Abstract:The discrete Fourier transform is among the most routine tools used in high-resolution scanning/transmission electron microscopy (S/TEM). However, when calculating a Fourier transform, periodic boundary conditions are imposed and sharp discontinuities between the edges of an image cause a cross patterned artifact along the reciprocal space axes. This artifact can interfere with the analysis of reciprocal lattice peaks of an atomic resolution image. Here we demonstrate that the recently developed Periodic Plus … Show more
“…An alternative way to suppress streaking in the discrete Fourier transform that both programs would probably benefit from is described in Ref. [ 45 ].…”
Three different algorithms, as implemented in three different computer programs, were put to the task of extracting direct space lattice parameters from four sets of synthetic images that were per design more or less periodic in two dimensions (2D). One of the test images in each set was per design free of noise and, therefore, genuinely 2D periodic so that it adhered perfectly to the constraints of a Bravais lattice type, Laue class, and plane symmetry group. Gaussian noise with a mean of zero and standard deviations of 10 and 50% of the maximal pixel intensity was added to the individual pixels of the noise-free images individually to create two more images and thereby complete the sets. The added noise broke the strict translation and site/point symmetries of the noise-free images of the four test sets so that all symmetries that existed per design turned into pseudo-symmetries of the second kind. Moreover, motif and translation-based pseudo-symmetries of the first kind, a.k.a. genuine pseudo-symmetries, and a metric specialization were present per design in the majority of the noise-free test images already. With the extraction of the lattice parameters from the images of the synthetic test sets, we assessed the robustness of the algorithms’ performances in the presence of both Gaussian noise and pre-designed pseudo-symmetries. By applying three different computer programs to the same image sets, we also tested the reliability of the programs with respect to subsequent geometric inferences such as Bravais lattice type assignments. Partly due to per design existing pseudo-symmetries of the first kind, the lattice parameters that the utilized computer programs extracted in their default settings disagreed for some of the test images even in the absence of noise, i.e., in the absence of pseudo-symmetries of the second kind, for any reasonable error estimates. For the noisy images, the disagreement of the lattice parameter extraction results from the algorithms was typically more pronounced. Non-default settings and re-interpretations/re-calculations on the basis of program outputs allowed for a reduction (but not a complete elimination) of the differences in the geometric feature extraction results of the three tested algorithms. Our lattice parameter extraction results are, thus, an illustration of Kenichi Kanatani’s dictum that no extraction algorithm for geometric features from images leads to definitive results because they are all aiming at an intrinsically impossible task in all real-world applications (Kanatani in Syst Comput Jpn 35:1–9, 2004). Since 2D-Bravais lattice type assignments are the natural end result of lattice parameter extractions from more or less 2D-periodic images, there is also a section in this paper that describes the intertwined metric relations/holohedral plane and point group symmetry hierarchy of the five translation symmetry types of the Euclidean plane. Because there is no definitive lattice parameter extraction algorithm, the outputs of computer programs that implemented such al...
“…An alternative way to suppress streaking in the discrete Fourier transform that both programs would probably benefit from is described in Ref. [ 45 ].…”
Three different algorithms, as implemented in three different computer programs, were put to the task of extracting direct space lattice parameters from four sets of synthetic images that were per design more or less periodic in two dimensions (2D). One of the test images in each set was per design free of noise and, therefore, genuinely 2D periodic so that it adhered perfectly to the constraints of a Bravais lattice type, Laue class, and plane symmetry group. Gaussian noise with a mean of zero and standard deviations of 10 and 50% of the maximal pixel intensity was added to the individual pixels of the noise-free images individually to create two more images and thereby complete the sets. The added noise broke the strict translation and site/point symmetries of the noise-free images of the four test sets so that all symmetries that existed per design turned into pseudo-symmetries of the second kind. Moreover, motif and translation-based pseudo-symmetries of the first kind, a.k.a. genuine pseudo-symmetries, and a metric specialization were present per design in the majority of the noise-free test images already. With the extraction of the lattice parameters from the images of the synthetic test sets, we assessed the robustness of the algorithms’ performances in the presence of both Gaussian noise and pre-designed pseudo-symmetries. By applying three different computer programs to the same image sets, we also tested the reliability of the programs with respect to subsequent geometric inferences such as Bravais lattice type assignments. Partly due to per design existing pseudo-symmetries of the first kind, the lattice parameters that the utilized computer programs extracted in their default settings disagreed for some of the test images even in the absence of noise, i.e., in the absence of pseudo-symmetries of the second kind, for any reasonable error estimates. For the noisy images, the disagreement of the lattice parameter extraction results from the algorithms was typically more pronounced. Non-default settings and re-interpretations/re-calculations on the basis of program outputs allowed for a reduction (but not a complete elimination) of the differences in the geometric feature extraction results of the three tested algorithms. Our lattice parameter extraction results are, thus, an illustration of Kenichi Kanatani’s dictum that no extraction algorithm for geometric features from images leads to definitive results because they are all aiming at an intrinsically impossible task in all real-world applications (Kanatani in Syst Comput Jpn 35:1–9, 2004). Since 2D-Bravais lattice type assignments are the natural end result of lattice parameter extractions from more or less 2D-periodic images, there is also a section in this paper that describes the intertwined metric relations/holohedral plane and point group symmetry hierarchy of the five translation symmetry types of the Euclidean plane. Because there is no definitive lattice parameter extraction algorithm, the outputs of computer programs that implemented such al...
