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.

中文翻译：

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来自二维周期图像的准确晶格参数，用于后续的Bravais晶格类型分配。

在三种不同的计算机程序中实现的三种不同算法的任务是，从每组设计在二维（2D）或多或少周期性的四组合成图像中提取直接空间晶格参数。每个设计中的每个测试图像中的每个图像均无噪声，因此是真正的2D周期性图像，因此它完全符合Bravais晶格类型，Laue类和平面对称性组的约束。将平均零的高斯噪声和最大像素强度的10％和50％的标准偏差分别添加到无噪声图像的各个像素中，以创建另外两个图像，从而完成设置。增加的噪声破坏了四个测试集的无噪声图像的严格平移和位点对称性，因此每个设计中存在的所有对称性都变成了第二种伪对称性。此外，在大多数无噪声测试图像中，每种设计都已经存在第一类基于基元和翻译的伪对称符号，也就是真正的伪对称符号，以及度量特化。通过从综合测试集的图像中提取晶格参数，我们评估了在存在高斯噪声和预先设计的伪对称性的情况下算法性能的鲁棒性。通过将三个不同的计算机程序应用于相同的图像集，我们还针对随后的几何推理（例如Bravais晶格类型分配）测试了程序的可靠性。部分由于归因于每个设计，现有的第一类伪对称性使得即使在没有噪声的情况下（即，在没有伪对称性的情况下），所使用的计算机程序在其默认设置中提取的晶格参数对于某些测试图像也不同。第二种，用于任何合理的误差估计。对于嘈杂的图像，算法中晶格参数提取结果的分歧通常更为明显。非默认设置和基于程序输出的重新解释/重新计算可以减少（但不能完全消除）这三种测试算法的几何特征提取结果中的差异。因此，我们的晶格参数提取结果说明了Kenani Kanatani的格言，即从图像中提取几何特征的算法都无法得出确定的结果，因为它们都针对所有现实应用中本质上不可能完成的任务（Syst Comput Jpn中的Kanatani 35：1-9，2004）。由于2D-Bravais晶格类型分配是从或多或少的2D周期图像中提取晶格参数的自然最终结果，因此本文中还有一节描述了五种相互交织的度量关系/水平面和点组对称层次欧几里得平面的平移对称类型。因为没有确定的晶格参数提取算法，所以实现这种算法的计算机程序的输出也不是确定的。因此，不应根据提取的晶格参数及其误差线的数值对实际图像进行高对称Bravais晶格类型的确定性分配。此类分配需要（在当前状态下）任意设置阈值，因此始终是主观的，因此它们无法主张客观的确定性。这就是Kenani Kanatani对绝大多数从噪声图像中提取对称性和其他层次几何特征的计算机化尝试的评论的本质（Kanatani在IEEE Trans Pattern Anal Mach Intell 19：246-247，1997）。对于嘈杂的和/或真正伪对称的图像，应该存在的所有分类是将相对对称性分类为更高对称的Bravais晶格类型，Laue类和平面对称性组。