Accurately reconstructing a radial refractive index (n) field is challenging in axisymmetric background oriented Schlieren (BOS) measurement. In this study, we systematically investigated several widely adopted inversion algorithms in BOS applications. To quantitatively assess the performance of each algorithm, a synthetic experiment mimicking a helium jet discharged into ambient air was established to provide the reference. Relying on the necessity to solve a Poisson equation for a line-of-sight projected variable, tested algorithms were categorized into two groups: direct and indirect. In the direct approach, the algorithm is applied directly to the light deflection angle ( $$\varepsilon $$ ) to reconstruct the radial $$\delta $$ field, defined as $$\delta =(n-n_{0})/n_{0}$$ where $$n_{0}$$ is the reference refractive index. In the indirect group, the Poisson equation is solved first. Then, an inversion algorithm is subsequently applied to the projected $${{\overline{\delta }}}$$ to obtain $$\delta $$ in the radial plane. The two approaches were compared with the synthetic experiment both using the adaptive Fourier–Hankel methods (AFH). The comparison showed that at the cost of introducing the additional step of solving the Poisson equation, the indirect approach performed more accurately when noises were present in the $$\varepsilon $$ measurements. To identify the proper inversion algorithm suitable for the indirect approach, we further compared four types of algorithms in the synthetic experiments including AFH, onion peeling (OP), three-point Abel (TPA), and filtered back projection tomography (FBPT). The results showed that TPA had the best performance in terms of the reconstruction accuracy with noisy $$\varepsilon $$ data. Finally, experiments on axisymmetric helium jets were conducted to confirm the effectiveness of the proposed TPA algorithm in the indirect approach.

中文翻译：

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面向轴对称流动的稳健 BOS 测量

在轴对称背景导向纹影 (BOS) 测量中，准确重建径向折射率 (n) 场具有挑战性。在这项研究中，我们系统地研究了 BOS 应用程序中几种广泛采用的反演算法。为了定量评估每种算法的性能，建立了一个模拟氦射流排放到环境空气中的合成实验，以提供参考。根据对视线投影变量求解泊松方程的必要性，测试算法分为两类：直接和间接。在直接方法中，该算法直接应用于光偏转角（ $$\varepsilon $$ ）以重建径向 $$\delta $$ 场，定义为 $$\delta =(n-n_{0}) /n_{0}$$ 其中 $$n_{0}$$ 是参考折射率。在间接组中，首先求解泊松方程。然后，随后将反演算法应用于投影 $${{\overline{\delta }}}$$ 以获得径向平面中的 $$\delta $$。将这两种方法与使用自适应傅立叶-汉克尔方法 (AFH) 的合成实验进行了比较。比较表明，以引入求解 Poisson 方程的额外步骤为代价，当 $$\varepsilon $$ 测量中存在噪声时，间接方法执行得更准确。为了确定适合间接方法的正确反演算法，我们进一步比较了合成实验中的四种算法，包括 AFH、洋葱皮 (OP)、三点 Abel (TPA) 和滤波反投影断层扫描 (FBPT)。结果表明，TPA 在噪声 $$\varepsilon $$ 数据的重建精度方面具有最佳性能。最后，对轴对称氦射流进行了实验，以证实所提出的 TPA 算法在间接方法中的有效性。