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.

本文研究了与在包含折射面的光学系统中使用不同相机模型相关的校准和测量不确定度。折射表面是具有不同光学特性的介质之间的界面，由于视线的折射，它会在成像过程中引入失真。这是所有关于在透明固体几何形状周围或内部流动的流体研究的共同问题，并且与固体/流体界面的强曲率相​​关。正如之前的工作（Paolillo 和 Astarita 2020 IEEE Trans. Pattern Anal. Mach. Intell.）。另一方面，具有纯数学基础的分析相机模型，如基于多项式或有理函数的模型，经典用于 PIV/PTV 社区。由于所涉及的失真的非线性特性，这些模型在表示存在折射几何的情况下成像过程的准确性在很大程度上取决于用于校准的数据的多项式阶数和噪声。目前的工作通过使用参考折射相机模型生成的数据，对分析相机模型固有的不确定性进行了数值估计。目前的结果表明，高精度需要高阶，这意味着大量的校准参数和对计算资源的高需求。特别是，与多项式相比，有理映射函数表现出优越的性能，尽管发现它们的校准对图像噪声敏感，并且可能会产生较大的外推误差。还报告了一个实验验证，它表明对于速度统计的估计，7 阶多项式模型提供的结果与折射相机模型的结果相当。