Abstract Measuring particles' three-dimensional (3D) positions using multi-camera images in fluid dynamics is critical for resolving spatiotemporally complex flows like turbulence and mixing. However, current methods are prone to errors due to camera noise, optical configuration and experimental setup limitations, and high seeding density, which compound to create fake measurements (ghost particles) and add noise and error to velocity estimations. We introduce a Bayesian Volumetric Reconstruction (BVR) method, addressing these challenges by using probability theory to estimate uncertainties in particle position predictions. Our method assumes a uniform distribution of particles within the reconstruction volume and employs a model mapping particle positions to observed camera images. We utilize variational inference with a modified loss function to determine the posterior distribution over particle positions. Key features include a penalty term to reduce ghost particles, provision of uncertainty bounds, and scalability through subsampling. In tests with synthetic data and four cameras, BVR achieved 95% accuracy with less than 3% ghost particles and an RMS error under 0.3 pixels at a density of 0.1 particles per pixel (ppp). This shows 57% to 97% less ghost particles and 1.3 to 2 times lower RMS error than standard MART and IPR reconstructions. In an experimental Poiseuille flow measurement, our method closely matched the theoretical solution. Additionally, in a complex cerebral aneurysm basilar tip geometry flow experiment, our reconstructions were dense and consistent with observed flow patterns. Our BVR method effectively reconstructs particle positions in complex 3D flows, particularly in situations with high particle image densities and camera distortions. It distinguishes itself by providing quantifiable uncertainty estimates and scaling efficiently for larger image dimensions, making it applicable across a range of fluid flow scenarios.
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