A Deep Convolutional Neural Network to Analyze Position Averaged Convergent Beam Electron Diffraction Patterns
AA Deep Convolutional Neural Network to Analyze Position AveragedConvergent Beam Electron Diffraction Patterns
W. Xu a , J. M. LeBeau a, ∗ a Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695, USA.
Abstract
We establish a series of deep convolutional neural networks to automatically analyze position averagedconvergent beam electron diffraction patterns. The networks first calibrate the zero-order disk size, centerposition, and rotation without the need for pretreating the data. With the aligned data, additional networksthen measure the sample thickness and tilt. The performance of the network is explored as a function of avariety of variables including thickness, tilt, and dose. A methodology to explore the response of the neuralnetwork to various pattern features is also presented. Processing patterns at a rate of ∼ Keywords:
Machine learning, Convolutional neural networks, Position averaged convergent beam electron diffraction(PACBED), Automation
1. Introduction
For a highly convergent and coherent, ˚angstr¨om-sized electron probe, the corresponding convergent beamelectron diffraction (CBED) disks strongly overlap to form a complex interference pattern. These patternsdepend sensitively upon the position of the probe within the unit cell, but by averaging these patternstogether, a position averaged CBED (PACBED) pattern is created [1]. The patterns then depend stronglyon sample thickness and tilt, and also reveal crystal polarity, changes in composition, octahedral distortions,and strain [2–8]. ∗ Corresponding author
Email address: [email protected] (J. M. LeBeau)
Preprint submitted to Ultramicroscopy August 4, 2017 a r X i v : . [ phy s i c s . d a t a - a n ] A ug hile PACBED patterns have been shown to be very sensitive to nanometer-level sample thicknessdifferences and sub-milliradian tilt [1], the patterns change with sample thickness in a non-intuitive way dueto dynamical diffraction. Even so, visual inspection is often sufficient to match experimental PACBED toa library of simulated ones. This process, however, is inherently subjective and time-consuming. To reducehuman error and enhance the repeatability of the measurements, a semi-automated approach is usuallyemployed. To this end, least square fitting (LSF) has been the primary tool [3, 4, 8–10]. The parameter ofinterest, e.g. thickness, tilt, polarity, etc., is found by searching for the best fit amongst a library of patterns.While LSF can precise and accurate [10], it can be time-consuming to avoid local minima during the searchacross a broad range of parameters. Beyond processing speed, patterns alignment is an additional limitingfactor. Generally, some pretreatment of the data is required by the user to locate the precise pattern centerand calibrate the pixel size/scale. Specimen tilt complicates this analysis by displacing the center of intensitymass from its true position. Furthermore, alignment is further obfuscated by significant CBED disk overlap,which precludes the use of the Hough transform [10–13].Instead of brute force methods, convolutional neural networks (CNNs) have enabled breakthrough im-age recognition performance, even within very complex scenes [14–16]. For example, CNN has become thestandard for applications ranging from face recognition to self-driving cars. By combining multiple, deepconvolutional layers with an appropriate training set, a CNN can automatically “learn” high-level repre-sentations needed for robust image classification. While neural networks have shown promise for electronmicroscopy analysis, these powerful tools have only recently begun to be applied [17, 18]. This is particularlyrelevant to automated PACBED analysis, as these networks have the potential to overcome many of thelimitations that occur with brute force methods.In this work, we develop a set of deep CNNs to automatically analyze PACBED patterns, extractingpattern size, center, rotation, specimen thickness, and specimen tilt. The training and processing speedsare accelerated by the implementation of GPU calculations. Further, we show that the network architec-ture enables fully automatic PACBED analysis without the need for human supervision. The approach iscompared to LSF using the same datasets and is found to be faster by orders of magnitude after training.Finally, we report various observations including application to 4D STEM datasets, generalizability of thetrained networks to other materials, and a hybrid approach to combine neural networks with LSF for fast,robust analysis of additional parameters.
