axes with respect to the object while acquiring the series. The object stays the same (an assumption underlying all of these series methods), as does the sample drift, but they appear to rotate in the scan reference frame. Flagging, skipping, and other scan distortions stay tied to the scan reference frame. Sang & LeBeau (2014) developed RevSTEM, which uses many scan rotations, and can correct very large linear drift as well as distortions. RevSTEM can also improve accuracy (as opposed to precision) by using the extra information from rotations to improve measurements of lattice spacings and angles. Ophus et al. (2016) used a similar approach that uses prior knowledge of the particular form of flagging and skipping errors (shifted lines and reordered lines, respectively) and two images with orthogonal scans to correction nonlinear drift, flagging and skipping. Ning et al. (2018) further refined the method for atomic resolution image pairs by emphasizing the matching of atomic site positions during drift and distortion corrections. These methods generally do not rely on prior knowledge of image features but do require additional steps, and sometimes software, in image series acquisition. With automated data acquisition growing rapidly, data acquisition for these purposes are getting easier with time.
Here, we focus on optimizing experimental and algorithmic parameters for obtaining sub-pm precision final images from the HAADF STEM image series of a SrTiO 3 [100] single-crystal sample aligned with the Berkels NRR algorithm. Experiment requirements for high-quality input image series are described in the "Experiment" section, with the NRR algorithm and the definition of precision introduced in the "NRR and Precision Analysis" section. Examples of NRR results are shown in the "Registration Results" section. Registration parameters are tested and discussed in the "Registration Parameters" section, and the "Visualization of Artifacts" section focuses on directly visualizing the registration between frames and possible artifacts from problematic parameter selections. In the end, the "Selection of Pixel Dwell Time/Frame Dose" section discusses how to use images series and Berkels NRR under a given fixed electron dose effectively.

Experiment
A SrTiO 3 [100] single-crystal sample was prepared by wedge polishing at a 1.6°angle using an Allied MultiPrep Polishing system with diamond lapping films, using lapping film particle sizes decreasing from 15 to 0.1 μm step by step. The sample was then ion milled from both sides at 5°angle in a Fischione Model 1050 TEM mill using an Ar ion beam. A 3 kV ion beam was used at the beginning step, followed by successively decreased voltages ending at 100 V. The sample was kept under vacuum and plasma cleaned before being inserted into the TEM column. The column and the inserted sample were cleaned with an on-column GV10x Gentle Asher plasma cleaner to reduce contamination during the long acquisition of the image series.
A Thermo-Fisher Titan STEM equipped with a CEOS probe aberration corrector operated at 200 kV was used to collect image series for NRR. STEM image series were collected with a 24.5 mrad probe semi-angle and 18.9 pA probe current. 256 by 256 pixel HAADF STEM image series were acquired on a Fischione Model 3000 detector spanning 53.9 and 270 mrad. 8 μs pixel dwell time (1.99 × 10 4 e − /Å 2 frame dose) was used as a standard dwell time for experiments on sample tilt and optimization of NRR parameters. 0.5-12 μs (1.24 × 10 3 -2.99 × 10 4 e − /Å 2 dose/frame) pixel dwell times were used to study the precision as a function of dwell time, with a variable number of frames to maintain constant dose. Line synchronization of the scan was not used during acquisition, as we want the distortion to be different in each image and thus average out of the series, rather than being the same because it is synchronized to the line frequency.
Large linear drift during the long total series exposure time required to achieve sub-pm precision will limit the field of view of the high-precision averaged image, as only the area that is inside every image in the image series will be used in the final registered image. The field of view of the final registered image can, of course, be adjusted by changing the number of points in the scan. Constant linear drift that is the same in every image of the series also violates the assumption of NRR that distortions have zero mean, resulting in systematic errors and poor accuracy in the registered image (Sang & LeBeau, 2014). Therefore, the sample was stabilized for more than 3 hours to minimize sample drift before image series acquisition. Drift better than 1 nm/min was routinely achieved, and drift better than 1 Å/min was achievable for an extremely stable sample if stabilizing long enough. A thin and uniform area with about 20 nm thickness was used to acquire the HAADF image series. The fast scan direction (horizontal rows in the images) is aligned with the [010] direction of SrTiO 3 lattice, and standard deviations of Sr column spacings along [010] and [001] directions are defined as image precisions on the fast scan and the slow scan direction. Residual aberrations were corrected manually to get round-shaped atomic columns with no elongations and a good contrast between atomic columns and the background on the atomic resolution image.

