# A variational approach to optical flow estimation of unsteady incompressible flows

## Abstract

We consider optical flow estimation of flows with vorticity governed by 2D incompressible Euler and Navier–Stokes equations . A vorticity-streamfunction formulation and optimization techniques are used. We use Helmholtz decomposition of the velocity field and prove existence of an unique velocity and vorticity field for the linearized vorticity equations. Discontinuous galerkin finite elements are used to solve the vorticity equation for Euler’s flow to efficiently track discontinuous vortices. Finally we test our method with two vortex flows governed by Euler and Navier–Stokes equations at high Reynolds number which support our theoretical results.

### Keywords

Euler Navier–Stokes Vorticity Streamfunction Discontinuous Galerkin Optimization Linearization Helmholtz decomposition Incompressible### Mathematics Subject Classification

76M## 1 Introduction

Optical flow method is the estimation of 2D velocities of objects in successive image sequences that are in apparent motion. The estimation is based on the changes in spatio-temporal brightness pattern recorded in successive image sequences. Our motivation for the present study is the problem of tracking the motion of clouds from satellite image sequences. This in turn will help us in understanding the movement of rain bearing clouds during the monsoon over the Indian subcontinent. Previous work in this direction are [9, 10, 11, 12, 13]. However a major problem in applying OFM to fluid flow, leave alone cloud motion is that the connection between optical flow in the image plane and fluid flow in the 3D world is yet to be understood satisfactorily [7]. Given this state of affairs we propose to apply OFM to images that are generated synthetically by solving the 2D incompressible Stokes and Navier–Stokes equation. Our aim is to track movement of vortex structures generated by solving the 2D incompressible Stokes and Navier–Stokes equation. Previous work in this direction includes the Horn–Schunck algorithm which implements a constraint free first order regularization approach with a finite differencing scheme [1], estimating optical flow involving prior knowledge that the flow satisfies Stokes equation [3] and higher order regularization with incompressibility constraint coupled with mimetic finite differencing scheme [4].

It is well known [1] that tracking rigid body motion by OFM can be done satisfactorily using nonlinear least squares technique whereas it is inadequate for fluid flow [8]. This is because rigid body motion has features like geometric invariance where local features such as corners, contours etc are usually stable over time [28]. However for fluid images these features are difficult to define leave alone being stable. This is one of the main problems in understanding the connection between optical flow and fluid flow [20, 22, 23, 24]. To recover fluid-type motions, a number of approaches have been proposed to integrate the basic optical flow solution with fluid dynamics constraints, e.g., the continuity equation that describes the fluid property [24, 25] or the divergence-curl (div-curl) equation [24, 26] to describe spreading and rotation. Such a work has its importance in determining atmospheric motion vectors (AMV), tracking smoke propagation, determining motion of tidal waves using floating buoys. Since the basic idea in the variational approach is not to estimate locally and individually but to estimate non-locally by minimizing a suitable functional defined over the entire image section, we therefore prefer a variational approach. We are interested in tracking vortex based high-Reynolds number incompressible flows. In [35] we had used such a variational approach by minimizing a functional with data obtained at a fixed time *t*. We found out that even though the method is efficient and recovers Stokes flow exactly, it fails to recover vortex structures for high Reynolds number flows. As a consequence, we formulate a minimization problem by penalizing the tracking or advection error over space as well as time. We use the vorticity-streamfunction formulation for the Euler and Navier–Stokes equations. This is because our main aim is to capture vortex structures and we can directly get the vorticity as an output rather than computing the curl of the velocity which could lead to numerical errors.

The paper is organised as follows. In Sect. 2 a variational formulation is presented. In Sect. 3 a first kind of modified variational formulation using linearized form of the Euler and Navier–Stokes is used. Existence and uniqueness of solutions is showed. In Sect. 4 continuous and discontinuous finite element formulations for the minimizing equations are presented. Section 5 deals with some numerical experiments for the modified variational problem. In Sects. 6 and 7 a second kind of modified variational formulation is discussed using linearized form of the Euler and Navier–Stokes along with some numerical results. In Sects. 8 and 9 the variational formulation as described in Sect. 2 is used and numerical tests are done with it. Finally in Sect. 10 the results are analyzed.

