Volumetric flow field measurement deflectometry via a geometry self-consistent framework

Accurate three-dimensional flow-field measurement is vital for diagnostics in aerospace, combustion, and thermal processes. We present a multi-view phase measurement deflectometry (PMD) method for volumetric temperature-field reconstruction. By encoding absolute screen coordinates in fringe phases, PMD determines ray directions on a pixel-wise basis, ensuring intrinsic geometric self-consistency between the deflection measurement and the 3D reconstruction without relying on cross-correlation operations. A chain-based joint calibration strategy with anchor cameras accommodates surround-view configurations lacking a global common field of view. The tomographic inverse problem is solved within an alternating direction method of multipliers framework incorporating total-variation regularization and visual-hull support constraints. Numerical simulations and experiments on a thermal plume validate the proposed method, with thermocouple comparison showing a mean absolute relative difference of 1.87% and a mean absolute difference of 26.3 K, offering a new, to the best of our knowledge, pathway toward high-resolution volumetric flow diagnostics.