NRR and Precision Analysis
The NRR algorithm registers a target frame to the keyframe by finding an optimal set of deformations ϕ NR , which consists of a separate vector for each pixel in the target frame. The deformation is supposed to capture the sample drift between the acquisition of the target frame and the keyframe. The deformation is determined by optimizing an energy function which contains the negative normalized cross-correlation between the registered/deformed target frame and the keyframe (first term) and the Dirichlet energy of the deformation (second term). The first term quantifies the difference between the keyframe and the registered target frame, and the second is a regularizer that penalizes abrupt changes in the deformation (Modersitzki, 2004). The smoothness term prevents NRR from simply duplicating the distortions that exist in the keyframe, as do other aspects of the algorithm described elsewhere Berkels & Liebscher, 2019). The smoothness term also means that the NRR is less effective at correcting discontinuous distortions like flagging and skipping than more specialized approaches (Braidy et al., 2012;Jones & Nellist, 2013), but in practice, these effects are canceled out by averaging without significant signs of loss of resolution (Yankovich et al., 2014). A loss of resolution could be more obvious if the flaggings and skippings are more serious. The image domain Ω is normalized to have the area of a unit square, so that the number of pixels on the image will not change the value of the energy function. A smoothness factor λ is used to balance the weights of the two terms. The energy function is minimized using a multilevel, regularized gradient descent method to find the optimal deformation . "Multilevel" means that the images are first downsampled, for example, from 256 × 256 to 64 × 64 px. The downsampled versions are registered, and the deformations from low pixel count registration are used as initial guesses for the registration at the next finer level, iterating back up to the original number of pixels. Deformations of large structures covering more pixels, caused by example by sample drift, are captured at low pixel count. Deformations of small structures affecting fewer pixels, caused for example by acoustic noise, are captured at high pixel count.
After the whole image series is registered, registered frames are averaged to produce one single high-quality frame with frame dose equal to the total dose of the image series. This frame is then cropped to retain only the region that remains in the STEM field of view throughout the acquisition, making it smaller than a single raw frame and often rectangular. Sr atom sites in the final image are used for precision analysis. The atomic column positions are determined by performing least-square fittings around each Sr site, which fits each Sr peak to an asymmetric 2D Gaussian function with a constant background (Yankovich et al., 2014).
Precision in this study is defined as the standard deviation of a repeated, known crystallographic distance in the image (Bals et al., 2006). Strain measurement has also been used as a metric for precision, in which case a small measured strain in a nominally strain-free crystal means high precision (Jones et al., 2017). Both metrics assume that the lattice is perfectly periodic, and any aperiodic structure is a result of an imperfect image. Both metrics can only be applied to perfect single-crystal samples without impurities or strain introduced by sample preparation.

