DeepCormack: Fermi surface tomography using model-based data-driven algorithms
Authors: Georg F. B. Lovric, Bryn Drury, Carola-Bibiane Schönlieb, Stephen B. Dugdale, Ander Biguri
Organizations: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK. · H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK
The experimental reconstruction of the 3D two-photon momentum density (TPMD) via angular correlation of electron-positron annihilation radiation (ACAR) is a particularly useful method for studying material Fermi surfaces. It does not rely on low temperatures, UHV conditions, or strong magnetic fields, and enables the study of the spin-resolved electronic structure of materials. Yet, it remains a challenging inverse problem. Typically, 10^8 positron annihilation events are measured for 3--6 projections of the TPMD at different angles. The standard reconstruction approach is an ACAR adaptation of Cormack's method (the MCM) that leverages the inherent symmetry in the crystal's structure. However, the poor signal-to-noise ratio means collecting data of sufficient quality for Fermi surface studies can take months per sample. We present DeepCormack, a family of data-driven model-based reconstruction algorithms that augments the MCM by integrating supervised deep-learning models (CNN, MLP, and UNet) at various stages. To overcome the lack of large experimental training sets, we propose a method which leverages singular value decomposition with dynamic mode decomposition to generate realistic synthetic TPMD volumes, requiring only a single reference momentum density computed via density functional theory. On test data, DeepCormack improves reconstruction quality over MCM by about 8.5 dB PSNR at 200M counts and remains stable at reduced counts, enabling significantly faster acquisition times. Generalisation to experimental data depends strongly on how well the training distribution from the reference momentum density matches the sample. We therefore recommend pairing DeepCormack with a DFT calculation of the target material to create sample-specific training data. Our proposed method offers either much higher quality reconstructions, or enables significantly faster ones, on the order of weeks.
Electron tomography (ET) plays an important role in the three-dimensional (3D) characterization of nanomaterials. However, under limited-angle and sparse-view conditions, conventional algorithms produce degraded reconstructions, which compromise the quality and interpretability of resulting 3D data. In this paper, we present deep image prior (DIP), an unsupervised deep learning (DL) approach, for highly degraded tomography acquisitions and demonstrate, using simulated data, that its performance is comparable to that of supervised approaches requiring training datasets, even for tilt ranges as limited as 60° and tilt increments of 10°. We then apply it to experimental data and show that it enables reliable 3D quantification under both sparse-view and limited-angle conditions, highlighting its potential for a wide range of materials and acquisition modalities.
Serge Brosset, Daniel del Pozo Bueno, Thomas David +3
Precise 3D characterization of nanomaterials is essential for unlocking structure-property relationships. However, standard electron tomography is fundamentally limited by the missing wedge problem. Consequently, conventional algorithms suffer from severe geometric distortions, a challenge further complicated by pervasive noise interference. Current learning-based methods either rely on physics-blind post-processing or employ end-to-end architectures constrained by local receptive fields, failing to capture complex 3D topologies. We propose NanoMorph-3D, a unified end-to-end framework grounded in a comprehensive Nanomorphological Taxonomy. Powered by a large-scale synthetic dataset explicitly modeling non-linear electron attenuation, we design a Physics-Driven Unrolled Network mapping proximal gradient descent into a learnable architecture. To capture complex internal topologies, we formulate a hierarchical attention mechanism with Physics-Normalization for long-range 3D dependencies and scale invariance. Crucially, our Dual-Domain strategy leverages Sinusoidal Attention to explicitly model physical projection trajectories, enforcing strict sinogram consistency to mitigate missing wedge artifacts. Finally, an unsupervised dual-stream mechanism bridges the simulation-to-reality gap. Experiments demonstrate NanoMorph-3D reconstructs diverse topologies with superior fidelity and speed.
Generative models, particularly Diffusion Models (DM), have shown strong potential for Computed Tomography (CT) reconstruction serving as expressive priors for solving ill-posed inverse problems. However, diffusion-based reconstruction relies on Stochastic Differential Equations (SDEs) for forward diffusion and reverse denoising, where such stochasticity can interfere with repeated data consistency corrections in CT reconstruction. Since CT reconstruction is often time-critical in clinical and interventional scenarios, improving reconstruction efficiency is essential. In contrast, Flow Matching (FM) models sampling as a deterministic Ordinary Differential Equation (ODE), yielding smooth trajectories without stochastic noise injection. This deterministic formulation is naturally compatible with repeated data consistency operations. Furthermore, we observe that FM-predicted velocity fields exhibit strong correlations across adjacent steps. Motivated by this, we propose an FM-based CT reconstruction framework (FMCT) and an efficient variant (EFMCT) that reuses previously predicted velocity fields over consecutive steps to substantially reduce the number of Neural network Function Evaluations (NFEs), thereby improving inference efficiency. We provide theoretical analysis showing that the error introduced by velocity reuse is bounded when combined with data consistency operations. Extensive experiments demonstrate that FMCT/EFMCT achieve competitive reconstruction quality while significantly improving computational efficiency compared with diffusion-based methods. The codebase is open-sourced at https://github.com/EFMCT/EFMCT.