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mpfit: a robust method for fitting atomic resolution images with multiple Gaussian peaks

mpfit: a robust method for fitting atomic resolution images with multiple Gaussian peaks The standard technique for sub-pixel estimation of atom positions from atomic resolution scanning transmission electron microscopy images relies on fitting intensity maxima or minima with a two-dimensional Gaussian function. While this is a widespread method of measurement, it can be error prone in images with non-zero aberrations, strong intensity differences between adjacent atoms or in situations where the neighboring atom positions approach the resolution limit of the microscope. Here we demonstrate mpfit, an atom finding algorithm that iteratively calculates a series of overlapping two-dimensional Gaussian functions to fit the experimental dataset and then subsequently uses a subset of the calculated Gaussian functions to perform sub-pixel refinement of atom positions. Based on both simulated and experimental datasets presented in this work, this approach gives lower errors when compared to the commonly used single Gaussian peak fitting approach and demonstrates increased robustness over a wider range of experimental conditions. Keywords: Peak refinement, BF-STEM imaging, Sub-pixel resolution, Aberration-corrected STEM Introduction STEM datasets [5, 9–14]. This technique has been used The development of spherical aberration-correction for for quantitative atomic displacement measurements scanning transmission electron microscopy (STEM) across thin films, 2D crystals, domain boundaries and has imaging has been one of the biggest triumphs of electron allowed the experimental observation of novel structural microscopy over the past several decades, allowing the phenomena such as polar vortices [15–20]. sub-ångström resolution imaging of crystal structures While the Gaussian function fitting approach is an [1–3]. Several pioneering STEM experiments have dem- extraordinarily powerful technique, one noted short- onstrated the feasibility of this technique for the direct coming is that it assumes well-separated atoms with no visualization of atom positions from aberration-corrected overlap, or negligible aberrations in the beam itself—con- STEM images and has proved itself an invaluable tool for ditions that are only available under a certain limited set sub-ångström resolution structural measurements [4–8]. of imaging conditions [16, 17]. Typically, such an imaging While the typical aberration-corrected STEM electron setup uses a ring shaped annular detector with the outer beam has a probe diameter approximately between 0.5 and inner detector collection circles centered along the and 1  Å, supersampling scanning positions below the microscope optic axis. Such a configuration will have an Nyquist–Shannon sampling limit and the subsequent fit - inner collection angle of approximately 85–90 mrad to ting of the probe image with a two-dimensional Gauss- capture only the incoherently scattered electrons, and ian function allows the sub-pixel precision assignment is conventionally referred to as high angle annular dark of atom column positions from aberration-corrected field STEM (HAADF-STEM) imaging [5, 21]. This mode of imaging is referred to as dark field imaging since atom columns themselves are bright due to electrons preferen- *Correspondence: alem@matse.psu.edu tially scattering from atomic nuclei as a consequence of Department of Materials Science & Engineering, The Pennsylvania State Rutherford scattering from proton–electron Coulombic University, University Park 16802, USA Full list of author information is available at the end of the article forces [22, 23]. Since this Coulombic force experienced © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 2 of 12 by the electron probe is directly proportional to the num- intensity distribution with accuracies approaching ber of protons in the nucleus (Z), atom column images in 0.5  pm [14, 17]. It is this combination of aberration- HAADF-STEM datasets generate peaks with an almost corrected imaging and Gaussian peak fitting that has linear relationship of intensity ∝ Z with the atomic enabled modern electron microscopy to reliably meas- number and is also referred to as Z-contrast imaging ure domain walls, grain boundaries, defects, and strain [24–26]. with single picometer precision, making STEM imaging Z-contrast imaging, however, is generally considered so powerful. unsuitable for imaging lighter elements such as oxygen, However, this approach runs into problems when boron or carbon [19–21]. However, structural metrol- applied to BF-STEM imaging. In Fig. 1a, we show a typ- ogy for many scientifically important material systems ical BF-STEM image of LiNbO with 4.9  pm scanning such as ferroelectrics needs the imaging and quantifica - pixel sizes. The darker regions in the image are the nio - tion of lighter atoms as well as heavier elements [27, 28]. bium and oxygen atom columns with the red dots cor- This problem can be significantly mitigated in bright field responding to the intensity minima. While the intensity STEM (BF-STEM) imaging, where rather than annular minima can be used as an initial estimate of atom posi- detectors a circular detector is used with the detector tions, the error in such a measurement is at least of center coinciding with the optic axis of the microscope the order of the pixel size, which is 5  pm in our case. [19, 29]. The conventional collection angle ranges in BF- This makes the error of measurement in BF-STEM an STEM imaging extend up to 15 mrad, significantly lower order of magnitude worse than the best HAADF-STEM than even the inner collection angle for HAADF-STEM results. Figure  1b demonstrates the same section of [29]. Because unscattered electron beams are imaged by the BF-STEM image with the refined atom positions this technique, in contrast to HAADF-STEM vacuum is obtained from fitting the intensity distribution with a bright, while the atom positions have comparatively lower single Gaussian peak with green dots next to the inten- intensity. The ideal BF-STEM image would thus have an sity minima (red dots). A visual estimation shows that intensity profile complementary to the images obtained the fitted Gaussians do not reliably converge on the from HAADF-STEM imaging. However, in reality owing atom positions, and are often tens of picometers away to the lower collection angles, atom positions are more when the intensity minima is weak, and the neigh- blurred from aberrations that are more prominent in BF- boring atom is close. In some cases, the refined atom STEM images [30]. Additionally, since BF-STEM images position is in the middle of the two neighboring atom capture both light and heavy atom positions the inter- columns with no definite atomic intensity. atomic distances are substantially smaller. These effects result in atom positions that are non-Gaussian in shape, and often have intensity overlaps and tails coming from their neighbors making position metrology challenging in BF-STEM images. Methods Fitting atom positions with Gaussians The best modern aberration-corrected microscopes can generate electron probes that are free of aber- rations up to 30  mrad, which corresponds to beam diameters that are of the order of 0.5 Å, or 50  pm at 200 kV [8, 10]. Super-sampling the beam by a factor of five results in scan positions that are spaced approxi - mately 10  pm apart from each other. For HAADF- STEM images where oxygen atoms are not observed, inter-atomic distances from the low index zones are mostly of the order of 1.5 Å, allowing enough distance Fig. 1 Error with single peak fitting on experimental data. a BF-STEM between atoms so that they are well separated and thus image of LiNbO with the red dots referring to the intensity minima. b an atom position can be reasonably approximated with BF-STEM image shown in a with the intensity minima and single peak fitting results overlaid in red and green, respectively. c Intensity profile a two-dimensional Gaussian intensity profile. Since along the arrow shown in a and b with the red arrows referring to the the FWHM of this Gaussian is around 50–75  pm, this intensity minima and the green arrows referring to the single peak fits allows the determination of the peak of the Gaussian Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 3 of 12 This can be quantitatively demonstrated by profiling refinement requires less stringent conditions, our algo - the summed intensity distribution (Fig.  1c) from the rithm extends equally well to such systems too. region shown along the white arrows in Fig.  1a, b. The red arrows in Fig.  1c correspond to the intensity min- The mpfit algorithm ima, while the green arrows correspond to the Gauss- The Gaussian curve is a centrosymmetric curve with ian refined atom positions. The presence of an intense wide uses in single processing for approximating sym- neighboring atom’s intensity tail gives rise to a dip in metric impulse functions [31, 32]. Moreover, it has been the intensity away from the original minima, right in demonstrated that given a sufficiently large number of the middle of two atom columns and the Gaussian peak Gaussians, any non-infinite signal can be approximated fitting technique converges to that local minima rather as a sum of overlapping Gaussians [31, 32]. We use this than the original position. Previous BF-STEM imaging insight and extend it into two dimensions