Registration Results
Figures 1a to 1c show the results from an on-zone high-quality image series acquired on an SrTiO 3 [100] single-crystal sample. The registration of 500 frames took 143 min using a singlethreaded implementation on a single Intel Xeon E5-2660 CPU processor. Figure 1a is an example of a single frame in the image series, which has limited SNR and some clear slicing artifacts from scan distortions. Figure 1b is the final image after NRR with improved SNR and no visual sign of residual scan distortion. It has 0.57 pm precision along the fast scan direction and 0.77 pm precision along the slow scan direction. The precision level is similar to the previous result under a similar high electron dose reported by Yankovich et al. (2014) on Si [110] single-crystal sample. The somewhat poorer precision along the slow scan direction could be from a long time between pixels (the line acquisition time, rather than the pixel dwell time), but this is inconsistent with the results in the "Selection of Pixel Dwell Time/Frame Dose" section below. Instead, we believe it may arise from skipping that is not corrected by the registration, or the fact that the drift was primarily in the y-direction. A further investigation would be required to determine the exact cause. Figure 1c shows the position averaged convergent beam electron diffraction (PACBED) pattern acquired from the same region together with the image series, which is very close to the [100] zone axis and shows a highly symmetric pattern. By estimating the distance between the center of Bragg diffractions and the center bright disk, we estimate the tilt from [100] zone axis to be less than 1 mrad. Figures 1d to 1f show a single frame, the final image, and the PACBED pattern acquired with the sample tilted by ∼6.3 mrad from the [100] zone axis, estimated by eye from the PACBED pattern. At this tilt, the atomic columns in Figures 1d and 1e show clear elongation along one direction and reduced contrast between the bright atomic columns and the dark background. Figure 1e has 1.67 pm precision along the fast scan direction and 1.79 pm precision along the slow scan direction. Precision is a measure of random displacements, so the reduced precision should not arise from the image distortion in Figure 1e. The loss of precision instead might be caused by the loss of resolution or reduced signal caused by the off-axis tilt. It is difficult to control the tilt precisely to perform a systematic experiment as a function of tilt along a particular direction, but experience across many experiments at different random tilts suggests that sub-pm precision is only obtained for off-zone tilts of 1 mrad or less.

Registration Parameters
The NRR algorithm has five tunable parameters: the smoothness factor λ shown in equation (1), and parameters controlling the multilevel registration scheme, the number of registration iterations, and the convergence criterion. A detailed description of these parameters can be found in Berkels et al. (2014) . Table 1 lists all the parameters with a brief definition, the allowed range of values, and a recommended value for each one derived from parameter optimizations by varying one parameter at a time with the other parameters fixed. Each parameter's effect on the precision of the final image after NRR on the 8 μs dwell time SrTiO 3 data was tested by varying one parameter at a time with the other parameters fixed at the recommended value. Figure 2 shows the effect of all five parameters on the precision of the final registered image. The smoothness factor has the most noticeable effect on the final precision, as shown in Figure 2a. Result suggests that a λ above 100 is enough to achieve sub-pm precision. A smaller λ causes worse precision and, as shown in the "Visualization of Artifacts" section, obvious artifacts in the deformations. Precision does not improve significantly with λ > 200. The fairly large optimal λ suggests that the dominant drift pattern in the images is smooth over a significant number of pixels. We recommend a default λ of 200. A similar selection of λ value has been reported in Berkels & Liebscher (2019). Figure 2b shows precision from different registration start levels. The start level controls the downsampling in the multilevel registration (Briggs et al., 2000;Modersitzki, 2004;Han et al., 2007;Hackbusch, 2013), which is used to avoid unwanted local minimum (described in detail in Berkels et al. (2014)). The NRR algorithm takes square images with side lengths in pixels a power of two (2 n along each side), and the downsampling procedure is processed by a factor of 2 at each step. With given start level k, the side length in number of pixels used at the beginning of the registration is where n original is the side length of the raw image series. Sub-pm precisions are achieved under all different start levels. When  Controls the relative importance of the smoothness term in the energy, equation (1) >0 200 Start level Images for the first level of registration will be downsampled to 2 start level × 2 start level pixels An integer between 1 and n for input images of 2 n × 2 n pixels n−1 λ multiplier Multiplier to λ for registration of data on a grid downsampled by a factor of 2 >0 5 Convergence criterion Stopping threshold for energy minimization > 0 10 −6

Total iterations
The number of total NRR iterations including the initial stage and following refining stages.