2. Materials & Methods
Single crystals of SrTiO , oriented along [100], and PbMg / Nb / O , oriented along [110], were usedthroughout this study. The crystals were thinned to electron transparency using wedge-polishing and low2nergy ion-milling using a Fischione 1050 Ar ion mill. The PMN sample was carbon coated to reducesample charging. A probe-corrected FEI Titan G2 STEM microscope was operated at 200 kV with probeconvergence semi-angle of either 13.6 mrad and 19.1 mrad. PACBED patterns were recorded using a GatanUltraScan 1000XP CCD camera.SrTiO PACBED patterns from experiment were used to create a database for performance testing.Using the 13.6 mrad probe, a total number of 231 PACBED patterns were recorded from regions 6-120nm thick. For the 19.1 mrad probe, a total of 156 PACBEDs were captured at thicknesses ranging from 8to 70 nm. In both cases, the patterns exhibited random tilts up to ∼ ×
10 4D STEM dataset was collected over a 60 ×
60 nm region of the sample with anacquisition time of 1 s/pattern. To establish a library for neural network training, PACBED patterns were simulated using the Bloch wavemethod. The Many-Beam dynamical-simulations and least-squares FITting (MBFIT) software was used forthis purpose [19]. Note that the original MBFIT source code was modified to generate the PACBED outputwith overlapping the diffraction disks [6]. Patterns were calculated in 1 nm increments with thicknessesranging from 1-120 nm at 13.6 mrad, and 1-80 nm for 19.1 mrad. At each thickness, a tilt series was alsosimulated with up to 4 mrad tilt along [100] and [010]. The tilts were separated by 0.25 mrad when tilt was < The convolutional neural networks applied here were based on the AlexNet architecture, a descriptionof which can be found in Ref. [20]. The network was trained via the MATLAB Neural Network Toolboxusing a Titan X Pascal GPU. The network was finely tuned to measure PACBED patterns using trainingdatasets and backpropagation through stochastic gradient descent (SGD) [21, 22]. The learning rate in thelast fully-connected layer was set to be 10 times faster than that of the other layers. To reduce overfittingand to better generalize the neural network, dropout was applied in the first two fully connected layers witha ratio of 0.5 [23]. Note that the original grayscale PACBED images were converted to 227 ×
227 RGB pixelimages to meet AlexNet input requirements. Although neural networks are usually considered “black boxes”,we evaluated the regions of the PACBED pattern that created the greatest neural response using a usingband-pass type mask. The width of the annular mask was set to the size of the first AlexNet convolutionalkernel size, 11 ×
11 pixels or 2.2/1.6 mrad for 13.6/19.1mrad PACBED, respectively. The pixels within theband-pass region were then set to the mean intensity of those pixels.3 .4. Least squares fitting
Least square fitting was employed to benchmark the results of thickness/tilt with neural network byfinding the minimum χ according to: χ = (cid:88) i (cid:88) j ( I exp,i,j − f · I sim,i,j ) (1)where I exp and I sim are the experiment and simulation intensities, respectively. The factor f is includedto match the intensity scale difference between the experiment and simulation. During the search of theglobal minimum of χ , the simulated PACBED patterns were automatically scaled, rotated and shifted. Theprecision of LSF was estimated using the method suggested in Ref. [10]. Further, when the actual specimenthickness was above ∼
70 nm, LSF tended to converge to a thinner value. This was found to be primarily dueto excess background due to inelastic scattering, which was empirically overcome by subtracting a uniformbackground intensity of 0.4 scaled intensity/nm from experiment.
3. Algorithm Description
The overall neural network configuration can be seen in Figure 1, which contains a total number of fiveCNNs combined for the tasks of PACBED thickness and tilt measurement. Specifications of the implementedCNN networks are listed in Table 1. In the first stage, the zero order disk size, disk center, and patternrotation angle are measured. The flow of the procedure is illustrated in Figure 2. There are a number ofautomated procedures that are applied before passing to the CNNs. Rough estimates of the of the patterncenter and size are provided by fitting the integrated intensity of the PACBED pattern along both horizontaland vertical direction to Gaussian functions. Precision here is not essential as these variables are iterated.The roughly centered and cropped patterns are then passed to subsequent CNNs to refine the center andshift measurements. To measure shift along both horizontal and vertical directions, the same CNN is used,but with the pattern rotated 90 ◦ . Updated center and size variables are used to realign the original PACBEDdataset until convergence. The rotation angle is then determined via another trained CNN, but without theneed for iteration. Prior to thickness and tilt measurements, uniform background is subtracted from thepattern to account for the contribution from inelastic scattering. This improves the network performance,particularly when determining sample thicknesses above ∼
70 nm. For more details, see Section 5.1. As partof the process, this background subtraction value is converged while determining determining thickness andtilt. It is also important to note that only positive tilt values from 0-4 mrad along [100] and [010] directionare measured using the trained CNN, which is justified by the four-fold symmetry for the zone considered.The sign of tilt along [100] and [010] can then be identified according to the relative image intensity in fourpattern quadrants. This treatment reduces the complexity of using a CNN to measure tilt.4 igure 1: The configuration of the convolutional neural networks used for automatic PACBED pattern measurements.
