Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications
Discrete & Computational Geometry
Sparse Recovery of Nonnegative Signals With Minimal Expansion
IEEE Transactions on Signal Processing
A Unique “Nonnegative” Solution to an Underdetermined System: From Vectors to Matrices
IEEE Transactions on Signal Processing
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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We analyze representative ill-posed scenarios of tomographic PIV particle image velocimetry 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 the 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. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing.