# High-resolution velocimetry from tracer particle fields using a wavelet-based optical flow method

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## Abstract

A wavelet-based optical flow method for high-resolution velocimetry based on tracer particle images is presented. The current optical flow estimation method (WOF-X) is designed for improvements in processing experimental images by implementing wavelet transforms with the lifting method and symmetric boundary conditions. This approach leads to speed and accuracy improvements over the existing wavelet-based methods. The current method also exploits the properties of fluid flows and uses the known behavior of turbulent energy spectra to semi-automatically tune a regularization parameter that has been primarily determined empirically in the previous optical flow algorithms. As an initial step in evaluating the WOF-X method, synthetic particle images from a 2D DNS of isotropic turbulence are processed and the results are compared to a typical correlation-based PIV algorithm and previous optical flow methods. The WOF-X method produces a dense velocity estimation, resulting in an order-of-magnitude increase in velocity vector resolution compared to the traditional correlation-based PIV processing. Results also show an improvement in velocity estimation by more than a factor of two. The increases in resolution and accuracy of the velocity field lead to significant improvements in the calculation of velocity gradient-dependent properties such as vorticity. In addition to the DNS results, the WOF-X method is evaluated in a series of two-dimensional vortex flow simulations to determine optimal experimental design parameters. Recommendations for optimal conditions for tracer particle seed density and inter-frame particle displacement are presented. The WOF-X method produces minimal error at larger particle displacements and lower relative error over a larger velocity dynamic range as compared to correlation-based processing.

## Notes

### Acknowledgements

This work was partially sponsored by the Air Force Office of Scientific Research under Grant FA9550-16-1-0366 (Chiping Li, Program Manager).

