Superresolution imaging across scales

<title>Abstract</title> <p>The Sparrow resolution limit defines the minimum separation at which two incoherent point sources can be distinguished based on the resulting intensity distribution. This limit arises when the photon detection probability distributions of the two sources overlap such that the combined profile exhibits no discernible dip between peaks. In this regime, individual photon emissions become statistically indistinguishable, and the sources appear as a single emitter. Surpassing this limit represents a critical advance in imaging, enabling the resolution of quantum systems and nanoscale structures beyond conventional spatial constraints. In this work, we provide the first rigorous mathematical proof that the explicit MeanShift vector can resolve Gaussian-distributed sources beyond the Sparrow limit, establishing a theoretical foundation for previously observed empirical results. These theoretical results are complemented with simulations and real imaging data from fluorescence microscopy, astronomy, and X-ray spectroscopy, illustrating the broad applicability of our approach. These findings establish a formal theoretical link between MS-based methods and classical resolution criteria, advancing theoretical insights for superresolution imaging.</p>