Date Published: March 28, 2018
Publisher: Springer International Publishing
Author(s): P. Moeck, P. DeStefano.
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 algorithms are also not definitive. Definitive assignments of higher symmetric Bravais lattice types to real-world images should, therefore, not be made on the basis of the numerical values of extracted lattice parameters and their error bars. Such assignments require (at the current state of affairs) arbitrarily set thresholds and are, therefore, always subjective so that they cannot claim objective definitiveness. This is the essence of Kenichi Kanatani’s comments on the vast majority of computerized attempts to extract symmetries and other hierarchical geometric features from noisy images (Kanatani in IEEE Trans Pattern Anal Mach Intell 19:246–247, 1997). All there should be instead for noisy and/or genuinely pseudo-symmetric images are rankings of the relative likelihoods of classifications into higher symmetric Bravais lattice types, Laue classes, and plane symmetry groups.
Direct space imaging techniques such as scanning tunneling microscopy (STM) and (scanning (S) electron probe and high-resolution (HR) parallel illumination) transmission electron microscopy (TEM) provide nowadays atomic resolution in detected images on a routine basis [1–4]. STEM and HRTEM images are typically projections from the third dimension and more or less 2D periodic when crystals are involved. Statistical precision of down to a few picometers is obtained in the case of STEM imaging . This allows for “parametric model based imaging” [2, 3] where the accuracies and precisions of extracted structural-geometric image parameters are statistically estimated on the basis of information theory (i.e., maximum likelihood, negative Boltzmann entropy  or maximum log-likelihood  methods) and geometric inferences  are possible.
A widespread misconception about 2D-Bravais lattice types in the wider scientific community is that they are considered to be independent of the origin and site symmetries of the plane symmetry groups. In Refs. [30, 31], for example, Bravais lattice types are considered to exist without any spatial relationships to the site symmetries of the motifs of the processed images that are more or less 2D periodic. (In Refs. [30, 32], the same has been done for 1D-periodic time series).
All of the 12 synthetic 2D-periodic images used for our lattice parameter extraction review are presented in Fig. 2. Eight of these images, i.e., #1, 3, 4, 6, 7, 9, 10, and 12, were also shown and analyzed in Ref. . Two of these images, i.e., #7 and #8 were also discussed in Ref. . As mentioned in the abstract, there is per design one noise-free (i.e., strictly 2D periodic) image and two noisy (i.e., more or less 2D periodic) images in each of the four sets of three images. These sets are arranged in columns in Fig. 2.
The first of the three algorithms/programs that we tested extracts the parameters of primitive 2D lattices in direct space . The other two programs utilize reciprocal (Fourier) space for the extraction of lattice parameters [20, 21] so that they possess the advantage of averaging over the periodic direct space information effectively as a byproduct. They are, therefore, both expected to perform better in the presence of noise than the first algorithm/program.
The default9 settings of the two programs/algorithms that extract lattice parameters in reciprocal/Fourier space [20, 21] were used in parts of this review and the corresponding results are reported in Tables 3 and 5. For the calculations of dFTs with the CRISP program, we also selected the maximal circular area of the images (i.e., a disk with a diameter of 256 pixels) as an alternative (non-default) setting for the least-squares extraction of lattice parameters. The corresponding results are reported separately in Tables 4 and 6.
Besides the fact that all three computer programs/algorithms aimed at an intrinsically impossible task [7, 16] in a real-world application when they extracted 2D-lattice parameters from the same sets of synthetic test images, one would naively expect that they still provide similar results in their default settings and without a re-interpretation/re-calculation that is indicated to be necessary by a program’s output such as the amplitude map of the dFT of an image. As Tables 3 and 5 show, this is often not the case.
Three different algorithms (as implemented in three different computer programs) were put to the task of extracting lattice parameters from four sets of synthetic test images that were 2D periodic per design but also contained images that were noisy so that all site and translation symmetries were broken. While one of the images in each of these sets was free of noise (and also free of systematic errors so that it was perfectly 2D periodic), independent Gaussian noise of mean zero and a standard deviation of 10 or 50% of the maximal pixel intensity was added to the individual pixels of that image in order to create two noisy images for each set of test images. While the added noise obscures the translation and site symmetries in these images, it obviously cannot change them in a systematic way. The presence of noise is supposed to present a greater challenge for any computer program to extract accurate lattice parameters with a high precision.