ReynoldsFlow: Exquisite Flow Estimation via Reynolds Transport Theorem

TL;DR

This work tackles the limitations of traditional and data-hungry deep optical flow approaches by introducing Reynolds flow, a training-free framework grounded in the Reynolds transport theorem that generalizes motion estimation beyond brightness constancy. By decomposing motion into irrotational and solenoidal components (via the Helmholtz decomposition) and deriving an irrotational Reynolds flow $\boldsymbol{v}_r$, the method augments conventional flow with lighting-variant residuals while remaining computationally efficient. It further introduces ReynoldsFlow+, an RGB-encoded representation that stacks $|\boldsymbol{v}_o|$, $|\boldsymbol{v}_r|$, and frame intensity $f^n$, improving motion visibility for downstream networks. Across UAVDB, Anti-UAV, and GolfDB, ReynoldsFlow+ achieves state-of-the-art performance and demonstrates robustness and efficiency suitable for real-time and edge deployments, offering a practical preprocessing module to enhance motion-related perception without additional training.

Abstract

Optical flow is a fundamental technique for motion estimation, widely applied in video stabilization, interpolation, and object tracking. Traditional optical flow estimation methods rely on restrictive assumptions like brightness constancy and slow motion constraints. Recent deep learning-based flow estimations require extensive training on large domain-specific datasets, making them computationally demanding. Also, artificial intelligence (AI) advances have enabled deep learning models to take advantage of optical flow as an important feature for object tracking and motion analysis. Since optical flow is commonly encoded in HSV for visualization, its conversion to RGB for neural network processing is nonlinear and may introduce perceptual distortions. These transformations amplify the sensitivity to estimation errors, potentially affecting the predictive accuracy of the networks. To address these challenges that are influential to the performance of downstream network models, we propose Reynolds flow, a novel training-free flow estimation inspired by the Reynolds transport theorem, offering a principled approach to modeling complex motion dynamics. In addition to conventional HSV-based visualization of Reynolds flow, we also introduce an RGB-encoded representation of Reynolds flow designed to improve flow visualization and feature enhancement for neural networks. We evaluated the effectiveness of Reynolds flow in video-based tasks. Experimental results on three benchmarks, tiny object detection on UAVDB, infrared object detection on Anti-UAV, and pose estimation on GolfDB, demonstrate that networks trained with RGB-encoded Reynolds flow achieve SOTA performance, exhibiting improved robustness and efficiency across all tasks.

Figures (3)

• Figure 1: The first column shows original frames from UAVDB chen2024uavdb (top), Anti-UAV jiang2021anti (middle), and GolfDB mcnally2019golfdb (bottom). The second and third columns depict Lucas-Kanade optical flow and ReynoldsFlow, both visualized using the HSV color space. The last column presents the proposed ReynoldsFlow+ visualization, which enhances motion features in complex scenes.
• Figure 2: Visualization of inputs and HiResCAM heatmaps for UAVDB and Anti-UAV, with the UAV highlighted in the green box region.
• Figure 3: Comparison of event probabilities in a golf swing video clip using original video and ReynoldsFlow+ inputs. (a) Displays the eight key events with corresponding RGB frames and ReynoldsFlow+ visualizations. (b) Shows the probability curves for each event throughout the entire video. The x-axis represents the frame number ($n$), while the y-axis denotes the event probability ($P_{\text{event}}$). The predicted event frame is determined by selecting the frame with the highest probability.