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Fluid experimental flow estimation based on an optical-flow scheme

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Abstract

We present in this paper a novel approach dedicated to the measurement of velocity in fluid experimental flows through image sequences. Unlike most of the methods based on particle image velocimetry (PIV) approaches used in that context, the proposed technique is an extension of “optical-flow” schemes used in the computer vision community, which includes a specific enhancement for fluid mechanics applications. The method we propose enables to provide accurate dense motion fields. It includes an image based integrated version of the continuity equation. This model is associated to a regularization functional, which preserve divergence and vorticity blobs of the motion field. The method was applied on synthetic images and on real experiments carried out to allow a thorough comparison with a state-of-the-art PIV method in conditions of strong local free shear.

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Acknowledgements

The authors would like to thank Joel Delville (University of Poitiers, France) and Beatriz Camano (Rio Grande do Sul Federal University, Brazil) for their valuable contribution on mixing layer experiments. The financial support by the Region Bretagne of France under grant no. 20048347, by the French Ministry of Research under grant no. 032593, and by the European Union under grant no. FP6-513663 are gratefully acknowledged.

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Author notes

  1. T. Corpetti

    Present address: Laboratoire COSTEL UMR 6554 LETG, Maison de la Recherche, Place du Recteur Henri Le Moal, 35043, Rennes Cedex, France

Authors and Affiliations

  1. avenue de Cucillé, Cemagref 17, 35044, Rennes Cedex, France

    T. Corpetti, D. Heitz, G. Arroyo & A. Santa-Cruz

  2. IRISA/INRIA Campus, Universitaire de Beaulieu, 35042, Rennes Cedex, France

    E. Mémin

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  1. T. Corpetti
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  3. G. Arroyo
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Correspondence to T. Corpetti.

Appendices

Appendix A: Div–Curl versus first-order regularization

By using the Euler–Lagrange condition of minimality, the equivalence between a standard first-order smoothness regularization, and a div–curl regularization with same weights for div and curl penalties can be readily demonstrated. Recall that Euler–Lagrange equation constitutes a necessary conditions for the minimization with respect to function g(x,y) of a functional

$$\int\limits_{\Omega} {\mathcal{F}}(g,g_{x},g_{y},x,y){\text{d}}{\user2{x}}. $$

It reads,

$$\frac{\partial {\mathcal{F}}}{\partial g} - \frac{\partial}{\partial x} \frac{\partial {\mathcal{F}}}{\partial g_{x}}- \frac{\partial}{\partial y} \frac{\partial {\mathcal{F}}}{\partial g_{y}} = 0 $$

Assuming a first-order regularization term

$${\mathcal{H}}_{\rm reg}({\user2{d}}) = \alpha \int\limits_{\Omega} (\left|{\varvec{\nabla}} u({\user2{x}}) \right|^{2} + \left|{\varvec{\nabla}} v({\user2{x}}) \right|^{2}){\text{d}}{\user2{x}}, $$

this condition amounts to the following coupled PDEs:

$$\left\{ {\begin{array}{*{20}c} {{ - 2\alpha u_{{xx}} - 2\alpha u_{{yy}} = 0}} \\ {{ - 2\alpha v_{{xx}} - 2\alpha v_{{yy}} = 0.}} \\ \end{array} } \right.$$
(19)

Now, considering a div–curl regularization

$${\mathcal{H}}_{\rm reg}({\user2{d}}) = \int\limits_{\Omega} \left(\alpha {\text{div}}^{2} {\user2{d}}({\user2{x}}) + \beta {\text{curl}}^{2} {\user2{d}}({\user2{x}}) \right){\text{d}}{\user2{x}}, $$
(20)

the Euler–Lagrange equations reads,

$$\left\{ {\begin{array}{*{20}c} {{ - 2\alpha u_{{xx}} - 2\beta u_{{yy}} - 2(\alpha - \beta )v_{{xy}} = 0}} \\ {{ - 2\beta v_{{xx}} - 2\alpha v_{{yy}} - 2(\alpha - \beta )v_{{xy}} = 0.}} \\ \end{array} } \right.$$
(21)

When α=β, these equations are the same as Eq. 19

Appendix B: Robust penalization

The main objective of robust estimators is to impose a different penalization for coherent and incoherent data: when the error to minimize is small (e.g. data are in accordance with underlying assumptions), the robust function tends to the L 2 quadratic norm; when this error is high (e.g. presence of outliers), it tends to attenuate the contribution of the error term (and then to be softer than the quadratic function). Figure 11 presents a possible shape of such function (the Leclerc penalty function: f 1(x)=1−exp(−τ1 x 2)).

Fig. 11
figure 11

Comparison of the shape of the Leclerc robust penalization (bottom) (with τ1=1) versus the quadratic one (top)

Full size image

The choice of robust penalty functions needs to define the parameter τ1 and generally makes the problem non-quadratic. The specific minimization problem we face is classically turned into an augmented half-quadratic minimization problem (Holland and Welsch 1977). Indeed, with all robust penalty functions f such that\(f(\sqrt{.})\) is concave (as the Leclerc one), we have the property that:

$$f(x) = {\mathop {\min }\limits_{z \in (m, M]} } z x^{2} + \psi(z), $$
(22)

where \(M = \lim_{0^{+}}\frac{f'(x)}{2x}, m = \lim_{+\infty}\frac{f'(x)}{2x},\) and ψ, for which an expression can be found in (Black and Rangarajan 1996; Geman and Reynolds 1992), is such that the minimizer on the right-hand-side is given by \(z = \frac{f'(x)}{2x}.\)

Using Eq. 22, each minimization problem of the generic form min x ∑ k f(g k (x)) can be replaced by the auxiliary problem \(\min_{x,\{z_{k}\}} \sum_{k} z_{k} g_{k}^{2}(x) + \psi(z_{k}) ,\) which can be solved by iteratively re-weighted least squares (IRLS) (Holland and Welsch 1977): for fixed auxiliary (weight) variables z k ∈(m,M], one faces a least-squares problem; for fixed x, the optimal value for each weight is known in closed form as \(\frac{f'(g_{k}(x))}{2g_{k}(x)}.\) This process is done until convergence.

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Corpetti, T., Heitz, D., Arroyo, G. et al. Fluid experimental flow estimation based on an optical-flow scheme. Exp Fluids 40, 80–97 (2006). https://doi.org/10.1007/s00348-005-0048-y

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