A new energy-based method for 3D motion estimation of incompressible PIV flows

doi:10.1016/j.cviu.2009.01.005

Computer Vision and Image Understanding

Volume 113, Issue 7, July 2009, Pages 802-810

https://doi.org/10.1016/j.cviu.2009.01.005 Get rights and content

Abstract

Motion estimation has many applications in fluid analysis, and a lot of work has been carried out using Particle Image Velocimetry (PIV) to capture and measure the flow motion from sequences of 2D images. Recent technological advances allow capturing 3D PIV sequences of moving particles. In the context of 3D flow motion, the assumption of incompressibility is an important physical property that is satisfied by a large class of problems and experiments. Standard motion estimation techniques in computer vision do not take into account the physical constraints of the flow, which is a very interesting and challenging problem. In this paper, we propose a new variational motion estimation technique which includes the incompressibility of the flow as a constraint to the minimization problem. We analyze, from a theoretical point of view, the influence of this constraint and we design a new numerical algorithm for motion estimation which enforces it. The performance of the proposed technique is evaluated from numerical experiments on synthetic and real data.

Introduction

Particle Image Velocimetry (PIV) is a technique which aims at obtaining image sequences of fluid motion in a variety of applications, in gaseous and liquid media, and at extracting the corresponding flow velocity information (see [1]). The typical setting of a PIV experiment includes the following components: the flow medium seeded with particles, droplets or bubbles, a double pulsed laser which illuminates the particles twice within a short interval of time, a light sheet optics guiding a thin light plane within the flow medium, one or several CCD cameras which capture the two frames exposed by the laser pulses and a timing controller synchronizing the laser and the camera. Standard PIV techniques estimate two planar components of the fluid motion from 2D images (2D-PIV). Using stereo techniques, dual-plane PIV or holographic recording (3C-PIV) the third spatial component of the flow can be also estimated, as in [2]. The extension of PIV techniques to full spatial domain (3D-PIV) is currently an active research area (see [3] for more details).

The most common technique for motion estimation in 2D-PIV is based on local correlation between two rectangular regions of two consecutive images, as in [4]. This technique has a straightforward extension to 3D images. Another approach to motion estimation widely used in computer vision is based on energy minimization, also called variational technique, where on the one hand, we assume the conservation of the image intensity of the scene objects across the image sequence (in PIV sequences, the scene objects are the particles); and on the other hand, we assume a certain regularity of the flow. In the context of 2D PIV, such an approach has been proposed by Corpetti et al. [5].

When dealing with 3D fluid flow estimation, sufficient information is captured to model physical constraints of the flow, such as the Navier–Stokes equations or the incompressibility. Currently, an interesting challenging problem is to include these constraints into the motion estimation model.

In this paper, we propose a new variational motion estimation technique which includes the incompressibility of the flow as a constraint in the minimization problem. In order to minimize the constrained energy, we use a generalized Lagrange multiplier approach.

The paper is organized as follows: in Section 2, we briefly describe related works on flow motion estimation, in Section 3, we present the classic Helmholtz vector field decomposition that we use to analyze the constrained energy problem. In Section 4, we propose a new variational model constrained by the incompressibility of the flow. In Section 5, we present our numerical scheme and implementation. In Section 6, the performance of the proposed method is evaluated from experimental results on synthetic and real data, before the discussion and conclusion.

Section snippets

Related works on motion estimation

Among the different existing approaches to estimate motion between two images, the methods based on local cross-correlation and the methods based on energy minimization (variational approaches) are widely used in the literature. In this section, we briefly review both approaches.

Helmholtz vector field decomposition

The result presented in this section can be found in classical books on fluid mechanics (see for instance [25], [26], [27]). First, we introduce the following notations: $H_{s} \equiv \{u_{s} \in C^{1} (Ω) \cap C (\bar{Ω}) : div (u_{s}) = 0 in Ω and u_{s} \cdot n = 0 in \partial Ω\},$ where $n (x)$ is the vector normal to the boundary of $Ω$ , and $H_{r} \equiv \{u_{r} \in C^{1} (Ω) \cap C (\bar{Ω}) : \exists p \in H^{1} (Ω) such that u_{r} = \nabla p in Ω\},$ where $H^{1} (Ω)$ is the usual Sobolev space.

