Reconstruction Properties of Sensing Matrices in Tomographic Particle Imaging Velocimetry

We analyze representative ill-posed scenarios of tomographic PIV with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine (i) the ability of unique reconstruction in terms of the imaging geometry and a single critical sparsity parameter, and (ii) sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Accordingly, we also comment on the role of various reconstruction algorithms currently used in PIV from the optimization point of view. Finally, we outline connections to major developments in other disciplines (compressed sensing) and indicate how the imaging set-up may be further improved. The present paper is a (very) short version of the forthcoming manuscrip