Table 1: EPE and AAE for the Square sequence.
Method EPE AAE
Spatial 0.071 0.629
o
Temporal 1 (β = 8) 0.049 0.204
o
Temporal 2 (δ = 25) 0.036 0.134
o
Temporal 3 (β = 1, δ = 25) 0.035 0.138
o
square sequence. Frame by frame, the optical ﬂows
are more accurate in the temporal methods. We also
observe that the results are very stable, especially in
the middle of the ’Temporal 2 (δ)’ line. Reasonably,
the frames at the beginning and end of the sequence
present higher errors, due to the Dirichlet boundary
conditions.
Figure 2: EPE in each optical ﬂow of the Square sequence.
5 CONCLUSIONS
In this paper we have presented a new spatio-temporal
coherence model for the consistent estimation of op-
tical ﬂows. We have focused on different nonlinear
ﬂow assumptions that are more conﬁdent in the es-
timation of motion ﬁelds than previous approaches.
These nonlinear assumptions correctly ﬁt with the
standard nonlinear brightness and gradient constancy
terms, can cope with general image sequences and
provide better solutions. In particular, we have pro-
posed two main contributions: on the one hand, we
have introduced the nonlinear ﬂow constancy assump-
tion (FCA) in the energy model. This term relates
ﬂow ﬁelds at different time instants and is consis-
tent with the rest of the energy terms. On the other
hand, we have proposed a nonlinear temporal diffu-
sion scheme at the PDE level, which produces con-
tinuous ﬂows in time. We have seen that this new
scheme is more general than using the continuous
temporal regularization of the ﬂow, with the advan-
tage that it conveniently deals with continuous and
non-continuous velocities. In fact, if the motion is
very small, this term approximates a continuous tem-
poral smoothing scheme. In the experimental results,
we have shown that the method provides important
accuracy improvements, specially in the presence of
large displacements. The results are promising in both
cases, although we observe a better performance for
the nonlinear temporal smoothing scheme in general.
Another interesting result of the temporal coherence
schemes is that the background motion oscillations
tend to disapper. These oscillations clearly appear in
the spatial method, in regions where there is no appar-
ent motion.
ACKNOWLEDGEMENTS
This work has been partly founded by the Spanish
Ministry of Science and Innovation through the re-
search project TIN2011-25488.
REFERENCES
´
Alvarez, L., Weickert, J., and S
´
anchez, J. (2000). Reli-
able estimation of dense optical ﬂow ﬁelds with large
displacements. International Journal of Computer Vi-
sion, 39(1):41–56.
Black, M. J. (1994). Recursive non-linear estimation of dis-
continuous ﬂow ﬁelds. In Proceedings of the third Eu-
ropean conference on Computer vision (vol. 1), ECCV
’94, pages 138–145, Secaucus, NJ, USA. Springer-
Verlag New York, Inc.
Nagel, H. H. and Enkelmann, W. (1986). An investigation
of smoothness constraints for the estimation of dis-
placement vector ﬁelds from image sequences. IEEE
Transanctions on Pattern Analysis and Machine Intel-
ligence, 8:565–593.
Papenberg, N., Bruhn, A., Brox, T., Didas, S., and Weick-
ert, J. (2006). Highly Accurate Optic Flow Compu-
tation with Theoretically Justiﬁed Warping. Interna-
tional Journal of Computer Vision, 67(2):141–158.
Salgado, A. and S
´
anchez, J. (2006). A temporal regularizer
for large optical ﬂow estimation. In IEEE Interna-
tional Conference on Image Processing ICIP, pages
1233–1236.
S
´
anchez, J., Monz
´
on, N., and Salgado, A. (2012). Robust
optical ﬂow estimation. IPOL: Image Processing On-
line, Preprint:1–18.
Sun, D., Sudderth, E., and Black, M. J. (2010). Lay-
ered Image Motion with Explicit Occlusions, Tempo-
ral Consistency, and Depth Ordering. In Lafferty, J.,
Williams, C. K. I., Shawe-Taylor, J., Zemel, R., and
Culotta, A., editors, Advances in Neural Information
Processing Systems 23, volume 23, pages 2226–2234.
Weickert, J. and Schn
¨
orr, C. (2001). Variational Optic
Flow Computation with a Spatio-Temporal Smooth-
ness Constraint. Journal of Mathematical Imaging
and Vision, 14(3):245–255.
OpticalFlowEstimationwithConsistentSpatio-temporalCoherenceModels
369