Critical Parameter Values and Reconstruction Properties of Discrete Tomography: Application to Experimental Fluid Dynamics

TL;DR

This paper investigates the conditions for unique volume reconstruction in discrete tomography for fluid dynamics, analyzing how sparsity and geometry affect reconstruction success and connecting it to compressed sensing theory.

Contribution

It provides a probabilistic analysis of unique reconstruction conditions and the transition to non-uniqueness in tomographic PIV, linking experimental parameters to mathematical guarantees.

Findings

Identifies critical sparsity thresholds for unique reconstruction.

Characterizes the transition from unique to non-unique solutions.

Connects PIV reconstruction to compressed sensing frameworks.

Abstract

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 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.