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Title: Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments

Abstract

Here, 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 (AF), 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 amore » 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.« less

Authors:
 [1]; ORCiD logo [1]; ORCiD logo [1]
  1. Purdue Univ., West Lafayette, IN (United States)
Publication Date:
Research Org.:
Purdue Univ., West Lafayette, IN (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Fusion Energy Sciences (FES)
OSTI Identifier:
1657789
Grant/Contract Number:  
SC0018156
Resource Type:
Accepted Manuscript
Journal Name:
Experiments in Fluids
Additional Journal Information:
Journal Volume: 61; Journal Issue: 6; Journal ID: ISSN 0723-4864
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING

Citation Formats

Rajendran, Lalit K., Bane, Sally M., and Vlachos, Pavlos P.. Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments. United States: N. p., 2020. Web. https://doi.org/10.1007/s00348-020-02978-8.
Rajendran, Lalit K., Bane, Sally M., & Vlachos, Pavlos P.. Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments. United States. https://doi.org/10.1007/s00348-020-02978-8
Rajendran, Lalit K., Bane, Sally M., and Vlachos, Pavlos P.. Fri . "Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments". United States. https://doi.org/10.1007/s00348-020-02978-8. https://www.osti.gov/servlets/purl/1657789.
@article{osti_1657789,
title = {Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments},
author = {Rajendran, Lalit K. and Bane, Sally M. and Vlachos, Pavlos P.},
abstractNote = {Here, 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 (AF), 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.},
doi = {10.1007/s00348-020-02978-8},
journal = {Experiments in Fluids},
number = 6,
volume = 61,
place = {United States},
year = {2020},
month = {5}
}

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