## 2 Variational formulation

*E*(

*x*,

*y*,

*t*) for \((x,y)\in \varOmega \) represents snapshots of the image of the scalars at various times \(t \in {\mathbb {R}}^+\). Here \(\varOmega \) is a bounded convex subset of \({\mathbb {R}}^2\). We assume our image \(E(x,y,t) \in W^{1,\infty }(\varOmega )\), for each

*t*and hence in \(L^2(\varOmega )\) (as \(\varOmega \) is bounded). The constant brightness assumption of the tracers gives us the optical flow equation

*E*is the inverse problem of determining \(\widetilde{\mathbf{U }}\) from the image sequence represented by

*E*. To estimate fluid flow, we need to include flow dynamics as constraints. We assume \( \widetilde{\mathbf{U }} \) satisfies the 2D incompressible Euler (Navier–Stokes) equations i.e the vorticity \(\omega = \nabla \times \widetilde{\mathbf{U }}\) satisfies

*Re*is the Reynolds number. We also impose the boundary condition

*n*is the unit normal on the boundary of \(\varOmega \). We assume that all vortex stay in the interior of the domain at all times \(t\in [0,T]\). Again, as \(\nabla \cdot \widetilde{\mathbf{U }}=0,\,\exists\;\psi{:}\;{\varOmega }\times {\mathbb {R}}^+\longrightarrow {\mathbb {R}}\), called streamfunction, such that

*E*, our aim is to determine appropriate

*g*and \(\omega _0\) so that (2), (3), (4), (5), (6) can be solved to determine \(\widetilde{\mathbf{U }}\). Such a pair \((g, \omega _0)\) can be obtained by minimizing the functional

## 3 Formulation 1: Linearized flow

*n*is the unit normal on the boundary of \(\varOmega \).

### 3.1 Existence and uniqueness of minimizer

We want to show existence of an unique minimizer of (P\(_1\)). Before that we state some standard definitions and results.

### 3.2 Preliminaries

Let \((Z,\Vert \cdot \Vert _Z)\) be a Banach space.

**Theorem 1**

*Let*\(J:Z\rightarrow {\mathbb {R}}\cup \lbrace -\infty ,\infty \rbrace \)*be a convex functional on**Z*. *If**J**is bounded from above in a neighbourhood of a point*\(U_0\in Z\), *then it is locally bounded i.e. each*\(U\in Z\)*has a neighbourhood on which**J**is bounded.*

**Definition 1**

*J*defined on

*Z*is said to be

**locally Lipschitz**if at each \(U\in Z\) there exists a neighbourhood \(N_{\epsilon }(U)\) and a constant

*R*(

*U*) such that if \(V,W \in N_\epsilon (U)\), then

*R*independent of

*U*then we say that

*J*is

**Lipschitz**on

*Y*.

**Theorem 2**

*Let**J**be convex on**Z*. *If**J**is bounded from above in a neighbourhood of one point of**X*, *then**J**is locally Lipschitz in**Z*.

**Theorem 3**

*Let**J**be convex on**Z*. *If**J**is bounded from above in a neighbourhood of one point of**Z*, *then**J**is continuous on**Z*.

Theorems 1, 2, 3 and Definition 1 can be found in [27]. We use the following theorem from [2] to establish an unique global minimizer for (P).

**Theorem 4**

*Let*\(J:Z\rightarrow {\mathbb {R}}\cup \lbrace -\infty ,\infty \rbrace \)

*be a lower semi-continuous strictly convex functional. Also let*

*J*

*be coercive i.e.*

*Let*

*C*

*be a closed and convex subset of*

*Z*.

*Then*

*J*

*has a unique global minimum over*

*C*.

We now verify conditions stated in Theorem (4) for the functional *J* in (P\(_1\)). Let \(Z=L^2([0,T]; H^\frac{1}{2}(\partial \varOmega ))\times L^2(\varOmega )\) with the norm \(\Vert (g,\omega _0)\Vert _Z =\left( {\int _0^T\int _{\partial \varOmega }\left| g\right| ^2 +\int _\varOmega \left| \omega _0\right| ^2}\right) ^{1/2}\).

