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Optical Stokes flow estimation: an imaging-based control approach

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Abstract

We present an approach to particle image velocimetry based on optical flow estimation subject to physical constraints. Admissible flow fields are restricted to vector fields satifying the Stokes equation. The latter equation includes control variables that allow to control the optical flow so as to fit to the apparent velocities of particles in a given image pair. We show that when the real unknown flow observed through image measurements conforms to the physical assumption underlying the Stokes equation, the control variables allow for a physical interpretation in terms of pressure distribution and forces acting on the fluid. Although this physical interpretation is lost if the assumptions do not hold, our approach still allows for reliably estimating more general and highly non-rigid flows from image pairs and is able to outperform cross-correlation based techniques.

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Notes

  1. Without loss of generality we take Δx = Δy = 1.

  2. Note that we confine ourselves to the time-independent case as we want to analyze image pairs only and therefore have no additional information about the temporal evolution of the velocity.

  3. Note that due to the regularizer, we will still get reliable velocity estimates at these locations.

  4. This is only true when solving the problem with the Stokes equation. If we had used the Navier–Stokes equations, the pressure distribution would read \({\frac{\partial p} {\partial r}} = {\frac{\sigma v^2}{r}}.\)

  5. We assume that the (imaginary) grid in out-of-plane direction has the same resolution as the in-plane grid.

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Acknowledgments

The authors thank Johan Carlier (Cemagref) for providing the turbulent image pairs and Rainer Hain (TU Braunschweig) for providing the cross-correlation estimates. Support by the Deutsche Forschungsgemeinschaft (DFG, SCHN 457/6) within the priority programme “Bildgebende Messverfahren in der Strömungsmechanik” ( www.spp1147.tu-berlin.de ) and by the EU project “Fluid Image Analysis and Description” ( http://www.//fluid.irisa.fr/ ) is gratefully acknowledged.

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  1. Computer Vision, Graphics, and Pattern Recognition Group, Department of Mathematics and Computer Science, University of Mannheim, 68131, Mannheim, Germany

    Paul Ruhnau & Christoph Schnörr

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  1. Paul Ruhnau
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  2. Christoph Schnörr
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Correspondence to Christoph Schnörr.

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Ruhnau, P., Schnörr, C. Optical Stokes flow estimation: an imaging-based control approach. Exp Fluids 42, 61–78 (2007). https://doi.org/10.1007/s00348-006-0220-z

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