AlexNet is configured with well-defined, initial convolutional kernels to extract local features for imageclassification. Utilizing these initial kernels not only speeds up the training process, but also improves thenetwork performance for PACBED pattern identification. To prepare the neural network for the training,the convolution filter bias/weights and first two fully connected layers are taken directly from AlexNet.In addition, we replace the classification layers (fully-connected and softmax) with image classificationcategories specific to PACBED.The training datasets are prepared using the simulated PACBED patterns. Two key factors must beconsidered for successful CNN training. First, both thickness and tilt variation are included in all CNNtraining sets. While ideally every PACBED pattern from experiment would be captured perfectly on-zone,patterns with sub-mrad tilt are difficult to avoid in practice. The inclusion of tilt generalizes the CNN moreeffectively and improves measurement accuracy. Second, and to further generalize the network, a wide offactors impacting experiment are also included in the simulated training sets through data augmentation.These include using random affine transformations to approximate geometric distortion introduced by theprojection lens aberrations, Gaussian blurring to account for inelastic scattering and the CCD point spreadfunction, and shot noise. Random flips of the training dataset are also included. A summary of networktraining for each CNN are included in Table 1.
4. Neural network performance
The trained CNNs are tested with a series of SrTiO PACBED patterns from experiment at differentthicknesses and tilts. To evaluate the network performance, the results from the CNN measurement are5 igure 2: Flow chart of the convolutional neural networks implemented during the automated alignment procedure. compared with the results from LSF and visual inspection. The overall performance is listed in Table 2,while typical thickness and tilt classification examples are presented in Table 3.The first neural networks – size, shift, and rotation – achieve near 100% accuracy within ± ± ± ◦ , respectively. Supplementary Video 1 highlights this procedure for a set of raw PACBED patternsdirectly from experiment, where the CNNs are able to center the patterns and match their size. Beyondsimple cases, the patterns are well-aligned even when the zero order disk is not well defined, i.e. for thicksample regions or when there is significant sample tilt. The accuracy of these automated measurementsguarantees robust pattern alignment for the subsequent thickness and tilt classification.In comparing the time required for processing, the trained CNNs vastly outperform the LSF method. Thetime for LSF to determine thickness and tilt is significant, and in some cases is more than 30 min/pattern.In contrast, an average rate of 0.1 s/pattern is achieved for the CNNs, as measured from the raw image6 able 1: Convolutional neural network training and data augmentation Neural Network Target Precision andApplication Range Augmentation TrainingPoints Learningrate
Size(zero order diskdiameter) ± ± ± ◦ (-44-45 ◦ ) distortion, shift, randomcrop, blur, intensity 4.3M Start 2e-4Final 1e-4Thickness (13.6 mrad) ± ± ± ± ± ± able 2: Validation results comparing the convolutional neural networks to least squares fitting. Neural Network Simulation Experiment
Size 96.5% 96.0% ( ±
1% size change)99.4% ( ±
2% size change)Center 99.8% 90.6% ( ± ± ± ◦ )100% ( ± ◦ )Thickness (13.6 mrad) 99.1% 81.0% ( ± * ± * ± * Tilt (13.6mrad) 97.2% 74.0% ( ± * ± * ± * Thickness (19.1 mrad) 98.0% 91.0% ( ± * ± * ± * Tilt (19.1 mrad) 91.3% 61.5% ( ± * ± * ± ** While LSF may not necessarily report the true value, all LSF results are confirmed by visual inspection. input to the output. For experiment thickness and tilt measurements, the CNN approach exhibits a highdegree of accuracy. Using LSF as a benchmark, near 95.7% CNN measurements are matched within ± ± ± ± able 3: Typical PACBED thickness and tilt measurements from the convolutional neural networks with corresponding classi-fication confidence. Experiment CNN LSF Visual MatchingThickness:
85 nm 79.6%104 nm 17.8%
Tilt ([100]/[010]):
Thickness:
86 nm (+6/-6 nm)
Tilt ([100]/[010]):
Thickness:
Thickness:
22 nm 58.8%21 nm 41.1%
Tilt ([100]/[010]): -2.5/0 mrad 80.4%-2/-0.5 mrad 9.5%
Thickness:
22 nm (+2/-2 nm)
Tilt ([100]/[010]): -2.5/0 mrad
Thickness:
Thickness:
28 nm 100%
Tilt ([100]/[010]):
Thickness:
28 nm (+4/-4 nm)
Tilt ([100]/[010]):
Thickness:
Thickness:
38 nm 59.2%39 nm 40.8%
Tilt ([100]/[010]): -1.5/1.5 mrad 52.8%-1/1.5 mrad 44.6%
Thickness:
39 nm (+4/-3 nm)
Tilt ([100]/[010]): -1.5/1.5 mrad
Thickness: . Factors Influencing Network Performance
The CNN thickness measurement precision is found to be dependent upon sample thickness, as shownin Figure 3. The CNN exhibits the best performance for thicknesses below 60 nm, where over 95% of thepatterns match within ± Figure 3: Percentage of CNN and LSF thickness and tilt measurements matching as a function of specimen thickness: (a, c)13.6 mrad and (b, d) 19.1 mrad
Close inspection of experiment and simulations suggests that the error for thicker regions is most likelydue to the pattern discrepancy. As seen in Figure 4a for example, a 100 nm pattern from experiment showsadditional background intensity at the periphery of the pattern and blurring. In addition, a discrepancyis also observed in the region of overlapped disks as shown in the difference map. These differences arise10argely from inelastic thermal diffuse and plasmon scattering [24, 25]. To minimize the effect of inelasticscattering on the CNN classification, a uniform background subtraction is applied, with the value foundthrough iteration within the network. This is shown in Figure 4b, where the accuracy of PACBED thicknessdetermination can be optimized with appropriate background subtraction. For the case of SrTiO , 0.7/nmand 0.3/nm are optimal for 13.6 mrad and 19.1 mrad, respectively. Figure 4: Percentage of CNN and LSF thickness measurement that match as function of uniform background intensity sub-traction strength. Abscissa values are on the same intensity scale as the image.)
Sensitivity of the CNNs to pattern features is explored further in Figures 5a,b, where the CNN/LSF matchrate as a function of band-pass mask position is provided. The performance degradation occurs at about18 mrad and 24 mrad for the 13.6 and 19.1 convergence semi-angle, respectively. As the band-pass mask isplaced closer to the PACBED center, the agreement between CNN and LSF decreases to a minimum roughlyat the convergence semi-angle. A similar trend can be seen in all the percentage of matching CNN/LSFresults within ± ±
2, and ± igure 5: Response of CNN ability to classify patterns when removing pattern features via the band-pass mask for thicknessmeasurements with (a) 13.6 mrad and (b) 19.1 mrad and tilt measurements with (c) 13.6 mrad and (d) 19.1 mrad the CNN automatically identified the most relevant local pattern features. For the thickness range of 1-20nm, however, the CNN performance is nearly flat because the patterns themselves do not exhibit strongfeatures within this regime. More importantly, the reduction of the CNN accuracy occurs for a broadrange of band-pass masks. This suggests the CNN utilizes multiple regions of the PACBED patterns whyclassifying, which leads to increased robustness. The CNN agreement with LSF tends to improve with increasing thickness due to the increasing patterndetail, as shown in Figures 3c-d. The CNN classification uncertainty within ± ± ≤
10 nm, theCNN is only reliable to within about ± igure 6: Response of CNN ability to classify patterns for indicated specimen thickness ranges. The range of band-pass maskpositions that lead to local minima are indicated by corresponding dashed lines (left) and circles (right). For 13.6 mrad tilt measurements, the greatest CNN response to the occurs with a band-pass mask witha starting semi-angle of 10-18 mrad. The minimum is located at about 13.5 mrad, which is near the edgesof zero order disk as marked by the dashed circle in the PACBED pattern in Figure 5b. A 20-30% drop isfound for the percentage of CNN/LSF results matching within ± ± ± The probe forming convergence semi-angle is a critical parameter for the neural network classificationas it controls the degree of diffraction disk overlap and pattern detail. To better understand the effect ofconvergence semi-angle on the CNN performance, the network activation is visualized in Figure 7 for boththe 13.6 and 19.1 mrad networks. A variety of bright and dark patterns are observed for the 13.6 mrad CNN,where the strongest activation occurs both inside the zero order disk and at its edge. The activation for 19.1mrad, however, is more localized at the zero-order disk edge. The localized region of network activation and13esponse of 19.1 mrad is attributed to the reduction of pattern features when the convergence semi-angleis large. As a result, the 19.1 mrad CNN trained for determining thickness cannot handle thicker samplemeasurement, which has also been reported for LSF methods [10]. In order to reduce the complexity ofthe network, the acceptable thickness range is reduced to 1-80 nm for SrTiO , which improves the networkreliability (87-92.3%) when compared to LSF. Figure 7: Thickness measurement neural network activation in the first and second convolutional layers.