by first mod - has attempted in circumventing such issues by using a eling our observed atom intensity as a sum of overlapping multi-parameter Gaussian peak, or performing image Gaussians. The second key idea is to recognize that not metrology through multivariate statistics rather than all Gaussian functions that are approximating the region fitting each individual atoms [19, 20]. Both approaches of interest are in fact originating from the atom whose require an initial knowledge of the crystal structure position we are trying to refine. Thus the Gaussian func - being imaged. Multi-parameter Gaussian fits need an tions are subsequently sorted and only a subset of them estimation of the number and location of the near- that approximate the atom position are used to refine est neighbors, and thus cannot be applied as a robust the atom. The flowchart of our algorithm is illustrated in technique as it necessitates custom fitting equations for Fig. 2. The steps of the mpfit algorithm can be described individual crystal structures. In particular, this restric- as: tion limits the application of this method where more than one crystal orientation may be present. Here, we 1. Get intensity minima/maxima The initial starting propose a novel multi-Gaussian refinement routine— point of this algorithm is the calculation of inten- mpfit—that does not require prior knowledge of the sity maxima for inverted contrast BF-STEM images crystal structures being imaged and can robustly refine or ADF-STEM images. This can be implemented a wider variety of images by deconvolution of a sub- through standard MATLAB or Python peak find - section of the image into multiple overlapping two- ing routines. However, in noisy images, sometimes a dimensional Gaussians. Since HAADF-STEM image single atom may generate multiple maxima. To pre- Fig. 2 Schematic of the procedure. Red circles correspond to intensity minima or maxima for BF-STEM and ADF-STEM images, respectively. The smaller squares surrounding the red dot refer to the nearest neighbor cutoff region while the yellow crosses refer to the refined atom positions Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 4 of 12 vent this, if there are more than one maxima with the advantage of simulated data is that the accurate atom distance between the maxima smaller than the reso- positions are already known and can be compared with lution of the microscope, the center of mass of this mpfit results. This allows the estimation of the relative cluster of points is chosen as the starting reference errors of the single Gaussian and the multiple Gauss- point. ian mpfit approaches, with the simulation parameters 2. Calculate median inter-neighbor distance Following outlined in Table  1. Following the steps of the algorithm the identification of intensity minima, the median outlined in Fig.  2, the intensity minima were first calcu - inter-peak distance is calculated, which is rounded to lated for the simulated image, with Fig.  3a demonstrat- the nearest integer, which we call η. ing the simulated BF-STEM image of LiNbO with the 3. Get region of interest The region of interest is a square intensity minima overlaid as blue dots. These intensity with the intensity minima as the central pixel, with minima are subsequently used to calculate the median the sides of the square given by s = 2η + 1 , where nearest neighboring distance (η) between the minima. s is the side of the square. Thus the (η + 1, η + 1) Based on the calculated η value, the region of interest for pixel in the square is the intensity minima that was this image is demonstrated for one of the atoms as a red the original starting point. Other regions of interest square in Fig.  3a. The region of interest for that atom is schemes, such as a Voronoi tessellation around the shown in Fig. 3b with the contrast inverted and the inten- intensity minima actually demonstrate comparatively sity minima for the atom in question overlaid as a blue worse results (see Fig. 8 in Appendix). dot. As could be ascertained from Fig.  3b, the intensity 4. Fit iteratively with Gaussians The region of interest distribution from the bottom left atom partially overlaps is then fit by a single 2D Gaussian function with a with the atom position we are aiming to refine, precisely user determined tolerance factor. The tolerance fac - indicating the scenario where single peak Gaussian fitting tor refers to the mean absolute difference in intensity approaches often give erroneous results. between the fitted Gaussian and the original data. Sixteen iteration steps were chosen to represent this The fitted Gaussian function is then subtracted from section of the image, as per step four of the algorithm. the original region of interest, and the residual is sub- The first 12 of these iteration steps and the evolution of sequently fitted again. This process continues for a the Gaussian summation are shown in Fig.  3c. The cal - pre-determined number of iterations, with the sum culation of the Gaussian is performed by taking in the of all the Gaussians then subsequently representing entire image, and calculating a two-dimensional Gaussian the original region of interest. In the authors’ experi- peak with the smallest absolute difference with the ini - ence, the tolerance factor is less important than the tial region of interest. Multiple different Gaussian fitting number of iterative Gaussians used, with reasonable approaches can be used, with the fitting equation used in accuracy and speed being obtained with a tolerance this approach expanded in Eq.  1. As could be observed −12 of 10 and 12 to 16 iterations for the mpfit exam - from Fig. 3c, the summation of the Gaussian peaks starts ples presented in this text. We deal with the back- to approximate the region of interest within only a few ground intensity by normalizing each ROI cell from iterations. This demonstrates that the iteration number 0 to 1 before starting the estimation of the individual chosen was sufficient enough to capture the complexities Gaussians. of the data being fitted. It is even more interesting to look 5 Sort peaks and get the refined position The Gaussian at the result of the first iteration, which is mathematically peaks are then subsequently sorted based on their equivalent to the single Gaussian peak fitting approach. distance from the original minima (step 1) with only As the first iteration in Fig.  3c shows, the single Gaussian those peak positions whose distances are less than peak fitting approach is a special case of the mpfit algo - from the minima/maxima (step 1) used for refine - rithm, where the number of iterations is one. According ment. The refined atom position is then the weighted to this image, the first Gaussian peak does not exist near average peak position of all the Gaussians that lie the center of the image, and is extracted towards the bot- within this selected region, with the peak amplitudes tom left corner. The central peak related to the atomic being the weights used. column of interest is captured in the second iteration rather than the first, thus visually demonstrating why the single Gaussian approach fails in some cases. While it may be possible to adjust the calculation Results and discussion of the region of interest to capture the atom position accurately, this approach necessitates tinkering with Results on simulated BF‑STEM images multiple different collection areas and a non-uniform The efficiency and accuracy of the mpfit algorithm was solution for all the atoms in the image. mpfit on the tested on simulated BF-STEM images of LiNbO . The 3 Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 5 of 12 Fig. 3 Evolution of Gaussian peaks for simulated data. a Simulated BF-STEM image of LiNbO with the intensity minima overlaid as blue dots. b Calculation region of interest, demonstrated as the red box in a of the simulated BF-STEM image with the intensity reversed, with the blue spot corresponding to the intensity minima. c Evolution of the sum of the Gaussian peaks over multiple iterations. d Contributions of the Gaussian peaks scaled to their amplitudes with larger spheres corresponding to peaks with higher amplitudes. The red circle refers to the region from which the Gaussian peaks were selected from. e Equivalent summation of multiple Gaussian peaks with the blue point corresponding to the intensity minima, the green point corresponding to the atom position calculated by the mpfit algorithm and the red point corresponding to the atom position. f Contribution from the atom whose positions are being measured. g Contribution from the nearest neighbors other hand, removes the necessity for such complicated Based on the final step of the algorithm, the Gaussian user modifications, allowing the estimation of all the peaks are assigned either to neighboring atoms or the Gaussian peaks that contribute to the final image. The central atom depending upon the distance of the peak individual peak positions are visually represented as a center from the initial intensity minima. As can be seen function of the iteration number in Fig.  3d where the in Fig.  3e, the intensity minima is not always a reliable radius of the spheres are scaled to the amplitude of the estimator of the actual atom position, but the mpfit algo - Gaussian calculated—with the X and Y position of the rithm converges extraordinarily close to the actual atom sphere referring to the location of the Gaussian peak. position (green and red dots overlapping), demonstrat- Only Gaussian peaks lying below a certain distance ing its superiority. The representative summation of the from the intensity minima (shown as the red circle in Gaussian summation can thus be broken down into two Fig.  3d) are used for the estimation of the refined posi - components—the Gaussian peaks that were used for tion. As could be observed, the peak obtained from the atom position refinement, the sum of which is visualized the most intense Gaussian in green is actually assigned in Fig.  3f and the Gaussian peaks that were further off, to the neighboring atom, and two other peaks are also and assigned as contributions of neighboring atom inten- assigned to two neighboring atoms, and only the subset sity—represented in Fig. 3g. u Th s, the combination of the of Gaussian peaks lying within the red circle are used main atom and the neighboring contributions gives rise for refinement. Summing all the Gaussian functions to the total intensity profile that was observed. together we obtain Fig. 3e, which shows close fidelity to We further evaluated the accuracy of the mpfit algo - the input data (Fig. 3b). rithm for an entire image rather than a single atom. Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 6 of 12 Figure  4 shows and compares the three different atom atom positions show close agreement, which generates position metrology techniques—intensity minima/max- the central cluster. However, there are also atom col- ima, single Gaussian peak fitting and the mpfit algorithm umns, where the neighboring atoms are either on the with each other, respectively. Figure 4a demonstrates the top left or the top right, giving rise to the two extra clus- intensity minima itself may not be coincident with the ters—demonstrating the shortcomings of this approach ideal atom positions due to minute intensity variations when the intensity distributions of neighboring atoms that are not accurately captured given a limited detec- approach the resolution limit of the electron microscope. tor dynamic range, with the errors of the order of a sin- The results from the mpfit algorithm, demonstrated in gle pixel. As a result, the intensity minima atom positions Fig.  4c, are on the other hand clustered in a region less are clustered at several different clustering values, which than 0.5  pm across from the known atom positions— can be understood based on the fact that results from the demonstrating it’s accuracy. However, in the authors’ intensity minima are always on the order of a pixel. Thus experience, the mpfit technique fails to converge for the compared to position refinement algorithms, just the edge atoms, which shows up as atom positions that are minima itself is incapable of sub-pixel precision metrol- not clustered and have a higher error. For the rest of the ogy. Figure  4b demonstrates the difference of the single atoms in the image, however, mpfit is significantly supe - peak approach from the ideal atom positions, with the rior to the other approaches. results being clustered into three distinct clusters. This can be understood based on the fact that there are three Results on experimental BF‑STEM images separate types of intensity distributions in the simulated Along with simulated datasets, we additionally per- data. For well-separated atoms, the single peak and the formed position metrology on experimental BF-STEM Fig. 4 Calculated positions. a Simulated LiNbO BF-STEM data with the original atom positions and the intensity minima overlaid in yellow and teal, respectively. b Simulated LiNbO BF-STEM data with the original atom positions and the atom positions obtained by fitting a single Gaussian peak overlaid in yellow and red, respectively. c Simulated LiNbO BF-STEM data with the original atom positions and the atom positions calculated via the mpfit algorithm overlaid in yellow and green, respectively Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 7 of 12 ¯ Gaussian peak fitting approach. As can be visually ascer - images of LiNbO viewed from the 1100 zone axis [15]. tained, the calculated atom position does not correspond The results were obtained through STEM imaging in to the atom position, and thus is an inaccurate represen- a spherical aberration-corrected FEI Titan transmis- tation. Following step 4 of the algorithm, and similar to sion electron microscope, corrected for upto third order the procedure outlined in Fig.  3c, the region of interest spherical aberrations. Imaging was performed at a cam- is represented by a succession of closely spaced two- era length of 145  mm and the BF-STEM images were dimensional Gaussian peaks over 16 iteration steps,with collected using Gatan detectors with an outer collection the contribution from the first 12 steps shown in Fig.  5c. semi-angle of 15 mrad, using scanning pixel step sizes of The individual Gaussian peaks that contribute to the final 9.8 pm. representation of the region of interest are pictorially In contrast to simulated datasets, the exact ideal atom represented in Fig. 5d, with the radius of the circle corre- positions are not known owing to specimen drift, ther- sponding to the amplitude of the Gaussian. Peaks that are mal vibrations, signal-to-noise ratio, and localized imper- further from the original intensity minima by more than fections in the crystal lattice. Figure  5a demonstrates a twice the median inter-peak distance (which is indicated region of interest in an experimental dataset, with the by the red circle) are assigned to the neighboring atoms, intensity reversed, with Fig.  5b showing a section of the and only peaks lying inside the red circle are used to cal- image marked by the red box in Fig.  5a. The blue dot in culate the refined atom position. Fig.  5b corresponds to the intensity minima, while the Figure  5b demonstrates the initial experimental data, green dot represents the position calculated by the single while Fig. 5e demonstrates the final summation from the Fig. 5 Evolution of Gaussian peaks for experimental data. a Experimental inverted contrast BF-STEM image of LiNbO . b Calculation region of interest of an experimental BF-STEM image with the intensity reversed from the region marked by the red box in a, with the blue point corresponding to the intensity minima, and the green point corresponding to the position calculated by fitting a single Gaussian peak. c Evolution of the sum of the Gaussian peaks over multiple iterations. d Contributions of the Gaussian peaks scaled as a function of their amplitude. Peaks lying outside the red circle are assigned to neighboring atoms. e Equivalent summation of multiple Gaussian peaks with the blue point representing the location of the intensity minima, the green point the position calculated by fitting a single Gaussian peak and the yellow point representing the atom position calculated by the mpfit algorithm. f Contribution from the atom whose positions is being measured. g Contribution from nearest neighbors Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 8 of 12 16 Gaussian peaks with visual inspection revealing close guess of local intensity maxima or minima. Using the correspondence between the experimental and repre- center of mass as the starting estimate, it then subse- sented data. Extending the number of iterations would quently approximates a two-dimensional Gaussian to allow progressively smaller Gaussian peaks resulting locate the estimated position of atoms. Atomap can in better correspondence, but would also increase the additionally sort the different species of atom columns demand for computational resources without a corre- in the image and analyze them individually. StatSTEM spondingly significant increase in precision. The inten - on the other hand is a model-based fitting algorithm for sity minima are overlaid on the images in blue, with the extraction of the atom position information from STEM results from the single peak fit approach in green and images. StatSTEM models the atoms in the images as the the mpfit results in yellow, respectively. u Th s the mpfit superposition of two-dimensional Gaussian peaks, and algorithm accurately determines the atom location rather since this is a model-based technique it requires prior than converging to saddle points created from intensity knowledge of the crystal structure of the sample being tails from neighboring atoms. Figure  5f represents the imaged to give a better estimation of the initial guess. sum of the Gaussians that represents the atom being After obtaining the initial guess, the algorithm will go refined and Fig.  5g represents the contribution from the through iterations to reach the least-squares estimation intensity tails from the neighboring atoms, and is calcu- of fitting parameters, and then determines the position. lated from the Gaussian peaks represented with red bor- oxygen octahedra picker is a software specialized in ders in Fig. 5d. identifying the octahedra rotations in the ABO perovs- Returning back to the original experimental BF-STEM kite oxides. It sorts out the oxygen and B atom positions image in Fig.  1a, we revisit that experimental data in and provides users the option of selecting a fast center of Fig.  6a, comparing the results obtained with the mpfit mass estimation or a slower peak fitting with two-dimen - approach. As could be visually ascertained, while the sin- sional gaussians. It exhibits an impressive accuracy of as gle peak fit approach fails in some of the cases, the mpfit small as 3 pm in simulated HAADF images. However, approach reliably