  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 $\sim10^6$ when compared to brute-force omnidirectional integration
methods, enabling the treatment of grids of $\sim 10^9$ 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. Finally, we show how the method can be extended from the regular
grids typical of particle image velocimetry to the unstructured meshes
characteristic of particle tracking velocimetry data.