**Theorem 5**

*The functional**J**given in* (P\(_1\)) *is strictly convex with respect to*\((\omega _0,g)\).

Before proving Thoerem 5 we show that \((\psi ,\omega )\) given by (8) and (9) is linear in \((\omega _0,g)\).

**Lemma 1**

\((\psi ,\omega )\)*given by* (8) *and* (9) *is linear in*\((g, \omega _0) \in L^2([0,T];H^\frac{1}{2}(\partial \varOmega ))\times L^2(\varOmega )\).

*Proof*

*Proof*

*J*is strictly convex with respect to \((\omega _0,g)\).

**Theorem 6**

*The constraint set*\(C=\{\omega \in L^2([0,T];L^2(\varOmega )): \omega \)*satisfies* (8) *and* (10)\(\}\)*is given by the level set of a convex function.*

*Proof*

*S*. Let

*A*is a bounded linear bijection. Consider the function

**Theorem 7**

*The constraint set* (9) *is given by the level set of a convex function.*

*Proof*

*B*is a bijection. Also \(B^{-1}\) is convex for if \(\psi _1\) and \(\psi _2\) satisfy (14) for boundary data \(g_1\) and \(g_2\) respectively then \(\lambda \psi _1 + (1-\lambda )\psi _2\) is a solution of (14) for boundary data \(\lambda g_1 + (1-\lambda )g_2\). By (18), \(B^{-1}\) is bounded. Hence \(B^{-1}\) is a bounded linear bijection. Now consider the function

*Z*. We now show that

*J*is continuous and coercive.

**Theorem 8**

*The functional**J**given in* (P\(_1\)) *is continuous*

*Proof*

*Q*. This gives us

*J*(

*U*) is bounded above in \(N_1\). As

*J*is convex (by Theorem 5) it implies

*J*is continuous for all \(U \in Z\) (by Theorem 3). \(\square \)

**Theorem 9**

*The functional**J**given in* (P\(_1\)) *is coercive for*\(\alpha >0\)*and*\(\beta >0\).

*Proof*

*J*is coercive.

*J* is a strictly convex continuous coercive functional on *Z* and the constraint set *C* given in (19) is convex. By Theorem 4, the convex minimization problem (7) has an unique global minimizer.

### 3.3 Optimization using Lagrange multipliers

*J*defined in (P\(_1\)). Now we determine the optimum solution. The functional

*J*is to be minimized subject to PDE constraints. This is done by the use of Lagrange multipliers [15]. We first write down the weak forms of (8) and (9). Multiplying (8) with a test function \(y\in H^1([0,T];L^2(\varOmega ))\), integrating by parts with respect to

*t*and incorporating initial conditions for \(\omega \), we get

*y*is the Lagrange Multiplier corresponding to the first constraint (8) and \(\phi \) is the Lagrange Multiplier corresponding to the second constraint (14). Here \(\omega ,y \in L^2([0,T];L^2(\varOmega ))\) and \(\psi ,\phi \in L^2([0,T];H^1_a(\varOmega ))\) where

### 3.4 PDE’s obtained after minimization of \(\widetilde{J}\)

## 4 Finite element method for problem (P\(_1\))

Equations (26), (27), (28), (29) and (30) are solved using space-time finite elements, considering time as the third dimension. The Eqs. (26) and (27) represent the forward vorticity equation and its backward adjoint equation. For Navier–Stokes flow, the solutions to the equations are smooth because of the presence of an extra diffusivity term on the right hand side. So continuous Galerkin finite elements are used. For Euler’s flow, vorticity is non-smooth. Since our aim is to capture vortex structures well, we use discontinuous Galerkin method. The equations (28) and (29) represent the streamfunction equation and its adjoint equation which are elliptic in nature and hence they are solved using continuous Galerkin finite elements.

