Variational Symmetric formulation 

for Optical Flow

Álvarez L., Castaño C., García M., Krissian K., Mazorra L., Salgado A., Sánchez J.

Departamento de Informática y Sistemas
Universidad de Las Palmas de Gran Canaria
Campus de Tafira, 35017 Las Palmas de Gran Canaria, Spain.
Tel:        +34 928 458708
Fax:       +34 928 458711
E-mails: lalvarez,ccastano,mgarcia,krissian,lmazorra,asalgado,jsanchez  @

This work was founded by the European Project FLUID (contract no. FP6-513663).


The problem of motion analysis or registration between two images is an important problem that has been widely addressed in the literature. One of the main technique used to solve this problem is optical flow, where the pixels of one image are matched to the pixels of the second image. Hence, the estimated motion vector field depends on the reference image and is asymmetric. However, in most application the solution should be independent of the reference image. Symmetrical formulations of the optical flow has been proposed in [1,2,3], where the solution is constraint to be symmetric using a combination of the flow in both directions. We propose a new symmetric variational formulation of the optical flow problem, where the flow is naturally symmetric. Results on the Yosemite sequence show an improved accuracy of our symmetric flow with respect to standard optical flow algorithm.

Symmetric Formulation

In order to find a displacement between 2 images I1 and I2 in a symmetric way, we consider an intermediate image Im at half way between  I1 and  I2 , so that there exists a displacement field  u which fullfils $ \forall \mathbf{x}, I_m(\mathbf{x})=I_1(\mathbf{x}-\mathbf{u}) = I_2(\mathbf{x}+\mathbf{u})$ .
To estimate this displacement, we minimize the energy: $\displaystyle E(\mathbf{u}) = \int_\Omega \left(I_1(\mathbf{x^-}) - I_2\left(\m...
where we denote  $ \mathbf{x^+}=\mathbf{x}+\mathbf{u}$ and  $ \mathbf{x^-}=\mathbf{x}-\mathbf{u}$ .
Euler-Lagrange equations lead to a system of equations, which  is solved using an iterative Gauss-Seidel scheme. A pyramidal multiscale approach is used to compute the flow and to avoid falling into local minima of the energy.


In order to compare our results with the synthetic data, we have to transform the flow  $ \mathbf{u}$ into a flow  $ \mathbf{h}$ defined as  $ I_1(\mathbf{x})=I_2(\mathbf{x+h(x)})$ , according to  $ \mathbf{h(x-u(x))=2u(x)}$$ \mathbf{h(x)}$ is computed as a weighted average of the values of  $ \mathbf{h(x-u(x)}$ in the neighbourhood of  $ \mathbf{x}$ . This transformation is similar to the one proposed in [4]. Figure 1 shows the mean angular error obtained on the Yosemite sequence as a function of the smoothing coefficient $ \alpha$ . The symmetric version of the optical flow reaches a better result with a mean angular error of 2.32 degrees compare to 2.765 degrees for the standard algorithm.

real flow for the yosemite sequence angular error for the 2d+t symmetric algorithm
Yosemite sequence real flow Angular error for the 2d+t symmetric algorithm

angular error for different techniques
Mean angular error as a function of the smoothing coefficient alpha.
Tested on 4 algorithms: standard/symmetric and 2D/2D+t.


Christensen, G., Johnson, H.:
Consistent image registration.
IEEE Transactions on Medical Imaging 20(7) (2001) 568-582
Cachier, P., Rey, D.:
Symmetrization of the non-rigid registration problem using inversion-invariant energies: Application to multiple sclerosis.
In: LNCS (MICCAI 2000). Volume 1935., Pittsburgh, USA, Springer Berlin/Heidelberg (2000) 472-481
Alvarez, L., Deriche, R., Papadopoulo, T., Sanchez, J.:
Symmetrical dense optical flow estimation with occlusions detection.
In: ECCV (1). (2002) 721-735
Salgado, A., Sánchez, J.:
Optical flow estimation with large displacements: A temporal reularizer.
Technical report, Instituto Universitario de Ciencias y Tecnologías Cibernéticas (2006)