Theorem 1 Helmholtz decomposition

Let $u \in C^{1} (Ω) \cap C (\bar{Ω})$ be a 3D vector field, then there exists $u_{s} \in H_{s}$ and $u_{r} \in H_{r}$ such as: $u = u_{s} + u_{r},$ and where $u_{s}$ and $u_{r}$ are orthogonal in $L^{2} (Ω),$ i.e. $\int_{Ω} u_{s} (x) u$

Incompressibility constrained model

A large class of real fluid flows satisfies the incompressibility constraint (i.e. $div (u) = 0$ ). In this section, we focus on how to include such an incompressibility constraint in the 3D flow motion estimation. To this end, we look for local minima of the constrained energy $\min_{u \in H_{s}} E (u),$ where $E (u)$ is defined in Eq. (2) and $H_{s}$ is defined in Eq. (5). Next, we show a generalization of the Lagrange multipliers technique applied to the constrained energy problem Eq. (10).

First, let us recall that the

Nonlinear discrete optimization problem

In order to solve numerically the constrained energy Eq. (10), we discretize it as described below.

We denote $Ω$ the discrete rectangular volume, $\hat{u} \equiv {{\hat{u}}_{i, j, k}}$ an approximation of $u (i \cdot δ x, j \cdot δ y, k \cdot δ z)$ , where u is a local minimum of Eq. (10) and $(δ x, δ y, δ z)$ the voxel size.

Next, the discretization of Eq. (14) yields the discrete nonlinear equation: $\nabla E (\hat{u}) = P_{r} (\nabla E (\hat{u})) .$

This challenging nonlinear system of equations is solved by the means of an iterative scheme. Let ${\hat{u}}^{n}$ be an estimation of a local minimum at

Experiments and results

In this section, we present experiments on synthetic and real data using both correlation and variational methods. The synthetic 3D flows are based on realistic flow models to check the performance of our methods. In these experiments, we first apply the standard correlation or variational methods to obtain a good approximation of the flow and then we refine the results with the new variational approach including the incompressibility constraint.

Since we focus on the influence of the

Conclusions

In this paper, we deal with the very challenging problem of including physical constraints (such as incompressibility) in the variational models of 3D fluid flow estimation. We have proposed a new variational motion estimation technique, which includes the incompressibility of the flow as a constraint in the minimization problem.

We have analyzed, from a theoretical point of view, the influence of the incompressibility constraint in the energy minimization problem, and we have derived a

Acknowledgments

This work has been funded by the European Commission under the Specific Targeted Research Project FLUID (Contract No. FP6-513663). We acknowledge the Cemagref laboratory (Centre for Agricultural and Environmental Engineering Research, Rennes, France) who kindly provided us with the simulated PIV flow sequence, LaVision GmbH who kindly provided to us the real PIV sequence, Prof. F. Scarano and the reviewers for their valuable comments.

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Computer Vision and Image Understanding

A new energy-based method for 3D motion estimation of incompressible PIV flows

Abstract

Introduction

Section snippets

Related works on motion estimation

Helmholtz vector field decomposition

Incompressibility constrained model

Nonlinear discrete optimization problem

Experiments and results

Conclusions

Acknowledgments

Particle image velocimetry

A Practical Guide

Three-dimensional particle velocimetry

Meas. Sci. Tech.

Stereoscopic and holographic approaches to get the third velocity component in PIV

Karman Inst. Fluid Dyn. Lect. Ser.

Iterative image deformation methods in PIV

Meas. Sci. Technol.

Fluid experimental flow estimation based on an optical-flow scheme

Exp. Fluids

Mathematical problems in image processing: partial differential equations and the calculus of variations

The computation of optical flow

ACM Comput. Surv.

Performance of optical flow techniques

IJCV

Reliable estimation of dense optical flow fields with large displacements

Int. J. Comput. Vis.

Symmetrical dense optical flow estimation with occlusions detection

Int. J. Comput. Vis.

Computing optical flow via variational techniques

SIAM J. Appl. Math.

A multigrid platform for real-time motion computation with discontinuity-preserving variational methods

Int. J. Comput. Vis.

Domain decomposition for variational optical flow computation

IEEE Trans. Image Process.