### 4.1 Discontinuous Galerkin formulation for vorticity equation for Euler’s flow

*Q*and consider the space of piecewise polynomials

*K*. The vorticity equation associated to Euler’s flow given in (26) can be rewritten in conservative form as

*K*is given as

For the adjoint equation (27) the same DG formulation in (33) is used with \(\omega ^h\) replaced \(y^h,\,B= (-u_0,-v_0,-1)\) and inflow boundary conditions as \(y^h=0\) using (34).

### 4.2 Continuous Galerkin formulation for vorticity equation for Navier–Stokes

*k*on

*K*. We choose \(k=1\) for our computations unless otherwise mentioned. The discrete problem is to find \(\omega ^h \in W_h^k\) such that

### 4.3 Continuous Galerkin formulation for streamfunction equation

*g*, we want to find \(\psi \in L^2([0,T];H^1_a(\varOmega ))\) such that

*g*and \(\omega _0\) from (30) in (33), (36) and (40), the discrete weak forms for the vorticity and streamfunction equations and their adjoints are combined and solved in a coupled way to obtain \({\mathbf{U}} \) and hence \(\widetilde{\mathbf{U }}\). Next we describe the procedure to determine \(E, E_t, \nabla {E}\).

### 4.4 Image data

*E*and try to recover the velocity given the information of the derivatives of

*E*. For this purpose

*E*is chosen whose analytic expression at \(t=0\) is

*E*at all times, we solve the advection equation

*E*and its derivatives to be differentiable. Hence the weak formulation can be stated as: find \(E\in H^2_c(Q)\) such that

*E*are evaluated by computing the derivatives of \(\phi _i\). The unsteady incompressible Euler equations can be written in vorticity-streamfunction form as

*Re*is the Reynolds number and \(\psi _b\) is a prescribed boundary condition. In practice, derivatives of images will be computed using some finite differences which will introduce errors in the computed velocity.

### 4.5 Test vortex flows

Our domain is \(Q=(0,T)\times \varOmega \), where \(\varOmega =[0,1]\times [0,1]\). Two types of vortex flows are considered.

**Vortex 1**: The tangential velocity distribution for the initial condition for vorticity is prescribed between an outer radius \(r=R_O\), and a core radius \(r=R_C\). For radius greater than \(R_O\) the tangential velocity is set to be zero. The tangential velocity of the vortex is expressed as follows:

**Vortex 2**: A vortex patch whose initial condition is as follows:

### 4.6 Mesh

*Q*is partitioned into tetrahedrons as shown in Fig. 1. The average mesh size is \(h=0.01\). There are 196365 triangles with 1640836 degrees of freedom.

Computations for Navier–Stokes flow are done with \(Re=1000\).

### 4.7 Solving equations (43) and (44)

*Q*for vector functions.

## 5 Numerical examples

### 5.1 Advection of vortex 1 and vortex 2 under Euler’s flow

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 2.5e\(-3\) | 1.6e\(-4\) |

1 | 1 | 2.8e\(-3\) | 1.6e\(-4\) |

10 | 10 | 2.7e\(-3\) | 1.5e\(-4\) |

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 3.1e\(-3\) | 1.1e\(-4\) |

1 | 1 | 3.2e\(-3\) | 1.1e\(-4\) |

10 | 10 | 3.2e\(-3\) | 1.1e\(-4\) |

### 5.2 Advection of vortex 1 and vortex 2 under Navier–Stokes flow

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 5.1e\(-3\) | 2.2e\(-4\) |

1 | 1 | 5.2e\(-3\) | 2.2e\(-4\) |

10 | 10 | 5.1e\(-3\) | 2.6e\(-4\) |

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 4.4e\(-3\) | 2.7e\(-4\) |

1 | 1 | 4.1e\(-3\) | 2.5e\(-4\) |

10 | 10 | 4.3e\(-3\) | 2.4e\(-4\) |

## 6 Formulation 2: Linearized flow

*J*given in (P\(_1\)). Our aim is to determine \({\mathbf{U}} \) and hence the total velocity \(\widetilde{\mathbf{U }}\) by minimizing

