Revisit Liu and Katz (2006) and Zigunov and Charonko (2024): On the Equivalency of Omni-directional Integration and Pressure Poisson Equation, and the Compatibility Condition

In this work, we demonstrate the equivalency of Omni-Directional Integration (ODI) and the Pressure Poisson Equation (PPE) for pressure field reconstruction from corrupted image velocimetry data. We show that ODI is equivalent to pursuing the minimal norm solution to a Poisson equation with Neumann boundary conditions and that a minimal norm solution automatically satisfies the compatibility condition required by the existence of the solution. We explain why some studies have reported poor robustness of the PPE and how this is often rooted in a lack of regularization to satisfy the compatibility consideration. Our new comprehensions on the equivalence of ODI and PPE not only provide insights into why ODI and the minimal norm solution of PPE are robust against random noise in the data, but, more importantly, reduce the immense computational cost of ODI to that of PPE. This work leads the way for further improvement to PPE/ODI-based pressure field reconstruction by leveraging the established fast and robust numerical methods for elliptic equations and the corresponding regularization methods. We conclude our analysis by providing a "minimalism" regularization for experimentalists for robust pressure field reconstruction, anything beyond which would require significantly more work.