On the PIV/PTV uncertainty related to calibration of camera systems with refractive surfaces

This paper investigates the calibration and measurement uncertainty related to the use of different camera models in optical systems that include refractive surfaces. A refractive surface is an interface between media with different optical properties which introduces distortions in the imaging process due to the refraction of the lines-of-sight. This is an issue common to all the investigations of fluids flowing around or inside transparent solid geometries and is of relevance for a strong curvature of the solid/fluid interface. Appropriate modelling of the refractive effects is possible by integrating the pinhole camera model with a ray-tracing method, as demonstrated in a previous work (Paolillo and Astarita 2020 IEEE Trans. Pattern Anal. Mach. Intell. ). On the other side, analytical camera models with a pure mathematical foundation, like those based on polynomials or rational functions, are classically used in the PIV/PTV community. Due to the non-linear nature of the involved distortions, the accuracy of these models in representing the imaging process in presence of refractive geometries depends strongly on the polynomial order and noise of the data used for the calibration. The current work provides a numerical estimate of the uncertainty inherent to the analytical camera models by using data generated via a reference refractive camera model. The present results show that high accuracy requires high orders, which implies a large number of calibration parameters and high demand for computational resources. In particular, the rational mapping functions exhibit superior performance compared to the polynomials, although their calibration is found to be sensitive to image noise and they might yield large extrapolation errors. An experimental verification is also reported, which shows that for the estimation of the velocity statistics a 7th order polynomial model offers results comparable to those of a refractive camera model.