refines to the atom position, which can the existing methods still possess limitations in practical also be ascertained by the intensity profile demonstrated cases—with neither the oxygen octahedra picker and in Fig. 6b. the Atomap software being able to process STEM images where atomic columns being measured will have inten- Comparisons with other algorithms sity contributions from their neighbors. Thus both these Several other specialized algorithms have been designed approaches work well for well-separated atom columns to quantify atom positions in electron microscope data- in HAADF images, but face accuracy penalties with BF- sets, such as Atomap [33], StatSTEM [34] and oxygen STEM images. While StatSTEM’s model-based algo- octahedra picker [35]. The Atomap algorithm uses rithm is able to solve the overlapping issue by assuming principal component analysis to obtain denoised STEM atom columns as overlapping 2D Gaussian peaks, its iter- images and finds the center of mass based on the initial ative model fitting process is computationally intensive, Fig. 6 Comparing mpfit with single peak fitting on experimental data. a Experimental LiNbO BF-STEM data with the red points referring to the intensity minima, green points referring to the fitted positions as calculated by the single peak approach, and yellow points being the points as calculated by the mpfit approach. b Intensity profile of the image along the white arrow, with the red arrows corresponding to intensity minima, green arrows to single peak refinement results, and yellow arrows referring to the mpfit refinement results Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 9 of 12 and requires prior knowledge of the crystal structure that even with parallelization implemented, the mpfit being imaged. As demonstrated in Fig.  7a, visually there algorithm solves for over ten Gaussian peaks in a batch is almost no difference between the fitting results of process. On the other hand, the single Gaussian approach StatSTEM versus mpfit, with StatSTEM’s results being solves for just one peak, thus making the single Gaussian slightly off-centered from mpfit’s estimation. Compari - approach faster by at around an order of magnitude. son of the results in Fig. 7b demonstrates that both tech- Future planned improvements include solving for nique give results that are less than a pixel apart from neighboring peaks simultaneously using the tail functions each other, with mpfit outperforming StatSTEM. The to deconvolve the full obtained image as an independ- standard deviation (σ ) of mpfit’s estimation from known ent set of impulse functions originating from individual atom positions is 1.49  pm compared to a σ of 3.31  pm atoms. Additionally, atom columns whose separation from StatSTEM. distances are below the resolution limit of the micro- scope may be particularly suited for this approach, by the Conclusions deconvolution of the observed impulse function into two While it may be possible to assume from the results pre- closely separated Gaussians and enabling the super-reso- sented here that the single Gaussian peak fitting approach lution metrology of atom positions from STEM datasets. fails to converge to atom solutions and gives erroneous u Th s, our results demonstrate that the mpfit algorithm results, it actually performs perfectly adequately for the can reliably and robustly refine the sub-pixel precision of majority of STEM experiments. However, for certain atoms even without a priori knowledge of the underlying non-ideal imaging conditions, the single Gaussian peak crystal structure. Additionally, since the single Gaussian fitting approach fails, while mpfit accurately obtains approach is a special case of the mpfit approach with the precise atom positions. For well-separated atoms, the total number of iterations as one, this approach will also results from mpfit and a single Gaussian refinement are work for ADF-STEM images, enabling a single approach to in fact identical. Additionally, it has to be kept in mind, the metrology of a wide variety of STEM data. The results Fig. 7 Comparison of mpfit and StatSTEM on simulated images. a Overlaid results from StatSTEM and mpfit on a simulated LiNbO dataset. b Distance from known atom positions and the calculated positions from StatSTEM (in blue) and mpfit (in orange). c Distance between StatSTEM and mpfit results in picometers Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 10 of 12 −8 are superior to existing algorithms, and exceeds the state of where τ is the tolerance, which was 10 for our the art—StatSTEM in accuracy, with the added advantage implementation. of being agnostic to the crystal structure being imaged. The equation itself is calculated through the least- squares approach using the trust-region reflective algo - rithm. Trust-region algorithms are an evolution of Abbreviations Levenberg–Marquardt (LM) algorithms. However, com- STEM: scanning transmission electron microscopy; ADF: annular dark field; BF: bright field; HAADF: high angle annular dark field. pared to the LM algorithms, this algorithm is curvature independent and is thus computationally significantly Acknowledgements faster [36–38]. The authors would like to acknowledge Dr. Jason Lapano and Prof. Venkatra- man Gopalan of Penn State for helpful discussions on atom fitting metrology. Simulation parameters Authors’ contributions The LiNbO images were simulated using the MacTem- DM and GS, advised by NA designed the algorithm. DM developed the MATLAB and Python subroutines for the algorithm implementation. DM pasX software, with the simulation parameters enumer- and LM, advised by NA analyzed the data and developed the algorithm. DM ated in Table 1 [39]. performed the electron microscopy simulations. DM wrote the manuscript. NA edited the manuscript. All authors commented on the final manuscript. All authors read and approved the final manuscript. Voronoi vs. square ROI Funding The authors acknowledge funding support from the Penn State Center One other region of interest method was tried, apart from of Nanoscale Science, an NSF MRSEC, funded under the grant number the standard box centered on the intensity minima—a DMR-1420620. ROI based on the Voronoi region around the intensity Availability of data and materials minima. A Voronoi region for a point is defined as the mpfit code in MATLAB is freely available from the mpfit Github repository section of the image, for which the Cartesian distance to (https ://githu b.com/dxm44 7/MPFit ). The Python codes are available in the the point is the lowest compared to all other points in afit module of stemtools (https ://githu b.com/dxm44 7/stemt ools/tree/ maste r/stemt ools/afit). The datasets used and/or analyzed during the cur - the image. However, we observed that the standard ROI rent study are available from the corresponding author (N.A.) on reasonable actually gave better results rather than the Voronoi tes- request. sellation. This is most clearly visible in Fig.  8b where the Competing interests distances between the two methods is scattered over even The authors declare that they have no competing interests. 20  pm, while the standard mpfit results are clustered less than 5  pm away from the known atom positions as Author details Department of Materials Science & Engineering, The Pennsylvania State Uni- could be observed in Fig.  7b. The standard deviation versity, University Park 16802, USA. Center for Nanophase Materials Sciences, (σ ) between the positions calculated with the two ROI Oak Ridge National Laboratory, Oak Ridge 37831, USA. Appendix Gaussian calculation parameters Table 1 BF-STEM simulation conditions in MacTempasX The Gaussian peaks were calculated based on Eq.1 Experimental condition Value (((x−x ) cos θ)+((y−y ) sin θ)) 0 0 Crystal structure LiNbO Z x, y = Ae × Debye–Waller parameters u = 0.67 Å (1) Li (((x−x ) sin θ)+((y−y ) cos θ)) 0 0 u = 0.3924 Å σy Nb e , u = 0.5 Å [40] where Z x, y is the Gaussian output as a function of x Lattice parameters a = 5.172 Å and y, σ and σ are the two normal distributions in the x x y b = 5.172 Å and y directions, x and y are the position of the Gauss- 0 0 c = 13.867 Å [41] ian peak, A is the amplitude of the Gaussian peak and θ Space group 161 (R3c) [42] is the rotation in the counter-clockwise direction of the Zone axis two-dimensional Gaussian peak. Accelerating voltage 200 kV u Th s given a set of x ,y, and z values from the experimen- Inner collection angle 0 mrad tal region of interest, a Gaussian curve is estimated from Outer collection angle 15 mrad Eq.1 such that: Cells 1 × 5 Frozen phonons 10 (z − Z) = τ , (2) Slices per unit cell 5 x,y Probe semi-angle 28 mrad Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 11 of 12 Fig. 8 Eec ff t of a Voronoi ROI. a Simulated BF-STEM image of LiNbO with the green dots referring to the mpfit calculated positions when the region of interest is a Voronoi region around the intensity minima, while the blue dots refer to refined positions calculated from the standard mpfit algorithm. b Distance between the two results in picometers columns by a spherical aberration-corrected electron microscope with a techniques is 10.93  pm, much higher than the standard 300 kV cold field emission gun. J. Electron Microsc 58(6), 357–361 (2009) deviation between the simulated peak positions and the 9. 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Watanabe, K., Asano, E., Yamazaki, T., Kikuchi, Y., Hashimoto, I.: Symmetries lished maps and institutional affiliations. in BF and HAADF STEM image calculations. Ultramicroscopy 102(1), 13–21 (2004) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advanced Structural and Chemical Imaging Springer Journals