## References

- Adrian RJ (2005) Twenty years of particle image velocimetry. Exp Fluids 39:159CrossRefGoogle Scholar
- Alvarez L, Castano CA, García M, Krissian K, Mazorra L, Salgado A, Sánchez J (2009) A new energy-based method for 3D motion estimation of incompressible PIV flows. Comput Vis Image Underst 113:802CrossRefGoogle Scholar
- Atcheson B, Heidrich W, Ihrke I (2009) An evaluation of optical flow algorithms for background oriented schlieren imaging. Exp Fluids 46(3):467CrossRefGoogle Scholar
- Beylkin G (1992) On the representation of operators in bases of compactly supported wavelets. SIAM J Numer Anal 6(6):1716MathSciNetCrossRefGoogle Scholar
- Beyou S, Cuzol A, Gorthi SS, Mémin E (2013) Weighted ensemble transform Kalman filter for image assimilation. Tellus A 65(1):18803CrossRefGoogle Scholar
- Black MJ, Anandan P (1996) The robust estimation of multiple motions: parametric and piecewise-smooth flow fields. Comput Vis Image Underst 63(1):75CrossRefGoogle Scholar
- Cai S, Mémin E, Dérian P, Xu C (2018) Motion estimation under location uncertainty for turbulent fluid flows. Exp Fluids 59:8CrossRefGoogle Scholar
- Carlier J, Wieneke B (2005) Report 1 on production and diffusion of fluid mechanics images and data. Fluid project deliverable 1.2. European Project “Fluid image analisys and description” (FLUID). http://www.fluid.irisa.fr
- Cassisa C, Simoens S, Prinet V, Shao L (2011) Subgrid scale formulation of optical flow for the study of turbulent flows. Exp Fluids 51(6):1739CrossRefGoogle Scholar
- Chen X, Zillé P, Shao L, Corpetti T (2015) Optical flow for incompressible turbulence motion estimation. Exp Fluids 56:8. https://doi.org/10.1007/s00348-014-1874-6 CrossRefGoogle Scholar
- Corpetti T, Heitz D, Arroyo G, Mémin E, Santa-Cruz A (2006) Fluid experimental flow estimation based on an optical-flow scheme. Exp Fluids 40(1):80. https://doi.org/10.1007/s00348-005-0048-y CrossRefGoogle Scholar
- Corpetti T, Mémin E, Pérez P (2002) Dense estimation of fluid flows. IEEE Trans Pattern Anal Mach Intell 24(3):365–380CrossRefGoogle Scholar
- Daubechies I, Sweldens W (1998) Factoring wavelet transforms into lifting steps. J Fourier Anal Appl 4(3):247–269MathSciNetCrossRefGoogle Scholar
- Dérian P, Almar R (2017) Wavelet-based optical flow estimation of instant surface currents from shore-based and UAV videos. IEEE Trans Geosci Remote Sens 55(10):5790CrossRefGoogle Scholar
- Dérian P, Héas P, Herzet C, Mémin É (2012) Wavelet-based fluid motion estimation. In: Bruckstein AM, ter Haar Romeny BM, Bronstein AM, Bronstein MM (eds) Scale space and variational methods in computer vision. SSVM 2011. Lecture notes in computer science, vol 6667. Springer, Berlin, HeidelbergzbMATHGoogle Scholar
- Dérian P, Héas P, Herzet C, Mémin E (2013) Wavelets and optical flow motion estimation. Num Math 6:116MathSciNetzbMATHGoogle Scholar
- Dérian P, Mauzey CF, Mayor SD (2015) Wavelet-based optical flow for two-component wind field estimation from single aerosol lidar data. J Atmos Ocean Technol 32(10):1759CrossRefGoogle Scholar
- Hart DP (1999) Super-resolution PIV by recursive local-correlation. J Vis 10:1–10Google Scholar
- Heitz D, Héas P, Mémin E, Carlier J (2008) Dynamic consistent correlation-variational approach for robust optical flow estimation. Exp Fluids 45:595CrossRefGoogle Scholar
- Heitz D, Mémin E, Schnörr C (2010) Variational fluid flow measurements from image sequences: synopsis and perspectives. Exp Fluids 48:369. https://doi.org/10.1007/s00348-009-0778-3 CrossRefGoogle Scholar
- Horn BKP, Schunck BG (1981) Determining optical flow. Artif Intell 17:185–203CrossRefGoogle Scholar
- Héas P, Herzet C, Mémin E, Heitz D, Mininni PD (2013) Bayesian estimation of turbulent motion. IEEE Trans Pattern Anal Mach Intell 35(6):1343–1356CrossRefGoogle Scholar
- Kadri-Harouna S, Dérian P, Héas P, Mémin E (2013) Divergence-free wavelets and high order regularization. Int J Comput Vis 103:80–99. https://doi.org/10.1007/s11263-012-0595-7 MathSciNetCrossRefzbMATHGoogle Scholar
- Keane RD, Adrian RJ (1990) Optimization of particle image velocimeters. I. Double pulsed systems. Meas Sci Technol 1:1202. https://doi.org/10.1088/0957-0233/1/11/013 CrossRefGoogle Scholar
- Kähler CJ, Scharnowski S, Cierpka C (2012) On the resolution limit of digital particle image velocimetry. Exp Fluids 52(6):1629CrossRefGoogle Scholar
- Liu T, Merat A, Makhmalbaf MHM, Fajardo C, Merati P (2015) Comparison between optical flow and cross-correlation methods for extraction of velocity fields from particle images. Exp Fluids 56:166CrossRefGoogle Scholar
- Liu T, Shen L (2008) Fluid flow and optical flow. J Fluid Mech 614:253. https://doi.org/10.1017/S0022112008003273 MathSciNetCrossRefzbMATHGoogle Scholar
- Mallat SG (1989) Multiresolution approximations and wavelet orthonormal bases of \(\text{ l }^2\text{(r) }\). Trans Am Math Soc 315(1):69–87zbMATHGoogle Scholar
- Mallat SG (2009) A wavelet tour of signal processing. Elsevier, New YorkzbMATHGoogle Scholar
- Plyer A, Besnerais GL, Champagnat F (2016) Massively parallel Lucas Kanade optical flow for real-time video processing applications. J Real Time Image Process 11(4):713CrossRefGoogle Scholar
- Pope SB (2001) Turbulent flows. IOP Publishing, BristolzbMATHGoogle Scholar
- Scarano F (2002) Iterative image deformation methods in PIV. Meas Sci Technol 13:R1CrossRefGoogle Scholar
- Schanz D, Gesemann S, Schröder A (2016) Shake-the-box: Lagrangian particle tracking at high particle image densities. Exp Fluids 57(5):70CrossRefGoogle Scholar
- Schmidt M (2005) minFunc: unconstrained differentiable multivariate optimization in matlab. https://www.cs.ubc.ca/~schmidtm/Software/minFunc.html
- Stanislas M, Okamoto K, Kähler CJ (2003) Main results of the first international PIV challenge. Meas Sci Technol 14:R63CrossRefGoogle Scholar
- Stanislas M, Okamoto K, Kähler CJ, Scarano F (2008) Main results of the third international PIV challenge. Exp Fluids 45(1):27CrossRefGoogle Scholar
- Stanislas M, Okamoto K, Kähler CJ, Westerweel J (2005) Main results of the second international PIV challenge. Exp Fluids 39(2):170CrossRefGoogle Scholar
- Susset A, Most JM, Honoré D (2006) A novel architecture for a super-resolution PIV algorithm developed for the improvement of the resolution of large velocity gradient measurements. Exp Fluids 40(1):70CrossRefGoogle Scholar
- Takehara K, Adrian RJ, Etoh GT, Christensen KT (2000) A Kalman tracker for super-resolution PIV. Exp Fluids 29(1):S034–S041Google Scholar
- Taubman D, Marcellin M (2002) JPEG2000: standard for interactive imaging. Proc IEEE 90:1336CrossRefGoogle Scholar
- Tokumaru PT, Dimotakis PE (1995) Image correlation velocimetry. Exp Fluids 19:1CrossRefGoogle Scholar
- Ullman S (1979) The interpretation of visual motion. Massachusetts Inst of Technology Pr, OxfordGoogle Scholar
- Westerweel J (1997) Fundamentals of digital particle image velocimetry. Meas Sci Technol 8:1379CrossRefGoogle Scholar
- Westerweel J, Elsinga GE, Adrian RJ (2013) Particle image velocimetry for complex turbulent flows. Ann Rev Fluid Mech 45:409MathSciNetCrossRefGoogle Scholar
- Wu Y, Kanade T, Li C, Cohn J (2000) Image registration using wavelet-based motion model. Int J Comput Vis 38(2):129CrossRefGoogle Scholar
- Yuan J, Schnörr C, Mémin E (2007) Discrete orthogonal decomposition and variational fluid flow estimation. J Math Imaging Vis 28:67MathSciNetCrossRefGoogle Scholar
- Zillé P, Corpetti T, Shao L, Chen X (2014) Observation model based on scale interactions for optical flow estimation. IEEE Trans Image Process 23(8):3281MathSciNetCrossRefGoogle Scholar