Uncertainty-based weighted least squares density integration for background-oriented schlieren
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RESEARCH ARTICLE
Uncertainty‑based weighted least squares density integration for background‑oriented schlieren Lalit Rajendran1 · Jiacheng Zhang2 · Sally Bane1 · Pavlos Vlachos2 Received: 4 April 2020 / Revised: 20 July 2020 / Accepted: 6 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose an improved density integration methodology for Background-Oriented Schlieren (BOS) measurements that overcomes the noise sensitivity of the commonly used Poisson solver. The method employs a weighted least-squares (WLS) optimization of the 2D integration of the density gradient field by solving an over-determined system of equations. Weights are assigned to the grid points based on density gradient uncertainties to ensure that a less reliable measurement point has less effect on the integration procedure. Synthetic image analysis with a Gaussian density field shows that WLS constrains the propagation of random error and reduces it by 80% in comparison to Poisson for the highest noise level. Using WLS with experimental BOS measurements of flow induced by a spark plasma discharge shows a 30% reduction in density uncertainty in comparison to Poisson, thereby increasing the overall precision of the BOS density measurements.
Lalit Rajendran and Jiacheng Zhang contributed equally to this work. * Pavlos Vlachos [email protected] 1
School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA
School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA
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Page 2 of 12
Experiments in Fluids
(2020) 61:239
Graphic abstract
Weighted Least Squares (WLS) Density Integration for Background Oriented Schlieren (BOS) 80% reduction in density random error
Formulate Integration as Least Squares Optimization
Poisson
WLS
Weight Matrix 30% reduction in density uncertainty Poisson
WLS
Weights based on Displacement/Density Gradient Uncertainty
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1 Introduction and methodology Background-Oriented Schlieren (BOS) is an optical technique used to measure density gradients by tracking the apparent distortion of a target dot pattern (Raffel 2015). The apparent displacement is obtained by comparing the distorted image and a reference image without the density gradients, and the estimation can be performed by cross-correlation, tracking, or optical flow algorithms (Raffel 2015; Atcheson et al. 2009; Rajendran et al. 2019). This displacement is related to the density gradient field and the optical layout parameters as given by
⃗= ΔX
MZD K ∇𝜌dz, n0 ∫
(1)
where ΔX�⃗ is the apparent displacement, M is the magnification of the dot pattern, ZD is the distance between the dot pattern and the mid-plane of the density gradient field, K is the Gladstone–Dale constant (= 0.225 × 10−3kg/m3 for air), n0 is the ambient refractive index, ∇𝜌 is the density gradient field
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and z is the co-ordinate along the viewing axis. The integral is over the depth/thickness of the density gradient field. Given the apparent displacement from the i
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