Robust electron counting for direct electron detectors with the Back-Propagation Counting method

TL;DR

Counting electron hits on direct detectors at high fluence is challenged by hit overlap and Landau tails. Back-Propagation Counting (BPC) models a fixed Gaussian single-electron response and fit per-pixel counts to raw frames via back-propagation, avoiding large training datasets. Across synthetic simulations and real 4D-STEM experiments on NaYF4 nanoparticles, BPC yields more consistent counts at high occupancy and produces stronger diffraction peaks with clearer images than a standard counting approach. This detector-agnostic method improves quantitative materials characterization under high electron flux and is complemented by open-source code for broader use.

Abstract

Electron microscopy (EM) is a foundational tool for directly assessing the structure of materials. Recent advances in direct electron detectors have improved signal-to noise ratios via single-electron counting. However, accurately counting electrons at high fluence remains challenging. We developed a new method of electron counting for direct electron detectors, Back-Propagation Counting (BPC). BPC uses machine learning techniques designed for mathematical operations on large tensors but does not require large training datasets. In synthetic data, we show BPC is able to count multiple electron strikes per pixel and is robust to increasing occupancy. In experimental data, frames counted with BPC are shown to reconstruct diffraction peaks corresponding to individual nanoparticles with relatively higher intensity and produce images with improved contrast when compared to a standard counting method. Together, these results show that BPC excels in experiments where pixels see a high flux of electron irradiation such as in situ TEM movies and diffraction.

Figures (4)

• Figure 1: Illustration of 4D-STEM and Back-Propagation Counting (BPC). (a) A scanning electron nanobeam incident on a sample produces a diffraction pattern captured by a pixelated APS-based detector. Simultaneously, high-angle scatterers are captured by a monolithic high-angle annular dark field (HAADF) detector. (b) An estimated counted frame is computed from the raw frame and convolved with the pre-computed 2D kernel to produce a reconstructed frame. Back-propagation is used to iteratively update the estimate according to a loss function. (c) The final optimized count grid is rounded to integer values. The ground-truth is shown for comparison but is not used in the algorithm.
• Figure 2: The BPC method counts electrons more consistently with increasing occupancy in synthetic data. A Gaussian ($\sigma = 3$ pixels) distribution of electron hits is simulated and counted for 1, 10, 100, and 1000 electrons per frame. The true number of electron hits summed over all frames in the Monte Carlo is shown in (a) for the different mean numbers of electron frames. The sums over all counted frames are shown using the BPC method in (b) and the standard method in (c). The final row (d) shows the counts projected along the y-axis, summed over all frames, for: the Monte Carlo truth, the standard method, a version of the BPC method with a maximum of 1 electron per hit (unweighted), the BPC method without application of the Landau correction described in Section \ref{['ss.prior']} (w/o prior), and the full BPC method (w/prior).
• Figure 3: The BPC method consistently counts experimental data under a constant electron flux, evaluated over varying aperture sizes. The constant flux of electrons was measured over four different camera lengths (corresponds to aperture size) to provide a stable number of electrons with different occupancies. Similar datasets at electron currents $I_0 \approx 30$ pA, $2I_0$, $4I_0$, and $8I_0$ were acquired. (a) The circular aperture datasets acquired, summed over 16512 frames of 576x576 pixels, for electron current $8I_0$. (b) The total number of electron counts summed over 16512 frames vs. camera length for each of the 4 electron currents. The curves shown for each electron current were normalized to the maximum number of counts obtained using the standard counting method. (c) Average difference in the number of counts per frame between variations of the BPC method and the standard method, and (d) variation [(max-min)/max] in the curves shown in panel (b) for each method, averaged over all electron currents.
• Figure 4: The BPC method enhances diffraction from nanoparticles, yielding higher intensity diffraction peaks and improved image clarity. The sum of counted diffraction patterns (a-c) and the real-space images constructed by summing the total diffraction pattern intensities at each probe position (d-f), as measured with a 4D-STEM detector and counted using the standard counting method and the BPC counting method, are shown. The diffraction patterns are shown with log-scale intensity, and all images in each row (a-c) and (d-f) are shown with equal contrast. An example diffraction pattern summed over real-space probe positions of a single nanoparticle, reconstructed with the standard (g) and BPC (h) counting methods, is also shown, as well as the difference $\mathrm{I}_{\mathrm{BPC}} - \mathrm{I}_{\mathrm{standard}}$ (i) between these two patterns. A single peak is highlighted to demonstrate a "halo" artifact observed in the standard counting for intense peaks. (j) Intensity difference for diffraction peaks selected from many nanoparticles, counted with the BPC and standard counting methods.