### 6.1 Optimization using Lagrange multipliers

*y*is the Lagrange multiplier corresponding to the first constraint set (8) and \(\phi \) is the Lagrange multiplier corresponding to the second constraint set (14). Here \(\omega ,y \in L^2([0,T];L^2(\varOmega ))\) and \(\psi ,\phi \in L^2([0,T];H^1_a(\varOmega ))\) where

### 6.2 PDE’s obtained after minimization of \(\widetilde{J}\)

## 7 Numerical examples

### 7.1 Advection of vortex 1 and vortex 2 under Euler’s flow

Relative \(L^2\) errors and advection errors for different values of \(\alpha ,\,\beta \) and \(\gamma \)

\(\alpha \) | \(\beta \) | \(\gamma \) | Relative \(L^2\) error | Advection error |
---|---|---|---|---|

0.01 | 0.01 | 0.01 | 2.8e\(-3\) | 1.3e\(-4\) |

1 | 1 | 1 | 2.6e\(-3\) | 1.4e\(-4\) |

10 | 10 | 10 | 2.7e\(-3\) | 1.3e\(-4\) |

Relative \(L^2\) errors and advection errors for different values of \(\alpha ,\,\beta \) and \(\gamma \)

\(\alpha \) | \(\beta \) | \(\gamma \) | Relative \(L^2\) error | Advection error |
---|---|---|---|---|

0.01 | 0.01 | 0.01 | 3.5e\(-3\) | 1.7e\(-4\) |

1 | 1 | 1 | 3.4e\(-3\) | 1.2e\(-4\) |

10 | 10 | 10 | 3.1e\(-3\) | 1.4e\(-4\) |

### 7.2 Advection of vortex 1 and vortex 2 under Navier–Stokes flow

Relative \(L^2\) errors and advection errors for different values of \(\alpha ,\,\beta \) and \(\gamma \)

\(\alpha \) | \(\beta \) | \(\gamma \) | Relative \(L^2\) error | Advection error |
---|---|---|---|---|

0.01 | 0.01 | 0.01 | 5.4e\(-3\) | 2.8e\(-4\) |

1 | 1 | 1 | 5.3e\(-3\) | 2.4e\(-4\) |

10 | 10 | 10 | 5.6e\(-3\) | 2.7e\(-4\) |

\(\alpha \) | \(\beta \) | \(\gamma \) | Relative \(L^2\) error | Advection error |
---|---|---|---|---|

0.01 | 0.01 | 0.01 | 4.3e\(-3\) | 2.2e\(-4\) |

1 | 1 | 1 | 4.4e\(-3\) | 2.8e\(-4\) |

10 | 10 | 10 | 4.6e\(-3\) | 2.5e\(-4\) |

## 8 Formulation 3: Non-linear flow

### 8.1 Optimization using Lagrange multipliers

*t*and incorporating initial conditions for \(\omega \), we get

*y*is the Lagrange multiplier corresponding to the first constraint set (8) and \(\phi \) is the Lagrange multiplier corresponding to the second constraint set (14). Here \(\omega ,y \in L^2([0,T];L^2(\varOmega ))\) and \(\psi ,\phi \in L^2([0,T];H^1_a(\varOmega ))\) where

### 8.2 PDE’s obtained after minimization of \(\widetilde{J}\)

## 9 Numerical examples

### 9.1 Advection of vortex 1 and vortex 2 under Euler’s flow

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 0.21 | 3.21e\(-2\) |

0.1 | 0.01 | 0.24 | 3.22e\(-2\) |

1 | 1 | 0.23 | 3.26e\(-2\) |

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 0.35 | 1.64e\(-2\) |

1 | 1 | 0.37 | 1.62e\(-2\) |

10 | 10 | 0.35 | 1.61e\(-2\) |

### 9.2 Advection of vortex 1 and vortex 2 under Navier–Stokes flow

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 0.51 | 6.6e\(-2\) |

1 | 1 | 0.55 | 6.1e\(-2\) |

10 | 10 | 0.52 | 6.7e\(-2\) |

Relative \(L^2\) errors and advection errors for different values of \(\alpha \) and \(\beta \)

\(\alpha \) | \(\beta \) | Relative \(L^2\) error | Advection error |
---|---|---|---|