“…The corresponding Fourier transforms (FT) are shown in Figures 3a–3d. The existence of irregular oscillations during the “fly back” time of the probe between successive line scans will invariably result in displacements of the rows of pixels (Braidy et al, 2012 a , 2012 b ; Jones & Nellist, 2013), and the assumption that the electron microscopy image has periodic boundary (Moisan, 2011; Hovden et al, 2015) produces a cross-pattern streaks running vertically and horizontally through the center of the 2D FT. For this reason, the images shown in Figure 2 are the FT of the “periodic component” of P+S algorithm (Hovden et al, 2015). Accordingly, several Fourier spots can be distinguished with enhanced clarity as a result of applying the proposed SR approach (Fig.…”
From simple averaging to more sophisticated registration and restoration strategies, such as super-resolution (SR), there exist different computational techniques that use a series of images of the same object to generate enhanced images where noise and other distortions have been reduced. In this work, we provide qualitative and quantitative measurements of this enhancement for high-angle annular dark-field scanning transmission electron microscopy imaging. These images are compared in two ways, qualitatively through visual inspection in real and reciprocal space, and quantitatively, through the calculation of objective measurements, such as signal-to-noise ratio and atom column roundness. Results show that these techniques improve the quality of the images. In this paper, we use an SR methodology that allows us to take advantage of the information present in the image frames and to reliably facilitate the analysis of more difficult regions of interest in experimental images, such as surfaces and interfaces. By acquiring a series of cross-sectional experimental images of magnetite (Fe3O4) thin films (111), we have generated interpolated images using averaging and SR, and reconstructed the atomic structure of the very top surface layer that consists of a full monolayer of Fe, with topmost Fe atoms in tetrahedrally coordinated sites.
“…2c) and expected from picometer periodic lattice displacements due to charge order in 1T-TaS 2 . A damping window along the image boundary was used to reduce periodic boundary artifacts in the FFT (Hovden et al, 2015). Figure 2Change in high-angle annular dark-field (HAADF) intensity due to stacking order.…”
Layered transition metal dichalcogenides (TMDs) have attracted interest due to their promise for future electronic and optoelectronic technologies. As one approaches the two-dimensional (2D) limit, thickness and local topology can greatly influence the macroscopic properties of a material. To understand the unique behavior of TMDs it is therefore important to identify the number of atomic layers and their stacking in a sample. The goal of this work is to extract the thickness and stacking sequence of TMDs directly by matching experimentally recorded high-angle annular dark-field scanning transmission electron microscope images and convergent-beam electron diffraction (CBED) patterns to quantum mechanical, multislice scattering simulations. Advantageously, CBED approaches do not require a resolved lattice in real space and are capable of neglecting the thickness contribution of amorphous surface layers. Here we demonstrate the crystal thickness can be determined from CBED in exfoliated 1T-TaS2 and 2H-MoS2 to within a single layer for ultrathin ≲9 layers and ±1 atomic layer (or better) in thicker specimens while also revealing information about stacking order-even when the crystal structure is unresolved in real space.
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