Approximate Bayesian framework for 3D reconstruction in a volumetric PIV/PTV measurement

Abstract

Non-invasive flow velocity measurement techniques like volumetric Particle Image Velocimetry (PIV) (Elsinga et al., 2006; Adrian and Westerweel, 2011) and Particle Tracking Velocimetry (PTV) (Maas, Gruen and Papantoniou, 1993) use multi-camera projections of tracer particle motion to resolve three-dimensional flow structures. A key step in the measurement chain involves reconstructing the 3D intensity field (PIV) or particle positions (PTV) given the projected images and known camera correspondence. Due to limited number of camera-views the projected particle images are non-unique making the inverse problem of volumetric reconstruction underdetermined. Moreover, higher particle concentration (>0.05 ppp) increases erroneous reconstructions or “ghost” particles and decreases reconstruction accuracy. Current reconstruction methods either use voxel-based representation for intensity reconstruction (e.g. MART (Elsinga et al., 2006)) or a particle-based approach (e.g. IPR (Wieneke, 2013)) for 3D position estimation. The former method is computationally intensive and has a lesser positional accuracy due to stretched shape of the reconstructed particle along the line of sight. The latter compromises triangulation accuracy (Maas, Gruen and Papantoniou, 1993) due to overlapping particle images for higher particle concentrations. Thus, each method has its own challenges and the error in 3D reconstruction significantly affects the accuracy of the velocity measurement. Though, other methods like maximum-a-posteriori (MAP) estimation have been previously developed (Levitan and Herman, 1987; Bouman and Sauer, 1996) for computed Tomography data, it has not been explored for PIV/ PTV 3D reconstruction. Here, we use a MAP estimation framework to model and solve the inverse problem. The cost function is optimized using a stochastic gradient ascent (SGA) algorithm. Such an optimization can converge to a better local maximum and also use smaller image patches for efficient iterations.