The "Intensity" Countoscope: Measuring particle dynamics in real space from microscopy images

Advances in intensity-based microscopy techniques have improved our ability to quantify particle motion at microscopic scales, enabling insight into diffusion and collective dynamics. Building on this foundation, we introduce a novel real-space approach that analyses intensity fluctuations within virtual observation boxes of variable size on microscopy images. By correlating these signals, we uncover distinct temporal regimes in the mean square changes of intensity, $\langle \Delta I^2(t) \rangle$, which are strongly dependent on the box size compared to the particle width. For small boxes or short timescales, $\langle \Delta I^2(t) \rangle$ scales with the mean-square displacement, while for longer timescales and larger boxes, it scales with its square root. We develop a general theoretical framework that captures these regimes and explicitly apply it to a dilute colloidal suspension imaged with confocal microscopy as an experimental model system. This allows for a robust extraction of diffusion coefficients and physical insights into particle dynamics. Our method complements intensity-based and real-space analysis, offering a tool for studying individual and potentially collective behaviour directly from image intensities, even in systems where individual particles cannot be resolved.