Integer ≥1 3 using a small start level, the image at the beginning of the registration may lose all the features that could be used for registration, especially for atomic resolution images on the periodic lattice structure. In this case, a possible registration artifact is a large jump by half or a full repeating unit (Savitzky et al., 2018). An example of this artifact will be discussed in the "Visualization of Artifacts" section. We recommend a default start level of n−1. However, if the pixel size is much smaller than what we are using (21.5 pm/px), a smaller start level might be beneficial or even necessary. Figure 2c shows the effect of the λ multiplier. Downsampled registrations are performed with a smoothness factor of (λ multiplier) r × λ, where r = log 2 (n/n downsampled ) is the number of times the original data have been downsampled by a factor of 2. A λ multiplier larger than 1 means the smoothness of deformation is emphasized more when registering with a coarser pixel grid. This enforces the prior knowledge that large spatial-scale deformations computed on downsampled images must be smoother than smaller spatial-scale deformations computed on the original pixel sampling. Result suggests that a λ multiplier larger than 1 is preferred, but that the final precision improves only slowly with λ multipliers larger than 1. We recommend a default λ -factor of 5. Figures 2d and 2e show the effects of the convergence criterion and the total iterations. The convergence criterion controls when to stop the optimization (Sundaramoorthi et al., 2007) of energy function [1]. The number of total iterations controls the number of times the algorithm registers the image series, including the initial stage, and re-registration stages using the output of the previous registration as the initial guess for the current registration. Results suggest that a convergence criterion of 10 −6 with a total of 3 iterations is enough for sub-pm precision. Figure 3 shows examples of nonrigid pixel deformations determined by NRR using different parameter settings. In each case, NRR seeks the deformation to register the target frame shown in Figure 3b to the keyframe shown in Figure 3a. Figures 3c, 3e, and 3g show the vector plots of nonrigid pixel deformations determined under the default parameter, a (too) small start level, and a (too) small lambda value, respectively. All three plots are shown at reduced pixel density (344 pm spacing between vectors) for the whole frame. Figures 3d, 3f, and 3h show the deformed frames after applying the deformation. Figures 3c and 3d show the deformation under optimal conditions. The vector field is smooth, primarily capturing the effects of sample drift, with small contributions from higher-frequency distortions that are not really visible in the figure. On a stable sample in thermal equilibrium with our microscope, the sample drift is a random walk with no consistent direction, which leads to the smoothly varying but not constant deformation field in Figure 3c. Figure 3d shows the registered image patch where the whole registered image patch moves a few pixels to match the keyframe shown in Figure 3a, but it does not simply duplicate all the features in the keyframe such as the slicing artifacts from scan distortions. Figures 3e and 3f show the deformation and registered image patch when the image was downsampled too much at the beginning step. The image with 256 px side length was downsampled to 8 px at the beginning of registration, resulting in an image series with 688 pm pixel size at the beginning stage of registration, larger than the lattice parameter of SrTiO 3 (390.5 pm). In this case, each pixel in the image was larger than a single unit cell, and the whole downsampled image had little to no features to be registered. As a result, the NRR algorithm made a half unit-cell jump that registered the Sr atom sites in the target frame to Ti sites in the keyframe. Note that this artifact does not occur for a reasonable choice of start level that is sufficiently large like the one we recommended in the "Registration Parameters" section, and also could be avoided if the image had a large-scale feature such as an interface, surface, or grain boundary, which provides registration references at large pixel sizes. Figures 3g and 3h show the deformation from a small λ value, in which NRR determined a deformation that is not smooth. The registered image patch in Figure 3h looks very similar to the keyframe, but it preserves all the scan distortions, as atomic columns in the registered image have exactly the same slicing artifact as the keyframe. This represents overfitting since we want to average out rather than preserve high-frequency distortions. We also suspect that some pixels could be shifted to match pixels that are bright in the keyframe due to random noise, not variations in sample scattering. Preserving scan distortion and noise reduces the precision in the averaged image.

Selection of Pixel Dwell Time/Frame Dose
High-precision STEM via series acquisition and NRR is inherently a high-dose method, but the total electron dose a given sample can sustain is often limited (e.g., Zhang et al., 2018;Kotakoski et al., 2010). There are two different ways to split a given dose into multiple frames at fixed pixel sampling: (1) Use a longer pixel dwell time and fewer frames. In this case, each frame has a higher SNR and clear features for registration, but each pixel can average over faster distortions that cannot be corrected by NRR, and the series contains a small number of samples of distortions.
