Revisiting Liu and Katz (2006) and Zigunov and Charonko (2024b): on the Equivalence of the Omnidirectional Integration and the Pressure Poisson Equation

Abstract

In this work, we demonstrate the equivalency of the Rotating Parallel Ray Omnidirectional Integration (RPR-ODI) and the Pressure Poisson Equation (PPE) for pressure field reconstruction from corrupted image velocimetry data (dubbed 'ODI $\equiv$ PPE'). Building on the work by Zigunov and Charonko (2024b), we show that performing the ODI is equivalent to pursuing the minimum norm least squares (MNLS) or minimum norm (MN) solution to a Poisson equation with all Neumann boundary conditions. By looking through the lens of linear algebra, regression, optimization, and the well-posedness of the Poisson equation, we provide a comprehensive and integrated framework to analyze ODI/PPE-based pressure field reconstruction methods. The new comprehensions on ODI $\equiv$ PPE provides theoretical and computational insights valuable to experimentalists beyond reducing the high computational cost of ODI to that of PPE. More importantly, we i) provide a comprehensive guideline for robust pressure reconstruction, and ii) unveil the shared strengths and limitations of ODI and PPE, which are elaborated in remarks and notes throughout this work. Some remarks suggest simple regularization strategies that serve as 'minimal reproducible examples' and provide a foundation for further refinement. This work paves the way for further improvements in ODI/PPE-based pressure field reconstruction by utilizing the extensive literature on fast and robust elliptic solvers as well as their associated regularization methods. Numerical experiments are presented to support and illustrate these arguments.