Physics-Informed Auto-Differentiation for Limited-Angle Tomography of Thick Amorphous Specimens Using BF-STEM

Abstract

Electron Tomography is a widely used 3D imaging tool for biological specimens because it offers higher resolution than optical imaging and greater accessibility than X-ray sources. While transmission electron microscopy (TEM) tomography can offer advantages for thin specimens under well-controlled imaging conditions, imaging thick, amorphous specimens becomes increasingly challenging due to reduced transmission and loss of usable contrast at large thicknesses. Alternatively, bright-field scanning transmission electron microscopy (BF-STEM) tomography, due to improved dose control and tolerance to multiple scattering, is preferred for thick samples, as it can image samples thicker than 400 nm while maintaining sufficient resolution and signal [1][2]. However, BF-STEM tomography of sheet-like laminar specimens remains strongly limited by incomplete tilt ranges, and the mismatch between conventional linear reconstruction algorithms and the underlying nonlinear image-formation physics limits reconstruction quality. In this work, we employ a physics-informed automatic differentiation (PIAD) based limited-angle tomography framework that uses a multislice TEM forward model to approximate the BF-STEM bright-field contrast, motivated by the reciprocity between TEM and STEM image formation in thick, amorphous specimens. By formulating reconstruction as a physics-informed inverse problem, the proposed approach enables stable three-dimensional recovery from severely limited angular data [3]. Comparisons with weighted back-projection (WBP) reconstruction demonstrate a substantial reduction of missing-wedge artifacts and improved morphological consistency across slices, highlighting the potential of PIAD-based multislice modeling for interpretable 3D reconstruction from BF-STEM data in regimes where conventional tomography fails. The physical basis for this improvement arises from the multislice model, which explicitly accounts for nonlinear multiple scattering in thick specimens. We chose multislice TEM, a simple plane-wave propagation, as our forward model for BF-STEM image formation, because the BF-STEM contrast can be approximated as the incoherent angular average of TEM multislice intensities over the probe convergence aperture [4]. We further simplify this weighted incoherent sum by noting that, for thick amorphous specimens, the dominant contributions arise from low-angle (near-zero-angle) components, as higher-angle components are preferentially scattered outside the bright-field acceptance. Under this assumption, BF-STEM image formation is well approximated by a blurred BF-TEM plane-wave multislice output, which we adopt as an effective forward model for thickness-dominated bright-field contrast in limited-angle tomography. To test this framework experimentally, we image a butterfly wing scale (Bicyclus anynana) with nanoscale features on a laminar sheet with lateral dimensions of a few hundred microns [5]. We acquired a tilt series of 35 BF-STEM images between -51° and 51°. In limited-angle reconstructions, conventional WBP reconstructions fail to resolve cross-rib structures and are affected by anisotropic smearing and missing-wedge artifacts (Fig. 1a & 1d). In contrast, the proposed PIAD reconstruction yields improved crossrib continuity and junction definition in both volume rendering (Fig. 1b) and orthogonal slices (Fig. 1c’ &1d’), enabling more interpretable 3D morphology of the crossrib architecture. To assess the predictive capability of the forward model, we perform a leave-one-out validation in which a single BF-STEM projection (Fig. 2a) is excluded from the reconstruction. The resulting volume is then forward-projected at the held-out angle using the TEM multislice model and compared with the unseen experimental projection. Despite this effective approximation for BF-STEM, the predicted projections capture the dominant contrast trends associated with the crossrib network (Fig. 2a-2b). The residual map shows differences in the cross-rib edges, as expected from blur, yet has an RMSE of 0.08 and a Pearson correlation coefficient of 0.86. The back-propagated loss gradients highlight spatial regions that are well constrained by the data. Although the cross-ribs are more clearly resolved in PIAD reconstructions, the lower lamina remains unresolved in both WBP and PIAD reconstructions due to the absence of projections near 90°; future PIAD reconstructions using a laminography geometry may address this limitation. Importantly, the present results demonstrate that PIAD reconstructions using an approximate forward model that accounts for multiple scattering outperform conventional linear projection methods for BF-STEM tomography under limited-angle acquisition. Physics-informed auto-differentiation improves recovery of crossrib morphology under limited-angle acquisition in BF-STEM tomography. a) Conventional weighted back-projection (WBP) reconstruction from a single-axis BF-STEM tilt series, showing preservation of coarse rib geometry but loss and anisotropic smearing of thin crossrib features (red dashed region). b) PIAD reconstruction using a TEM multislice forward model improves cross-rib continuity and reduces missing-wedge artifacts. The inset shows a schematic of the actual scale structure. (c,d) Representative xy- and xz-slices from the WBP reconstruction. (c′,d′) Corresponding slices from the PIAD reconstruction, demonstrating enhanced cross-rib definition and thickness consistency. Scale bar = 300 nm. Leave-one-out projection validation of a TEM multislice forward model for BF-STEM tomography. One BF-STEM projection (a) is excluded from reconstruction, and the resulting volume is forward-projected at the held-out angle using the TEM multislice model. The simulated projection (b) is compared with the unseen experimental projection after fitting a per-projection gain and offset (c), (RMSE 0.08; Pearson correlation 0.86). (d) Back-propagated loss gradients (projection alone z-axis) from the leave-one-out error localize data-supported regions of the volume, including crossrib features, distinguishing them from underconstrained regions dominated by missing-wedge artifacts. Scale bar = 500 nm.

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