mpfit: a robust method for fitting atomic resolution images with multiple Gaussian peaks

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Abstract

The standard technique for sub-pixel estimation of atom positions from atomic resolution scanning transmission electron microscopy images relies on fitting intensity maxima or minima with a two-dimensional Gaussian function. While this is a widespread method of measurement, it can be error prone in images with non-zero aberrations, strong intensity differences between adjacent atoms or in situations where the neighboring atom positions approach the resolution limit of the microscope. Here we demonstrate mpfit, an atom finding algorithm that iteratively calculates a series of overlapping two-dimensional Gaussian functions to fit the experimental dataset and then subsequently uses a subset of the calculated Gaussian functions to perform sub-pixel refinement of atom positions. Based on both simulated and experimental datasets presented in this work, this approach gives lower errors when compared to the commonly used single Gaussian peak fitting approach and demonstrates increased robustness over a wider range of experimental conditions. Keywords: Peak refinement, BF-STEM imaging, Sub-pixel resolution, Aberration-corrected STEM Introduction STEM datasets [5, 9–14]. This technique has been used The development of spherical aberration-correction for for quantitative atomic displacement measurements scanning transmission electron microscopy (STEM) across thin films, 2D crystals, domain boundaries and has imaging has been one of the biggest triumphs of electron allowed the experimental observation of novel structural microscopy over the past several decades, allowing the phenomena such as polar vortices [15–20]. sub-ångström resolution imaging of crystal structures While the Gaussian function fitting approach is an [1–3]. Several pioneering STEM experiments have dem- extraordinarily powerful technique, one noted short- onstrated the feasibility of this technique for the direct coming is that it assumes well-separated atoms with no visualization of atom positions from aberration-corrected overlap, or negligible aberrations in the beam itself—con- STEM images and has proved itself an invaluable tool for ditions that are only available under a certain limited set sub-ångström resolution structural measurements [4–8]. of imaging conditions [16, 17]. Typically, such an imaging While the typical aberration-corrected STEM electron setup uses a ring shaped annular detector with the outer beam has a probe diameter approximately between 0.5 and inner detector collection circles centered along the and 1  Å, supersampling scanning positions below the microscope optic axis. Such a configuration will have an Nyquist–Shannon sampling limit and the subsequent fit - inner collection angle of approximately 85–90 mrad to ting of the probe image with a two-dimensional Gauss- capture only the incoherently scattered electrons, and ian function allows the sub-pixel precision assignment is conventionally referred to as high angle annular dark of atom column positions from aberration-corrected field STEM (HAADF-STEM) imaging [5, 21]. This mode of imaging is referred to as dark field imaging since atom columns themselves are bright due to electrons preferen- *Correspondence: alem@matse.psu.edu tially scattering from atomic nuclei as a consequence of Department of Materials Science & Engineering, The Pennsylvania State Rutherford scattering from proton–electron Coulombic University, University Park 16802, USA Full list of author information is available at the end of the article forces [22, 23]. Since this Coulombic force experienced © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 2 of 12 by the electron probe is directly proportional to the num- intensity distribution with accuracies approaching ber of protons in the nucleus (Z), atom column images in 0.5  pm [14, 17]. It is this combination of aberration- HAADF-STEM datasets generate peaks with an almost corrected imaging and Gaussian peak fitting that has linear relationship of intensity ∝ Z with the atomic enabled modern electron microscopy to reliably meas- number and is also referred to as Z-contrast imaging ure domain walls, grain boundaries, defects, and strain [24–26]. with single picometer precision, making STEM imaging Z-contrast imaging, however, is generally considered so powerful. unsuitable for imaging lighter elements such as oxygen, However, this approach runs into problems when boron or carbon [19–21]. However, structural metrol- applied to BF-STEM imaging. In Fig. 1a, we show a typ- ogy for many scientifically important material systems ical BF-STEM image of LiNbO with 4.9  pm scanning such as ferroelectrics needs the imaging and quantifica - pixel sizes. The darker regions in the image are the nio - tion of lighter atoms as well as heavier elements [27, 28]. bium and oxygen atom columns with the red dots cor- This problem can be significantly mitigated in bright field responding to the intensity minima. While the intensity STEM (BF-STEM) imaging, where rather than annular minima can be used as an initial estimate of atom posi- detectors a circular detector is used with the detector tions, the error in such a measurement is at least of center coinciding with the optic axis of the microscope the order of the pixel size, which is 5  pm in our case. [19, 29]. The conventional collection angle ranges in BF- This makes the error of measurement in BF-STEM an STEM imaging extend up to 15 mrad, significantly lower order of magnitude worse than the best HAADF-STEM than even the inner collection angle for HAADF-STEM results. Figure  1b demonstrates the same section of [29]. Because unscattered electron beams are imaged by the BF-STEM image with the refined atom positions this technique, in contrast to HAADF-STEM vacuum is obtained from fitting the intensity distribution with a bright, while the atom positions have comparatively lower single Gaussian peak with green dots next to the inten- intensity. The ideal BF-STEM image would thus have an sity minima (red dots). A visual estimation shows that intensity profile complementary to the images obtained the fitted Gaussians do not reliably converge on the from HAADF-STEM imaging. However, in reality owing atom positions, and are often tens of picometers away to the lower collection angles, atom positions are more when the intensity minima is weak, and the neigh- blurred from aberrations that are more prominent in BF- boring atom is close. In some cases, the refined atom STEM images [30]. Additionally, since BF-STEM images position is in the middle of the two neighboring atom capture both light and heavy atom positions the inter- columns with no definite atomic intensity. atomic distances are substantially smaller. These effects result in atom positions that are non-Gaussian in shape, and often have intensity overlaps and tails coming from their neighbors making position metrology challenging in BF-STEM images. Methods Fitting atom positions with Gaussians The best modern aberration-corrected microscopes can generate electron probes that are free of aber- rations up to 30  mrad, which corresponds to beam diameters that are of the order of 0.5 Å, or 50  pm at 200 kV [8, 10]. Super-sampling the beam by a factor of five results in scan positions that are spaced approxi - mately 10  pm apart from each other. For HAADF- STEM images where oxygen atoms are not observed, inter-atomic distances from the low index zones are mostly of the order of 1.5 Å, allowing enough distance Fig. 1 Error with single peak fitting on experimental data. a BF-STEM between atoms so that they are well separated and thus image of LiNbO with the red dots referring to the intensity minima. b an atom position can be reasonably approximated with BF-STEM image shown in a with the intensity minima and single peak fitting results overlaid in red and green, respectively. c Intensity profile a two-dimensional Gaussian intensity profile. Since along the arrow shown in a and b with the red arrows referring to the the FWHM of this Gaussian is around 50–75  pm, this intensity minima and the green arrows referring to the single peak fits allows the determination of the peak of the Gaussian Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 3 of 12 This can be quantitatively demonstrated by profiling refinement requires less stringent conditions, our algo - the summed intensity distribution (Fig.  1c) from the rithm extends equally well to such systems too. region shown along the white arrows in Fig.  1a, b. The red arrows in Fig.  1c correspond to the intensity min- The mpfit algorithm ima, while the green arrows correspond to the Gauss- The Gaussian curve is a centrosymmetric curve with ian refined atom positions. The presence of an intense wide uses in single processing for approximating sym- neighboring atom’s intensity tail gives rise to a dip in metric impulse functions [31, 32]. Moreover, it has been the intensity away from the original minima, right in demonstrated that given a sufficiently large number of the middle of two atom columns and the Gaussian peak Gaussians, any non-infinite signal can be approximated fitting technique converges to that local minima rather as a sum of overlapping Gaussians [31, 32]. We use this than the original position. Previous BF-STEM imaging insight and extend it into two dimensions by first mod - has attempted in circumventing such issues by using a eling our observed atom intensity as a sum of overlapping multi-parameter Gaussian peak, or performing image Gaussians. The second key idea is to recognize that not metrology through multivariate statistics rather than all Gaussian functions that are approximating the region fitting each individual atoms [19, 20]. Both approaches of interest are in fact originating from the atom whose require an initial knowledge of the crystal structure position we are trying to refine. Thus the Gaussian func - being imaged. Multi-parameter Gaussian fits need an tions are subsequently sorted and only a subset of them estimation of the number and location of the near- that approximate the atom position are used to refine est neighbors, and thus cannot be applied as a robust the atom. The flowchart of our algorithm is illustrated in technique as it necessitates custom fitting equations for Fig. 2. The steps of the mpfit algorithm can be described individual crystal structures. In particular, this restric- as: tion limits the application of this method where more than one crystal orientation may be present. Here, we 1. Get intensity minima/maxima The initial starting propose a novel multi-Gaussian refinement routine— point of this algorithm is the calculation of inten- mpfit—that does not require prior knowledge of the sity maxima for inverted contrast BF-STEM images crystal structures being imaged and can robustly refine or ADF-STEM images. This can be implemented a wider variety of images by deconvolution of a sub- through standard MATLAB or Python peak find - section of the image into multiple overlapping two- ing routines. However, in noisy images, sometimes a dimensional Gaussians. Since HAADF-STEM image single atom may generate multiple maxima. To pre- Fig. 2 Schematic of the procedure. Red circles correspond to intensity minima or maxima for BF-STEM and ADF-STEM images, respectively. The smaller squares surrounding the red dot refer to the nearest neighbor cutoff region while the yellow crosses refer to the refined atom positions Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 4 of 12 vent this, if there are more than one maxima with the advantage of simulated data is that the accurate atom distance between the maxima smaller than the reso- positions are already known and can be compared with lution of the microscope, the center of mass of this mpfit results. This allows the estimation of the relative cluster of points is chosen as the starting reference errors of the single Gaussian and the multiple Gauss- point. ian mpfit approaches, with the simulation parameters 2. Calculate median inter-neighbor distance Following outlined in Table  1. Following the steps of the algorithm the identification of intensity minima, the median outlined in Fig.  2, the intensity minima were first calcu - inter-peak distance is calculated, which is rounded to lated for the simulated image, with Fig.  3a demonstrat- the nearest integer, which we call η. ing the simulated BF-STEM image of LiNbO with the 3. Get region of interest The region of interest is a square intensity minima overlaid as blue dots. These intensity with the intensity minima as the central pixel, with minima are subsequently used to calculate the median the sides of the square given by s = 2η + 1 , where nearest neighboring distance (η) between the minima. s is the side of the square. Thus the (η + 1, η + 1) Based on the calculated η value, the region of interest for pixel in the square is the intensity minima that was this image is demonstrated for one of the atoms as a red the original starting point. Other regions of interest square in Fig.  3a. The region of interest for that atom is schemes, such as a Voronoi tessellation around the shown in Fig. 3b with the contrast inverted and the inten- intensity minima actually demonstrate comparatively sity minima for the atom in question overlaid as a blue worse results (see Fig. 8 in Appendix). dot. As could be ascertained from Fig.  3b, the intensity 4. Fit iteratively with Gaussians The region of interest distribution from the bottom left atom partially overlaps is then fit by a single 2D Gaussian function with a with the atom position we are aiming to refine, precisely user determined tolerance factor. The tolerance fac - indicating the scenario where single peak Gaussian fitting tor refers to the mean absolute difference in intensity approaches often give erroneous results. between the fitted Gaussian and the original data. Sixteen iteration steps were chosen to represent this The fitted Gaussian function is then subtracted from section of the image, as per step four of the algorithm. the original region of interest, and the residual is sub- The first 12 of these iteration steps and the evolution of sequently fitted again. This process continues for a the Gaussian summation are shown in Fig.  3c. The cal - pre-determined number of iterations, with the sum culation of the Gaussian is performed by taking in the of all the Gaussians then subsequently representing entire image, and calculating a two-dimensional Gaussian the original region of interest. In the authors’ experi- peak with the smallest absolute difference with the ini - ence, the tolerance factor is less important than the tial region of interest. Multiple different Gaussian fitting number of iterative Gaussians used, with reasonable approaches can be used, with the fitting equation used in accuracy and speed being obtained with a tolerance this approach expanded in Eq.  1. As could be observed −12 of 10 and 12 to 16 iterations for the mpfit exam - from Fig. 3c, the summation of the Gaussian peaks starts ples presented in this text. We deal with the back- to approximate the region of interest within only a few ground intensity by normalizing each ROI cell from iterations. This demonstrates that the iteration number 0 to 1 before starting the estimation of the individual chosen was sufficient enough to capture the complexities Gaussians. of the data being fitted. It is even more interesting to look 5 Sort peaks and get the refined position The Gaussian at the result of the first iteration, which is mathematically peaks are then subsequently sorted based on their equivalent to the single Gaussian peak fitting approach. distance from the original minima (step 1) with only As the first iteration in Fig.  3c shows, the single Gaussian those peak positions whose distances are less than peak fitting approach is a special case of the mpfit algo - from the minima/maxima (step 1) used for refine - rithm, where the number of iterations is one. According ment. The refined atom position is then the weighted to this image, the first Gaussian peak does not exist near average peak position of all the Gaussians that lie the center of the image, and is extracted towards the bot- within this selected region, with the peak amplitudes tom left corner. The central peak related to the atomic being the weights used. column of interest is captured in the second iteration rather than the first, thus visually demonstrating why the single Gaussian approach fails in some cases. While it may be possible to adjust the calculation Results and discussion of the region of interest to capture the atom position accurately, this approach necessitates tinkering with Results on simulated BF‑STEM images multiple different collection areas and a non-uniform The efficiency and accuracy of the mpfit algorithm was solution for all the atoms in the image. mpfit on the tested on simulated BF-STEM images of LiNbO . The 3 Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 5 of 12 Fig. 3 Evolution of Gaussian peaks for simulated data. a Simulated BF-STEM image of LiNbO with the intensity minima overlaid as blue dots. b Calculation region of interest, demonstrated as the red box in a of the simulated BF-STEM image with the intensity reversed, with the blue spot corresponding to the intensity minima. c Evolution of the sum of the Gaussian peaks over multiple iterations. d Contributions of the Gaussian peaks scaled to their amplitudes with larger spheres corresponding to peaks with higher amplitudes. The red circle refers to the region from which the Gaussian peaks were selected from. e Equivalent summation of multiple Gaussian peaks with the blue point corresponding to the intensity minima, the green point corresponding to the atom position calculated by the mpfit algorithm and the red point corresponding to the atom position. f Contribution from the atom whose positions are being measured. g Contribution from the nearest neighbors other hand, removes the necessity for such complicated Based on the final step of the algorithm, the Gaussian user modifications, allowing the estimation of all the peaks are assigned either to neighboring atoms or the Gaussian peaks that contribute to the final image. The central atom depending upon the distance of the peak individual peak positions are visually represented as a center from the initial intensity minima. As can be seen function of the iteration number in Fig.  3d where the in Fig.  3e, the intensity minima is not always a reliable radius of the spheres are scaled to the amplitude of the estimator of the actual atom position, but the mpfit algo - Gaussian calculated—with the X and Y position of the rithm converges extraordinarily close to the actual atom sphere referring to the location of the Gaussian peak. position (green and red dots overlapping), demonstrat- Only Gaussian peaks lying below a certain distance ing its superiority. The representative summation of the from the intensity minima (shown as the red circle in Gaussian summation can thus be broken down into two Fig.  3d) are used for the estimation of the refined posi - components—the Gaussian peaks that were used for tion. As could be observed, the peak obtained from the atom position refinement, the sum of which is visualized the most intense Gaussian in green is actually assigned in Fig.  3f and the Gaussian peaks that were further off, to the neighboring atom, and two other peaks are also and assigned as contributions of neighboring atom inten- assigned to two neighboring atoms, and only the subset sity—represented in Fig. 3g. u Th s, the combination of the of Gaussian peaks lying within the red circle are used main atom and the neighboring contributions gives rise for refinement. Summing all the Gaussian functions to the total intensity profile that was observed. together we obtain Fig. 3e, which shows close fidelity to We further evaluated the accuracy of the mpfit algo - the input data (Fig. 3b). rithm for an entire image rather than a single atom. Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 6 of 12 Figure  4 shows and compares the three different atom atom positions show close agreement, which generates position metrology techniques—intensity minima/max- the central cluster. However, there are also atom col- ima, single Gaussian peak fitting and the mpfit algorithm umns, where the neighboring atoms are either on the with each other, respectively. Figure 4a demonstrates the top left or the top right, giving rise to the two extra clus- intensity minima itself may not be coincident with the ters—demonstrating the shortcomings of this approach ideal atom positions due to minute intensity variations when the intensity distributions of neighboring atoms that are not accurately captured given a limited detec- approach the resolution limit of the electron microscope. tor dynamic range, with the errors of the order of a sin- The results from the mpfit algorithm, demonstrated in gle pixel. As a result, the intensity minima atom positions Fig.  4c, are on the other hand clustered in a region less are clustered at several different clustering values, which than 0.5  pm across from the known atom positions— can be understood based on the fact that results from the demonstrating it’s accuracy. However, in the authors’ intensity minima are always on the order of a pixel. Thus experience, the mpfit technique fails to converge for the compared to position refinement algorithms, just the edge atoms, which shows up as atom positions that are minima itself is incapable of sub-pixel precision metrol- not clustered and have a higher error. For the rest of the ogy. Figure  4b demonstrates the difference of the single atoms in the image, however, mpfit is significantly supe - peak approach from the ideal atom positions, with the rior to the other approaches. results being clustered into three distinct clusters. This can be understood based on the fact that there are three Results on experimental BF‑STEM images separate types of intensity distributions in the simulated Along with simulated datasets, we additionally per- data. For well-separated atoms, the single peak and the formed position metrology on experimental BF-STEM Fig. 4 Calculated positions. a Simulated LiNbO BF-STEM data with the original atom positions and the intensity minima overlaid in yellow and teal, respectively. b Simulated LiNbO BF-STEM data with the original atom positions and the atom positions obtained by fitting a single Gaussian peak overlaid in yellow and red, respectively. c Simulated LiNbO BF-STEM data with the original atom positions and the atom positions calculated via the mpfit algorithm overlaid in yellow and green, respectively Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 7 of 12 ¯ Gaussian peak fitting approach. As can be visually ascer - images of LiNbO viewed from the 1100 zone axis [15]. tained, the calculated atom position does not correspond The results were obtained through STEM imaging in to the atom position, and thus is an inaccurate represen- a spherical aberration-corrected FEI Titan transmis- tation. Following step 4 of the algorithm, and similar to sion electron microscope, corrected for upto third order the procedure outlined in Fig.  3c, the region of interest spherical aberrations. Imaging was performed at a cam- is represented by a succession of closely spaced two- era length of 145  mm and the BF-STEM images were dimensional Gaussian peaks over 16 iteration steps,with collected using Gatan detectors with an outer collection the contribution from the first 12 steps shown in Fig.  5c. semi-angle of 15 mrad, using scanning pixel step sizes of The individual Gaussian peaks that contribute to the final 9.8 pm. representation of the region of interest are pictorially In contrast to simulated datasets, the exact ideal atom represented in Fig. 5d, with the radius of the circle corre- positions are not known owing to specimen drift, ther- sponding to the amplitude of the Gaussian. Peaks that are mal vibrations, signal-to-noise ratio, and localized imper- further from the original intensity minima by more than fections in the crystal lattice. Figure  5a demonstrates a twice the median inter-peak distance (which is indicated region of interest in an experimental dataset, with the by the red circle) are assigned to the neighboring atoms, intensity reversed, with Fig.  5b showing a section of the and only peaks lying inside the red circle are used to cal- image marked by the red box in Fig.  5a. The blue dot in culate the refined atom position. Fig.  5b corresponds to the intensity minima, while the Figure  5b demonstrates the initial experimental data, green dot represents the position calculated by the single while Fig. 5e demonstrates the final summation from the Fig. 5 Evolution of Gaussian peaks for experimental data. a Experimental inverted contrast BF-STEM image of LiNbO . b Calculation region of interest of an experimental BF-STEM image with the intensity reversed from the region marked by the red box in a, with the blue point corresponding to the intensity minima, and the green point corresponding to the position calculated by fitting a single Gaussian peak. c Evolution of the sum of the Gaussian peaks over multiple iterations. d Contributions of the Gaussian peaks scaled as a function of their amplitude. Peaks lying outside the red circle are assigned to neighboring atoms. e Equivalent summation of multiple Gaussian peaks with the blue point representing the location of the intensity minima, the green point the position calculated by fitting a single Gaussian peak and the yellow point representing the atom position calculated by the mpfit algorithm. f Contribution from the atom whose positions is being measured. g Contribution from nearest neighbors Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 8 of 12 16 Gaussian peaks with visual inspection revealing close guess of local intensity maxima or minima. Using the correspondence between the experimental and repre- center of mass as the starting estimate, it then subse- sented data. Extending the number of iterations would quently approximates a two-dimensional Gaussian to allow progressively smaller Gaussian peaks resulting locate the estimated position of atoms. Atomap can in better correspondence, but would also increase the additionally sort the different species of atom columns demand for computational resources without a corre- in the image and analyze them individually. StatSTEM spondingly significant increase in precision. The inten - on the other hand is a model-based fitting algorithm for sity minima are overlaid on the images in blue, with the extraction of the atom position information from STEM results from the single peak fit approach in green and images. StatSTEM models the atoms in the images as the the mpfit results in yellow, respectively. u Th s the mpfit superposition of two-dimensional Gaussian peaks, and algorithm accurately determines the atom location rather since this is a model-based technique it requires prior than converging to saddle points created from intensity knowledge of the crystal structure of the sample being tails from neighboring atoms. Figure  5f represents the imaged to give a better estimation of the initial guess. sum of the Gaussians that represents the atom being After obtaining the initial guess, the algorithm will go refined and Fig.  5g represents the contribution from the through iterations to reach the least-squares estimation intensity tails from the neighboring atoms, and is calcu- of fitting parameters, and then determines the position. lated from the Gaussian peaks represented with red bor- oxygen octahedra picker is a software specialized in ders in Fig. 5d. identifying the octahedra rotations in the ABO perovs- Returning back to the original experimental BF-STEM kite oxides. It sorts out the oxygen and B atom positions image in Fig.  1a, we revisit that experimental data in and provides users the option of selecting a fast center of Fig.  6a, comparing the results obtained with the mpfit mass estimation or a slower peak fitting with two-dimen - approach. As could be visually ascertained, while the sin- sional gaussians. It exhibits an impressive accuracy of as gle peak