This work was founded by the European Project FLUID (contract no. FP6513663).
Introduction
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 I
_{1}
and I
_{2}
in a symmetric way,
we consider an intermediate image I
_{m}
at half way between I
_{1}
and I
_{2}
, so that there exists a displacement field
u
which fullfils
.
To estimate this displacement, we minimize the energy:
where we denote
and
.
EulerLagrange equations lead to a system of equations, which is solved using an iterative GaussSeidel
scheme. A pyramidal multiscale approach is used to compute the flow and
to avoid falling into local minima of the energy.
Experiments
In order to compare our results with the synthetic data, we have to transform the flow
into a flow
defined as
, according to
.
is computed as a weighted average of the values of
in the neighbourhood of
. 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
. 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.


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

Mean angular error as a function of the smoothing coefficient alpha.
Tested on 4 algorithms: standard/symmetric and 2D/2D+t. 
Bibliography
 1

Christensen, G., Johnson, H.:
Consistent image registration.
IEEE Transactions on Medical Imaging 20(7) (2001) 568582
 2

Cachier, P., Rey, D.:
Symmetrization of the nonrigid registration problem using
inversioninvariant energies: Application to multiple sclerosis.
In: LNCS (MICCAI 2000). Volume 1935., Pittsburgh, USA, Springer
Berlin/Heidelberg (2000) 472481
 3

Alvarez, L., Deriche, R., Papadopoulo, T., Sanchez, J.:
Symmetrical dense optical flow estimation with occlusions detection.
In: ECCV (1). (2002) 721735
 4

Salgado, A., Sánchez, J.:
Optical flow estimation with large displacements: A temporal
reularizer.
Technical report, Instituto Universitario de Ciencias y Tecnologías
Cibernéticas (2006)