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Fast and robust volumetric refractive index measurement by unified background-oriented schlieren tomography

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Abstract

We propose a novel approach to background-oriented schlieren (BOS) tomography (BOST) that unifies the deflection sensing and reconstruction algorithms. BOS is a 2D flow visualization technique that renders light deflections due to refraction in the fluid. Simultaneous BOS measurements from unique views can be reconstructed by tomography to estimate the fluid’s 3D refractive index field. The cameras are focused through the fluid on textured background patterns. Deflections between an undistorted reference image and distorted image are typically determined by gradient-based optical flow (OF), which is a complex inverse problem and potential source of error in BOST. This paper presents an alternative approach to BOST that unifies the OF equations and deflection model. Our new operator simultaneously calculates the image distortions seen by each camera for a discrete refractive index distribution. Unified BOST (UBOST) thus reconstructs observed image distortions instead of inferred deflections, which are influenced by user-selected OF parameters. The UBOST operator has one third as many equations as the classical BOST operator. We show that our formulation reduces the effects of model error and the computational cost of reconstruction. These advantages are demonstrated with a numerical experiment using phantoms of varied complexity. Best practice UBOST reconstructions were more accurate than classical reconstructions of the exact deflections for each phantom. Moreover, UBOST estimates converged substantially faster, resulting in a \(\ge \) 62.5% speedup with our solver.

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Notes

  1. PIV algorithms are sometimes considered to be distinct from OF. However, since both methods yield the apparent displacement field described by Davies (2004), we opt for the consolidated taxonomy of Atcheson et al. (2009).

  2. See Chapter 2, Eqs. (2.3) to (2.5), and Appendix A.6 of Settles (2001) for an alternative derivation of Eq. (6) in terms of the angular deflection of a ray.

  3. Axisymmetric BOST uses a single camera but implies the same set of deflections for infinite rotations of this view, so the same argument applies in that case.

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Acknowledgements

The authors thank Dr. Andreas Rittler and Prof. Khadijeh Mohri for their assistance with the swirl flame phantom. The authors also gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC).

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  1. Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Samuel J. Grauer & Adam M. Steinberg

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  1. Samuel J. Grauer
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Correspondence to Samuel J. Grauer.

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Grauer, S.J., Steinberg, A.M. Fast and robust volumetric refractive index measurement by unified background-oriented schlieren tomography. Exp Fluids 61, 80 (2020). https://doi.org/10.1007/s00348-020-2912-1

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