Image Velocimetry using Direct Displacement Field estimation with Neural Networks for Fluids

TL;DR

This work addresses the challenge of achieving high-spatial-resolution velocity fields from image pairs without extensive training data. It introduces an unsupervised neural-network that directly estimates a displacement field δ(x,y) from image coordinates, using Fourier-feature embeddings to capture multi-frequency content, and derives the velocity via $U(x,y) = (C/Δt) δ(x,y)$. The approach yields a continuous velocity field at full image resolution, validated on synthetic and experimental data, achieving sub-pixel accuracy and physically meaningful spectra (e.g., Kolmogorov −5/3 in PSD). Its simplicity, lack of pretraining, and ability to operate on arbitrary image sizes make it a practical tool for high-resolution PIV and potentially physics-informed extensions.

Abstract

An important tool for experimental fluids mechanics research is Particle Image Velocimetry (PIV). Several robust methodologies have been proposed to perform the estimation of velocity field from the images, however, alternative methods are still needed to increase the spatial resolution of the results. This work presents a novel approach for estimating fluid flow fields using neural networks and the optical flow equation to predict displacement vectors between sequential images. The result is a continuous representation of the displacement, that can be evaluated on the full spatial resolution of the image. The methodology was validated on synthetic and experimental images. Accurate results were obtained in terms of the estimation of instantaneous velocity fields, and of the determined time average turbulence quantities and power spectral density. The methodology proposed differs of previous attempts of using machine learning for this task: it does not require any previous training, and could be directly used in any pair of images.

Figures (9)

• Figure 1: (a) Diagram of the architecture used and (b) iterative learning diagram.
• Figure 2: Learning example. From left to right columns, the first image, the inferred displacement, the deformed image, and a superposition of the particles on the deformed image (blue) and the second image (red). Rows are in increasing order of iteration, with the last row showing the ground-truth
• Figure 3: In the middle is the mean displacement field of 10 different initialisations of the method, an example of these results can be seen on the left, and on the right the standard deviation of the results. The effect of $\beta$ can be seen from top to bottom.
• Figure 4: Visual comparison, for illustration purposes between the ground truth (known displacements) and the estimated displacement vectors using the proposed methodology. For better comparison, the vorticity field is shown with the same scale for the colormap for both cases.
• Figure 5: On the top we have the loss for the 50 runs on each frame, in the middle the loss of the 43 runs that converged at the start, in the bottom the RMSE for the displacement of the particles on each frame for the 43 runs that converged. In red, the frames shown on Fig. \ref{['fig:examplejapan']}.