0.01 | 0.01 | 0.60 | 7.2e\(-2\) |

1 | 1 | 0.64 | 7.4e\(-2\) |

10 | 10 | 0.65 | 7.1e\(-2\) |

## 10 Conclusions

To determine vortex based flows, we have assumed they satisfy Euler or Navier–Stokes equations. The vorticity-streamfunction formulation for the Euler and Navier–Stokes equations were introduced and the Helmholtz decomposition of the velocity field was used to segregate the translational and rotational part of the velocity field. To determine the velocity and vorticity field a variational approach to minimize a functional, which penalized the tracking or advection error of the scalar image field and the initial vorticity and boundary condition for the streamfunction, was used. For the linearized case we have shown existence of an unique velocity field. We also exploited the advantages of the discontinuous Galerkin finite elements for the vorticity equation in case of Euler’s flow to capture discontinuous vortices effectively. Two types of vortex movement under Euler and Navier–Stokes flows were investigated and it was observed that the results in the cases where we assumed Helmholtz decomposition of the velocity vector field gave significantly good results even for higher Reynolds number flows. In the case where there was no assumption on the velocity field, occurrence of vortex shredding and high diffusion took place. A reason for this could be the non-uniqueness of solutions to the vorticity equation which is itself an interesting theoretical problem. The authors plan to address this in future. It also suggests that to capture vortex structures, boundary information of the velocity or the initial vorticity is needed or else rotational dynamics should be introduced into the model. Further work will be to use other penalties in our objective functional *J* and try to capture other types of non-linear flows.

