Diffeomorphic Particle Image Velocimetry

The existing particle image velocimetry (PIV) techniques do not consider the curvature effect of the nonstraight particle trajectory because it seems to be impossible to obtain the curvature information from a pair of particle images. In this work, the particle curved trajectory between two recordings is first explained with the streamline segment of a steady flow (diffeomorphic transformation) instead of a single vector, and this novel idea is termed diffeomorphic PIV. Specifically, a deformation field is introduced to describe the particle displacement along the streamline, i.e., we try to find the optimal velocity field, of which the corresponding deformation vector field agrees with the particle displacement. Because the variation of the deformation function can be approximated with the variation of the velocity function, the diffeomorphic PIV can be implemented as special iterative PIV. This says that the diffeomorphic PIV warps the images with deformation vector field instead of velocity field and keeps the rest procedures as the same as a conventional iterative PIV. Two diffeomorphic deformation schemes-forward diffeomorphic deformation interrogation (FDDI) and central diffeomorphic deformation interrogation (CDDI)-are proposed in this article. Tested on synthetic PIV images, the FDDI achieves significant accuracy improvement across different one-pass displacement estimators (cross correlation, optical flow (OF), and deep learning flow). Besides, the results on three real PIV image pairs demonstrate the nonnegligible curvature effect for central difference interrogation (CDI)-based measurement, and our FDDI provides larger velocity estimation-more accurate-in the fast curvy streamline areas. The significant accuracy improvement of the combination of FDDI and accurate dense estimator (e.g., OF) means that our diffeomorphic PIV paves a completely new way for complex flow field measurement.