Calculating small-angle scattering intensity functions from electron-microscopy images

We outline procedures to calculate small-angle scattering (SAS) intensity functions from 2-dimensional electron-microscopy (EM) images. Two types of scattering systems were considered: (a) the sample is a set of particles confined to a plane; or (b) the sample is modelled as parallel, infinitely long cylinders that extend into the image plane. In each case, an EM image is segmented into particle instances and the background, whereby coordinates and morphological parameters are computed and used to calculate the constituents of the SAS-intensity function. We compare our results with experimental SAS data, discuss limitations, both general and case specific, and outline some applications of this method which could potentially complement experimental SAS.

We outline procedures to calculate small-angle scattering (SAS) intensity functions from 2-dimensional electron-microscopy (EM) images for two types of scattering systems.

The structures of nanoparticulate systems are commonly characterized by various forms of electron microscopy (EM) and small-angle scattering (SAS) methods. The size and shape of nanoparticles, as well as their spatial-distribution functions, are of particular interest since they govern their structure–function relationships and thus their nanotechnological prospects.^1–6 EM and SAS data are highly complementary. For example, the former images a specific section of a nanomaterial, while the latter realizes its bulk structure by averaging signals obtained from a larger overall area and depth reflective of the sample thickness and beam size. There exists a high degree of overlap in the length scale that is interrogated by EM and SAS data on the same nanomaterial. Yet, these data are necessarily acquired separately and they are analyzed independently. Nevertheless, if suitably processed, the data from one metrology could be used to reconstruct the other. This could draw out the maximum possible structural information about a nanomaterial, or allow data from both sources to be fused to obtain more accurate insights or even highlight processes that result in discrepancies between data from the two methods.This work presents two case studies in which we calculate SAS data from 2-D EM images where (1) the particles being characterized exist on a plane; (2) the sample being imaged can be modelled as parallel, infinitely long cylinders that extend into the image plane. In both cases, we discuss limitations that result in discrepancies between image-obtained SAS intensities and those obtained experimentally. Despite these limitations, we discuss how this method can be complementary to small-angle scattering measurements, by informing experimental design decisions and aiding in model selection. The second case that we present was partially explored by Worthington and Inouye,⁷ and later Meek and Quantock⁸ as well as Quantock et al.⁹ They studied the interfibril distance of collagen fibres in animal corneas by calculating an interference function from pairwise distances of points obtained from an EM image. Their interference function is related to the structure factor which we include in our calculation of SAS intensities, along with form factors which we additionally compute from images. Grubb et al.¹⁰ studied the effect of the orientation of lamellar stack structures on SAXS patterns. The authors did this by generating synthetic images of arrays of lamellar stacks and simulating SAXS data using the 2-D Fourier transform, where they use the Fourier Slice theorem to obtain a 2-D slice of the 3-D transform.¹¹ Afsari et al.¹² and Kim et al.¹³ outline a procedure for calculating small-angle X-ray scattering (SAXS) data from cryo-EM images. Their work makes use of the fact that averaging the correlation functions of many cryo-EM images is equivalent to the Abel transform of SAXS data. Their work is complementary to ours as both methods can be applied under different circumstances. Our work is relevant in situations where image-processing and computer-vision techniques can be employed to segment single EM images and determine morphological and structural information about the scatterers; theirs is relevant when one has numerous cryo-EM images of the same sample.