(2) Use a shorter dwell time and more frames (Jones et al., 2018). In this case, each frame has lower SNR and less clear features, but each one provides a better sampling of the distortions at different time scales, and the series contains a larger number of distortion samples. Dose can also be varied by using large pixels to reduce the dose or smaller pixels to increase the dose (Jones, 2016). High signal-to-noise images at a much lower dose than used here can still be achieved with very large pixels, but the precision suffers: ∼4 pm, rather than <1 pm (Yankovich et al., 2015). The effect of pixel size has not been studied systematically here.
Different ways to use a fixed total electron dose budget have been studied on a SrTiO 3 [100] image series acquired with frame doses ranging from 1.2 × 10 3 to 2.5 × 10 4 e − /Å 2 , varied by using pixel dwell times from 0.5 to 12 μs at a fixed beam current. Pixel dwell times shorter than 0.5 μs/px were not tested as they cause large, systematic scan distortion on our microscope (Buban et al., 2010), and images could be affected by the scintillator afterglow Krause et al., 2016). Five different total electron doses ranging from 2 × 10 5 to 12 × 10 5 e − /Å 2 were studied by varying the total numbers of frames used. All the image series were registered with recommended parameters as described in Table 1. Figure 4 shows precision along the fast scan direction and the slow scan direction versus pixel dwell time for the five different total doses. All these data were acquired with a relatively low electron dose compare to the results shown in the "Registration Results" and "Registration Parameters" sections, which is why we do not have sub-pm precision on all of these image series. Both scan directions show improved precisions with a higher total dose under all different frame doses. Under a fixed total electron dose, a short pixel dwell time of 0.5 μs/px (1.25 × 10 3 e − /Å 2 frame dose) is best for the fast scan direction precision. Even shorter dwell time might be better, but cannot be tested on our microscope. A similar trend has been reported in Jones et al. (2017) on a different instrument and using a different NRR algorithm. We suspect that faster scanning means improved precision because faster scanning means that the images capture, rather than averaging over, a broader frequency range of distortions. A longer pixel dwell time of 6 μs/px (1.5 × 10 4 e − /Å 2 frame dose) is best for the slow scan direction precision. We speculate that this effect arises because of some imperfection in the scan that grows worse at short pixel dwell time, like the random vertical offset of the pixels created by skipping, but understanding the exact reason requires further investigation. The fast scan direction has a local minimum in the precision at 8 μs, which might arise from the same source. At longer than 10 μs/px, precision along both scan directions grows worse. We speculate that this is a result of longer acquisition times averaging over fast distortions within each frame, making them uncorrectable by NRR. On balance, we recommend 6 or 8 μs pixel dwell time as optimal for highprecision imaging using NRR. However, if the optimal pixel dwell time is influenced by microscope nonidealities and instability as we guess, it will vary from instrument to instrument and lab to lab. This work here outlines an approach to determine this for other microscopes.

Conclusion
Experimental requirements and NRR algorithm parameters to achieve sub-pm precision have been studied using atomic resolution HAADF image series acquired on an SrTiO 3 [100] singlecrystal sample. Sub-pm precision requires less than 1 mrad offzone tilt and less than 1 nm/min sample drift. The smoothness factor parameter is the most important to precision after NRR, and a sufficiently large value (larger than 100 in this study) is shown to be the key factor for high precision. We derive the recommended default values for all the parameters as a result of our parameter study. Large deviations from these parameters can create artifacts in the registration, while in general, NRR precisions are robust over a wide range of parameter selections. When applying NRR to image series with fixed total electron dose, the precision is best along the fast scan direction at the shortest accessible pixel dwell time, but the precision along the slow scan direction is best at a pixel dwell time of 6 μs/px. On balance, 6-8 μs/px is the optimal pixel dwell time. With these conditions and careful acquisition, sub-picometer precision on samples that can sustain a high electron dose is routinely achieved.

Availability of data and materials
All the raw data, registration parameter files, registration results, and code for data analysis are available via the Materials Data Facility https://doi.org/10.18126/1h93-n56i. Code for data analysis are also available on Github (https://doi.org/10.5281/zenodo. 3723363). The NRR software can be accessed on the Nanohub online app: https://nanohub.org/tools/nrr. NRR source code is owned by BB and is released on https://github.com/berkels/ match-series.