No abstract

Modern scanning tunneling microscopes (STMs) are often employed in the imaging of crystalline surfaces at atomic resolution, Fig. 1. In recent years, we adapted crystallographic image processing (CIP) techniques that originated in the field of electron crystallography some 50 years ago to the imaging with scanning probe microscopes [1][2][3]. In cases of images from STMs and atomic force microscopes, our CIP method allows for the removal of the effects of a blunt scanning probe tip [2,3].Recent improvements to our method allow for fully objective, i.e. completely researcher independent, classifications of noisy 2D periodic images into Bravais lattice types [3,4], Laue classes [4], and plane symmetry groups [4]. The preconditions that need to be fulfilled for the application of our new information theory based method are a sufficiently long-range ordered crystalline material, a sufficient number of repeats of the unit cell in the image, and a relatively modest amount of generalized Gaussian noise [4,5]. Because discrete Fourier transforms are involved, noise in the images reduces to "noise per unit cell" with the square root of the number of unit cells that are processed [4]. Generalized noise arises from multiple sources in all imaging and image processing steps as well as from structural defects in the crystalline sample itself. None of these sources is allowed to dominate so that the central limit theorem applies and the sum total of all errors possesses approximately a Gaussian distribution [4,5]. The type of the microscope is not important; it is only important that its operation is reasonably stable and uncorrected systematic errors are small compared to the sum total of generalized random errors.We applied our method to a STM image from a graphite sample that is openly available (in the on-line supporting material of [6]), Fig. 1. Utilizing sets of ratios of geometric-bias corrected squared residuals for the involved symmetry hierarchy branches, see right hand side of Fig. 1, the plane symmetry group of the selected region of the STM image was determined to be h31m with confidence levels of approximately 26 % over c11m (on the basis of the primitive sub-units of oc lattices [8] averaged over three settings) as well as 95 % over h3, respectively. There are, thus, no serious doubts that the prevailing translation symmetry of the STM-imaged graphite sample is that of the 2D hexagonal Bravais lattice type (as also obtained in [6]). Note in passing that the h31m result is indicative of a rhombohedral layer stacking that may be due to the sample preparation. Note also that h31m on a triple hexagonal unit cell features the same plane symmetries as p3m1 on a three times smaller cell [5], since the former is a minimal non-isomorphic supergroup of index 3 of the latter. Our classification enabling geometric-bias corrected residuals are in the form of geometric Akaike information criteria [9] as derived in [4]. Our confidence levels are based on the information content equation in [9] as derived in [3] and recently ge...

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) continues to mutate during the ongoing COVID-19 pandemic, and some of the nucleotide polymorphisms may result in diagnostic detection failures.…

No abstract

Crystallographic Image Processing (CIP) allows for the extraction of good estimates of the phases of the structure factors of many inorganic crystalline materials when the conditions of the (pseudo) weak phase object approximation are met in parallel-illumination high-resolution transmission electron microscopy (TEM) [1]. In atomic-resolution scanning TEM, CIP has been employed to remove noise from images and to confirm the average alkali-metal content per unit cell in a type I silicon clathrate [2]. The CIP technique has also been utilized to improve the signal to noise ratio (S/N) of scanning probe microscopy (SPM) images, reveal individual molecules more clearly [3,4], and remove the effects of multiple scanning probe mini-tips on a blunt scanning tunneling microscope (STM) tip from the images [5]. Because spatial averaging in CIP is done over all "asymmetric fraction" of all unit cells of regular 2D periodic arrays, this kind of noise removal from 2D periodic images is up to twelve times more effective [3,4] than traditional Fourier filtering [6].The effectiveness of crystallographic averaging by CIP is demonstrated in Figure 1 on noisy images that were created with the popular freeware program GIMP on the basis of STM data that were published in ref. [7]. The first column of this figure shows the effect of traditional Fourier filtering (p1 enforcing) on images with from top to bottom progressively worsening S/Ns. The second and third columns show the effects of crystallographic averagings in Fourier space that take plane symmetries p2 ("extra averaging factor" of 2 gained) and p4 ("extra averaging factor" of 4 gained) into account. The program CRISP [1] was used for these purposes and also for the determination of the entries in the table to the right of Fig. 1.The conclusion on the most probable* plane symmetry (p4) that underlies this STM data [7] was reached on the basis of the application of our geometric Akaike information criterion (G-AIC) to the detection of the underlying 2D translation lattice [4]. These kinds of criteria are only applicable when there are negligible systematic errors [8], which should be the case for experiments with well calibrated modern STMs and TEMs. Traditional plane symmetry deviation quantifiers for symmorphic 2D space groups are the residuals of the Fourier coefficient amplitude (A res ) and phase (φ res ) [1,3,4] as listed in the table to the right of Fig. 1 for the (not shown) test images that underlie this demonstration. Note the increase of the difference between the phase residuals for p2 and p4 with reduced S/Ns. The square root of the ratio of two G-AIC values (that were calculated on the basis of squared plane symmetry deviation quantifiers) can be utilized as a measure of the noise dependent information content [8] of a STM, SPM, TEM, or scanning TEM image with respect to the detection of the underlying plane symmetry of the regular periodic array from which it was recorded [9].The raw STM image [7] that served as foundation of Fig. 1 was run through a low-pass (G...