Generative-model-based null-space iterative reconstruction for atomic electron tomography with sparse data

Atomic electron tomography (AET) is a powerful technique for determining the three-dimensional atomic structure of matter in real space. However, conventional AET requires numerous projections across a wide angular range. The high dose and prolonged acquisition severely limit its application. Here, we propose an interpretable and universal algorithm for low-dose, fast and tilt-constrained AET: null-space iterative reconstruction (NSIRE), which uses an unsupervised diffusion model to iteratively compute null-space solutions of projection equations, thereby generating tomograms consistent with both projection constraints and atomic potential prior. NSIRE can resolve a wide range of complex materials under 6-12° sparse projection or within ±29° small tilt-range without retraining. Using the NSIRE, we determine the three-dimensional atomic structures of a 4-nm Pt nanoparticle with grain boundaries and a 3-nm PtCo nanoalloy (the smallest one resolved so far), achieving a root-mean-square displacement of <20 pm and high projection consistency, overcoming the scale, dose and time limitations of AET.