Model-independent machine learning approach for nanometric axial localization and tracking

Accurately tracking particles and determining their coordinate along the optical axis is a major challenge in optical microscopy, especially when extremely high precision is needed. In this study, we introduce a deep learning approach using convolutional neural networks (CNNs) that can determine axial coordinates from dual-focal-plane images without relying on predefined models. Our method achieves an axial localization precision of 40 nanometers — six times better than traditional single-focal-plane techniques. The model's simple design and strong performance make it suitable for a wide range of uses, including dark matter detection, proton therapy for cancer, and radiation protection in space. It also shows promise in fields like biological imaging, materials science, and environmental monitoring. This work highlights how machine learning can turn complex image data into reliable, precise information, offering a flexible and powerful tool for many scientific applications.