One-Shot Omnidirectional Pressure Integration Through Matrix Inversion

In this work, we present a method to perform 2D and 3D omnidirectional pressure integration from velocity measurements with a single-iteration matrix inversion approach. This work builds upon our previous work, where the rotating parallel ray approach was extended to the limit of infinite rays by taking continuous projection integrals of the ray paths and recasting the problem as an iterative matrix inversion problem. This iterative matrix equation is now ``fast-forwarded'' to the ``infinity'' iteration, leading to a different matrix equation that can be solved in a single iteration, thereby presenting the same computational complexity as the Poisson equation. We observe computational speedups of ~10⁶ when compared to brute-force omnidirectional integration methods, enabling the treatment of grids of ~ 10⁹ points and potentially even larger in a desktop setup at the time of publication. Further examination of the boundary conditions of our one-shot method shows that omnidirectional pressure integration implements a new type of boundary condition, which treats the boundary points as interior points to the extent that information is available.