The PACBED pattern clarity is affected by electron dose. Figure 8a, for example, shows patterns fromthe same area, but with a dose ranging from 4.5 × to 9.3 × e − /pattern. While the pattern alignmentis robust for all dose levels, the impact on the thickness measurements is shown in Figure 8b. When the doseis ≥ × e − /pattern, the CNN reports the same thicknesses. Similar results are observed for both 13.6and 19 mrad patterns. Below this dose, the thickness error increases. A similar dose effect is also seen in theCNN tilt measurement, but the dose threshold for robust measurements is around ∼ × e − /pattern.This dose sensitivity for tilt measurements is due to tilt determination being heavily weighted by featurechanges on the pattern periphery as highlighted in Section 5.2.14 igure 8: Effect of noise level on 13.6 mrad CNN thickness and tilt measurements. (a) Example patterns from one specimenregion at different dose levels: × , × , × , × , × , × electrons/e − .(b) and (c) CNN measurement of thickness and tilt at various thicknesses and dose.
6. Observations & Opportunities
The CNN processing speed is well suited for PACBED quantification of big, 4D STEM datasets. Asan example, Figure 9 shows the result of thickness and tilt measurements over a 60 ×
60 nm area. SeeSupplementary Video 2 for the collection of PACBED patterns from this region exhibiting changes in boththickness and sample distortion. It is important to emphasize that PACBED analysis from this region ischallenging due to the combination of a thickness gradient with local crystal distortion.The CNN identified thicknesses are presented in Figure 9b, which shows the smooth, wedge shape of thespecimen from the top left to the bottom right. As shown, the thickness changes by about 5 nm across theanalysis region with an average thickness of 23 nm. This is particularly important when quantifying STEMimages where a 5 nm difference in thickness introduces a 25% change in the image signal [26–28].15 igure 9: (a) HAADF STEM image of a local 60 ×
60 nm area and corresponding local (b) thickness and tilt along (c) [001]and (d) [100]. The tilt magnitude and the azimuthal angle are shown in (e) and (f), respectively. In addition to thickness, the crystal is found to be distorted by ± While the neural network is trained for automatically aligning 13.6 mrad and 19.1 mrad datasets, theneural network is sufficiently generalized to align patterns acquired at other convergence semi-angles. A testusing simulated patterns shows that the current CNN can align PACBED pattern size and shift for conver-gence semi-angles greater than 13 mrad when captured at 200 kV, see Supplementary Video 3. Furthermore,the neural networks achieve good generalization for other directions and different structures even thoughthey are trained to align [001] SrTiO patterns. For example, the CNNs align [011] PbMg / Nb / O (alsoa perovskite) patterns without additional training. In fact, the pattern rotation CNN aligns [100] and [0¯11]along the pattern horizontal and vertical, respectively. This is particularly fortuitous, as the [011] exhibits2-fold symmetry, whereas the training set is 4-fold. Thickness and tilt of the aligned pattern can then beestimated using the SrTiO datasets, but precise and accurate measurements require additional training.16 .3. Hybrid CNN+LSF Analysis The limitations outlined in the introduction of the LSF approach can be overcome through a hybridapproach with CNN methods. As illustrated in Figure 10, the CNN approach developed here can supportLSF via its automatic PACBED pattern alignment and thickness/tilt measurements as initial parameters,e.g. the search range is then limited within ± ± > Figure 10: Hybrid CNN+LSF architecture for PACBED measurement.
7. Conclusions
A convolutional neural network approach has been developed to automatically measure key parametersfrom PACBED patterns. This includes the zero-order disk size, pattern center, rotation, thickness, andtilt. The network has been successfully transfer-trained using thousands of simulated patterns that areaugmented with additional variables to account for random geometric distortion, size, shift, noise, andintensity variations. The trained networks show excellent accuracy for measuring PACBED patterns whencompared with brute force methods. Through GPU acceleration, an overall processing rate of 0.1 s perpattern has been achieved, enabling fast analysis of 4D STEM data. Furthermore, a methodology has beendeveloped to explore how the CNN responds to features across the PACBED patterns. Overall, the approachprovides a critical demonstration of how neural networks can be successfully implemented for the automatedanalysis of electron diffraction data. 17 cknowledgments
The authors gratefully acknowledge the Air Force Office of Scientific Research (FA9550-14-1-0182) forsupport of this research. We thank Dr. Rohan Dhall for discussion. This work was performed in part at theAnalytical Instrumentation Facility (AIF) at North Carolina State University, which is supported by theState of North Carolina and the National Science Foundation (award number ECCS-1542015). The AIF isa member of the North Carolina Research Triangle Nanotechnology Network (RTNN), a site in the NationalNanotechnology Coordinated Infrastructure (NNCI).
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