fit approach fails in some of the cases, the mpfit small as 3 pm in simulated HAADF images. However, approach reliably refines to the atom position, which can the existing methods still possess limitations in practical also be ascertained by the intensity profile demonstrated cases—with neither the oxygen octahedra picker and in Fig. 6b. the Atomap software being able to process STEM images where atomic columns being measured will have inten- Comparisons with other algorithms sity contributions from their neighbors. Thus both these Several other specialized algorithms have been designed approaches work well for well-separated atom columns to quantify atom positions in electron microscope data- in HAADF images, but face accuracy penalties with BF- sets, such as Atomap [33], StatSTEM [34] and oxygen STEM images. While StatSTEM’s model-based algo- octahedra picker [35]. The Atomap algorithm uses rithm is able to solve the overlapping issue by assuming principal component analysis to obtain denoised STEM atom columns as overlapping 2D Gaussian peaks, its iter- images and finds the center of mass based on the initial ative model fitting process is computationally intensive, Fig. 6 Comparing mpfit with single peak fitting on experimental data. a Experimental LiNbO BF-STEM data with the red points referring to the intensity minima, green points referring to the fitted positions as calculated by the single peak approach, and yellow points being the points as calculated by the mpfit approach. b Intensity profile of the image along the white arrow, with the red arrows corresponding to intensity minima, green arrows to single peak refinement results, and yellow arrows referring to the mpfit refinement results Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 9 of 12 and requires prior knowledge of the crystal structure that even with parallelization implemented, the mpfit being imaged. As demonstrated in Fig.  7a, visually there algorithm solves for over ten Gaussian peaks in a batch is almost no difference between the fitting results of process. On the other hand, the single Gaussian approach StatSTEM versus mpfit, with StatSTEM’s results being solves for just one peak, thus making the single Gaussian slightly off-centered from mpfit’s estimation. Compari - approach faster by at around an order of magnitude. son of the results in Fig. 7b demonstrates that both tech- Future planned improvements include solving for nique give results that are less than a pixel apart from neighboring peaks simultaneously using the tail functions each other, with mpfit outperforming StatSTEM. The to deconvolve the full obtained image as an independ- standard deviation (σ ) of mpfit’s estimation from known ent set of impulse functions originating from individual atom positions is 1.49  pm compared to a σ of 3.31  pm atoms. Additionally, atom columns whose separation from StatSTEM. distances are below the resolution limit of the micro- scope may be particularly suited for this approach, by the Conclusions deconvolution of the observed impulse function into two While it may be possible to assume from the results pre- closely separated Gaussians and enabling the super-reso- sented here that the single Gaussian peak fitting approach lution metrology of atom positions from STEM datasets. fails to converge to atom solutions and gives erroneous u Th s, our results demonstrate that the mpfit algorithm results, it actually performs perfectly adequately for the can reliably and robustly refine the sub-pixel precision of majority of STEM experiments. However, for certain atoms even without a priori knowledge of the underlying non-ideal imaging conditions, the single Gaussian peak crystal structure. Additionally, since the single Gaussian fitting approach fails, while mpfit accurately obtains approach is a special case of the mpfit approach with the precise atom positions. For well-separated atoms, the total number of iterations as one, this approach will also results from mpfit and a single Gaussian refinement are work for ADF-STEM images, enabling a single approach to in fact identical. Additionally, it has to be kept in mind, the metrology of a wide variety of STEM data. The results Fig. 7 Comparison of mpfit and StatSTEM on simulated images. a Overlaid results from StatSTEM and mpfit on a simulated LiNbO dataset. b Distance from known atom positions and the calculated positions from StatSTEM (in blue) and mpfit (in orange). c Distance between StatSTEM and mpfit results in picometers Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 10 of 12 −8 are superior to existing algorithms, and exceeds the state of where τ is the tolerance, which was 10 for our the art—StatSTEM in accuracy, with the added advantage implementation. of being agnostic to the crystal structure being imaged. The equation itself is calculated through the least- squares approach using the trust-region reflective algo - rithm. Trust-region algorithms are an evolution of Abbreviations Levenberg–Marquardt (LM) algorithms. However, com- STEM: scanning transmission electron microscopy; ADF: annular dark field; BF: bright field; HAADF: high angle annular dark field. pared to the LM algorithms, this algorithm is curvature independent and is thus computationally significantly Acknowledgements faster [36–38]. The authors would like to acknowledge Dr. Jason Lapano and Prof. Venkatra- man Gopalan of Penn State for helpful discussions on atom fitting metrology. Simulation parameters Authors’ contributions The LiNbO images were simulated using the MacTem- DM and GS, advised by NA designed the algorithm. DM developed the MATLAB and Python subroutines for the algorithm implementation. DM pasX software, with the simulation parameters enumer- and LM, advised by NA analyzed the data and developed the algorithm. DM ated in Table 1 [39]. performed the electron microscopy simulations. DM wrote the manuscript. NA edited the manuscript. All authors commented on the final manuscript. All authors read and approved the final manuscript. Voronoi vs. square ROI Funding The authors acknowledge funding support from the Penn State Center One other region of interest method was tried, apart from of Nanoscale Science, an NSF MRSEC, funded under the grant number the standard box centered on the intensity minima—a DMR-1420620. ROI based on the Voronoi region around the intensity Availability of data and materials minima. A Voronoi region for a point is defined as the mpfit code in MATLAB is freely available from the mpfit Github repository section of the image, for which the Cartesian distance to (https ://githu b.com/dxm44 7/MPFit ). The Python codes are available in the the point is the lowest compared to all other points in afit module of stemtools (https ://githu b.com/dxm44 7/stemt ools/tree/ maste r/stemt ools/afit). The datasets used and/or analyzed during the cur - the image. However, we observed that the standard ROI rent study are available from the corresponding author (N.A.) on reasonable actually gave better results rather than the Voronoi tes- request. sellation. This is most clearly visible in Fig.  8b where the Competing interests distances between the two methods is scattered over even The authors declare that they have no competing interests. 20  pm, while the standard mpfit results are clustered less than 5  pm away from the known atom positions as Author details Department of Materials Science & Engineering, The Pennsylvania State Uni- could be observed in Fig.  7b. The standard deviation versity, University Park 16802, USA. Center for Nanophase Materials Sciences, (σ ) between the positions calculated with the two ROI Oak Ridge National Laboratory, Oak Ridge 37831, USA. Appendix Gaussian calculation parameters Table 1 BF-STEM simulation conditions in MacTempasX The Gaussian peaks were calculated based on Eq.1 Experimental condition Value (((x−x ) cos θ)+((y−y ) sin θ)) 0 0 Crystal structure LiNbO Z x, y = Ae × Debye–Waller parameters u = 0.67 Å (1) Li (((x−x ) sin θ)+((y−y ) cos θ)) 0 0 u = 0.3924 Å σy Nb e , u = 0.5 Å [40] where Z x, y is the Gaussian output as a function of x Lattice parameters a = 5.172 Å and y, σ and σ are the two normal distributions in the x x y b = 5.172 Å and y directions, x and y are the position of the Gauss- 0 0 c = 13.867 Å [41] ian peak, A is the amplitude of the Gaussian peak and θ Space group 161 (R3c) [42] is the rotation in the counter-clockwise direction of the Zone axis two-dimensional Gaussian peak. Accelerating voltage 200 kV u Th s given a set of x ,y, and z values from the experimen- Inner collection angle 0 mrad tal region of interest, a Gaussian curve is estimated from Outer collection angle 15 mrad Eq.1 such that: Cells 1 × 5 Frozen phonons 10 (z − Z) = τ , (2) Slices per unit cell 5 x,y Probe semi-angle 28 mrad Mukherjee et al. Adv Struct Chem Imag (2020) 6:1 Page 11 of 12 Fig. 8 Eec ff t of a Voronoi ROI. a Simulated BF-STEM image of LiNbO with the green dots referring to the mpfit calculated positions when the region of interest is a Voronoi region around the intensity minima, while the blue dots refer to refined positions calculated from the standard mpfit algorithm. b Distance between the two results in picometers columns by a spherical aberration-corrected electron microscope with a techniques is 10.93  pm, much higher than the standard 300 kV cold field emission gun. J. Electron Microsc 58(6), 357–361 (2009) deviation between the simulated peak positions and the 9. 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