### References

- 1.Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artif. Intell.
**17**(1–3), 185–203 (1981)CrossRefGoogle Scholar - 2.Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. In: Dilcher, K., Taylor, K. (eds.) CMS Books in Mathematics. Springer, New York. ISBN: 978-1-4419-9466-0 (2011)Google Scholar
- 3.Ruhnau, P., Schnörr, C.: Optical Stokes flow: an imaging based control approach. Exp. Fluids
**42**, 61–78 (2007)CrossRefGoogle Scholar - 4.Yuan, J., Ruhnau, P., Mémin, E., Schnörr, C.: Discrete orthogonal decomposition and variational fluid flow estimation. In: Scale-Space 2005, volume 3459 of Lecture Notes Computer Science, pp. 267–278. Springer (2005)Google Scholar
- 5.Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics. Springer, Berlin (1986)Google Scholar
- 6.Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112, American Mathematical Society, USA (2010)Google Scholar
- 7.Liu, T., Shen, L.: Fluid flow and optical flow. J. Fluid Mech
**614**, 253–291 (2008)MATHMathSciNetCrossRefGoogle Scholar - 8.Heitz, D., Mémin, E., Schnörr, C.: Variational fluid flow measurement from image sequences: synopsis and perspectives. Exp. Fluids
**48**, 369–393 (2010)CrossRefGoogle Scholar - 9.Cayula, J.-F., Cornillon, P.: Cloud detection from a sequence of SST images. Remote Sens. Environ.
**55**, 80–88 (1996)CrossRefGoogle Scholar - 10.Leese, J.A., Novak, C.S., Taylor, V.R.: The determination of cloud pattern motions from geosynchronous satellite image data. Pattern Recognit.
**2**, 279–292 (1970)CrossRefGoogle Scholar - 11.Fogel, S.V.: The estimation of velocity vector-fields from time varying image sequences. CVGIP: Image Underst.
**53**, 253–287 (1991)MATHCrossRefGoogle Scholar - 12.Wu, Q.X.: A correlation-relaxation labeling framework for computing optical flow—template matching from a new perspective. IEEE Trans. Pattern Anal. Mach. Intell.
**17**, 843–853 (1995)CrossRefGoogle Scholar - 13.Parikh, J.A., DaPonte, J.S., Vitale, J.N., Tselioudis, G.: An evolutionary system for recognition and tracking of synoptic scale storm systems. Pattern Recognit. Lett.
**20**, 1389–1396 (1999)CrossRefGoogle Scholar - 14.Logg, A., Mardal, K.-A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)MATHCrossRefGoogle Scholar
- 15.Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Netherlands. ISBN 978-1-4020-8839-1 (2009)Google Scholar
- 16.Aubert, G., Kornprobst, P.: A mathematical study of the relaxed optical flow problem in the space. SIAM J. Math. Anal.
**30**(6), 1282–1308 (1999)MATHMathSciNetCrossRefGoogle Scholar - 17.Aubert, G., Deriche, R., Kornprobst, P.: Computing optical flow via variational techniques. SIAM J. Math. Anal.
**60**(1), 156–182 (1999)MATHMathSciNetCrossRefGoogle Scholar - 18.Nagel, H.-H.: Displacement vectors derived from second-order intensity variations in image sequences. CGIP
**21**, 85–117 (1983)Google Scholar - 19.Nagel, H.-H.: On the estimation of optical flow: relations between different approaches and some new results. AI
**33**, 299–324 (1987)Google Scholar - 20.Nagel, H.-H.: On a constraint equation for the estimation of displacement rates in image sequences. IEEE Trans. PAMI
**11**, 13–30 (1989)MATHCrossRefGoogle Scholar - 21.Nagel, H.-H., Enkelmann, W.: An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Trans. PAMI
**8**, 565–593 (1986)CrossRefGoogle Scholar - 22.Papadakis, N., Mémin, E.: Variational assimilation of fluid motion from image sequence. SIAM J. Imaging Sci.
**1**(4), 343–363 (2008)MATHMathSciNetCrossRefGoogle Scholar - 23.Mukawa, N.: Estimation of shape, reflection coefficients and illuminant direction from image sequences. In: ICCV90, pp. 507–512 (1990)Google Scholar
- 24.Corpetti, T., Mémin, E., Pérez, P.: Dense estimation of fluid flows. IEEE Trans. Pattern Anal. Mach. Intell.
**24**(3), 365380 (2002)CrossRefGoogle Scholar - 25.Nakajima, Y., Inomata, H., Nogawa, H., Sato, Y., Tamura, S., Okazaki, K., Torii, S.: Physics-based flow estimation of fluids. Pattern Recognit.
**36**(5), 1203–1212 (2003)CrossRefGoogle Scholar - 26.Arnaud, E., Mémin, E., Sosa, R., Artana, G.: A fluid motion estimator for schlieren image velocimetry. In: ECCV06, I, pp. 198–210 (2006)Google Scholar
- 27.Wayne Roberts, A., Varberg, Dale E.: Convex Functions. Academic Press, New York (1973)MATHGoogle Scholar
- 28.Haussecker, H.W., Fleet, D.J.: Computing optical flow with physical models of brightness variation. IEEE Trans. Pattern Anal. Mach. Intell.
**23**(6), 661–673 (2001)CrossRefGoogle Scholar - 29.Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys.
**141**(2), 199–224 (1998)MATHMathSciNetCrossRefGoogle Scholar - 30.Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput.
**16**(3), 173–261 (2001)MATHMathSciNetCrossRefGoogle Scholar - 31.Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin methods. Theory, computation and applications. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2000)Google Scholar
- 32.Arlotti, L., Banasiak, J., Lods, B.: A new approach to transport equations associated to a regular field: trace results and well-posedness. Mediterr. J. Math.
**6**, 367–402 (2009)MATHMathSciNetCrossRefGoogle Scholar - 33.Blanc, V.L.: \(L^1\)-stability of periodic stationary solutions of scalar convection-diffusion equations. J. Differ. Equ.
**247**, 1746–1761 (2009)MATHCrossRefGoogle Scholar - 34.Roy, S.: Reconstruction of a class of fluid flows by variational methods and inversion of integral transforms in tomography, Ph.D. dissertation, Tata Institute of Fundamental Research, CAM, Bangalore. https://www.dropbox.com/s/cbxblk3dg7kwxp5/Souvik_phd_thesis_compressed.pdf?dl=0 (2015)
- 35.Roy, S., Chandrashekar, P., Vasudeva Murthy, A.S.: A variational approach to optical flow estimation of incompressible fluid flow. J. Comput. Vis. Sci. (2015, submitted) Google Scholar