Abstract We theoretically analyze the effect of density/refractive index gradients on the measurement precision of background-oriented schlieren (BOS) experiments by deriving the Cramer–Rao lower bound (CRLB) for the 2D centroid estimation process. A model is derived for the diffraction limited image of a dot viewed through a medium containing density gradients that includes the effect of the experimental parameters such as the magnification and f-number. It is shown using the model that nonlinearities in the density gradient field lead to blurring of the dot image. This blurring amplifies the effect of image noise on the centroid estimation process, leading to an increase in the CRLB and a decrease in the measurement precision. The ratio of position uncertainties of a dot in the reference and gradient images is shown to be a function of the ratio of the dot diameters and dot intensities. We termed this parameter the amplification ratio ( $$A_{\text{F}}$$ A F ), and a methodology for reporting position uncertainties in tracking-based BOS measurements is proposed. The theoretical predictions of the dot position estimation variance from the CRLB are compared to ray tracing simulations, and agreement is obtained. The uncertainty amplification is also demonstrated on experimental BOS images of flow induced by a spark discharge, where it is seen that regions of high amplification ratio correspond to regions of density gradients. This analysis elucidates the dependence of the position uncertainty on density and refractive index gradient-induced distortion parameters, provides a methodology for accounting its effect on uncertainty quantification and provides a framework for optimizing experiment design. Graphic abstract

中文翻译：

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背景导向纹影实验中密度/折射率梯度引起的不确定度放大

摘要 我们通过推导二维质心估计过程的 Cramer-Rao 下界 (CRLB)，从理论上分析了密度/折射率梯度对背景导向纹影 (BOS) 实验测量精度的影响。通过包含密度梯度的介质观察到的点的衍射受限图像推导出模型，该介质包括实验参数（如放大率和 f 值）的影响。使用模型表明，密度梯度场中的非线性导致点图像模糊。这种模糊放大了图像噪声对质心估计过程的影响，导致 CRLB 增加和测量精度下降。参考图像和梯度图像中点的位置不确定性比率显示为点直径和点强度比率的函数。我们将此参数称为放大率 ($$A_{\text{F}}$$ AF )，并提出了一种在基于跟踪的 BOS 测量中报告位置不确定性的方法。将来自 CRLB 的点位置估计方差的理论预测与光线追踪模拟进行比较，并获得一致。在由火花放电引起的流动的实验 BOS 图像上也证明了不确定性放大，其中可以看到高放大率区域对应于密度梯度区域。该分析阐明了位置不确定性对密度和折射率梯度引起的畸变参数的依赖性，提供了一种计算其对不确定性量化影响的方法，并提供了优化实验设计的框架。图形摘要