## Topological Methods in Nonlinear Analysis

### On the nonlinear analysis of optical flow

#### Abstract

We utilize the methods of computational topology to the database of optical flow created by Roth and Black from range images, and demonstrate a qualitative topological analysis of spaces of $3 \times 3, 5 \times 5$ and $7 \times 7$ optical flow patches. We experimentally prove that there exist subspaces of the spaces of the three sizes high-contrast patches that are topologically equivalent to a circle and a three circles model, respectively. The Klein bottle is the quotient space described as the square $[0,1] \times [0,1]$ with sides identified by the relations $(0, y)\sim (1, y)$ for $y\in [0, 1]$ and $(x, 0) \sim (1-x, 1)$ for $x\in [0, 1]$. For the space of $3 \times 3$ optical flow patches we found a subspace having the same homology as that of the Klein bottle. As the size of patches increases, the Klein bottle feature of the spaces of $5 \times 5$ and $7 \times 7$ optical flow patches gradually disappears.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 661-676.

Dates
First available in Project Euclid: 21 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1482289234

Digital Object Identifier
doi:10.12775/TMNA.2016.054

Mathematical Reviews number (MathSciNet)
MR3642778

Zentralblatt MATH identifier
1364.68351

#### Citation

Xia, Shengxiang; Yin, Yanmin. On the nonlinear analysis of optical flow. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 661--676. doi:10.12775/TMNA.2016.054. https://projecteuclid.org/euclid.tmna/1482289234

#### References

• H. Adams, A. Atanasov and G. Carlsson, Nudged elastic band in topological data analysis, Topol. Methods Nonlinear Anal. 45 (2015), 247–272.
• H. Adams and G. Carlsson, On the nonlinear statistics of range image patches, SIAM J. Imag. Sci. 2 (2009), 110–117.
• H. Adams and A. Tausz, Javaplex tutorial, http://javaplex.googlecode.com/svn/trunk /reports/javaplex$\_$tutorial/javaplex$\_$tutorial.pdf.
• S. Baker, D. Scharstein, J.P. Lewis, S. Roth, M. J. Black and R. Szeliski, A database and evaluation methodology for optical flow, Internat. J. Comput. Vision 92 (2011), 1–31.
• J.L. Barron, D.J. Fleet and S.S. Beauchemin, Performance of optical flow techniques, Internat. J. Comput. Vision 12 (1994), 43–77.
• G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.) 46 (2009), 255–308.
• G. Carlsson, T. Ishkhanov, V. de Silva and A. Zomorodian, On the local behavior of spaces of natural images, Internat. J. Comput. Vision, 76 (2008), 1–12.
• V. de Silva and G. Carlsson, Topological estimation using witness complexes, Proc. Sympos. Point-Based Graphics (2004), 157–166.
• H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28 (2002), 511–533.
• D.J. Field, Relations between the statistics of natural images and the response properties of cortical cells, J. Opt. Soc. Amer. 4 (1987), 2379–2394.
• A. Geiger, P. Lenz and R. Urtasun, Are we ready for autonomous driving? the kitti vision benchmark suite, CVPR (2012), 3354–3361.
• R. Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc. 45 (2008), 61–75.
• J.J. Gibson, The Perception of the Visual World, Riverside Press, Cambridge, 1950.
• K. Jia, X. Wang and X. Tang, Optical flow estimation using learned sparse model, 2011 IEEE International Conference on Computer Vision, November (2011), 2391–2398.
• A.B. Lee, K.S. Pedersen and D. Mumford, The non-linear statistics of high-contrast patches in natural images, Internat. J. Comput. Vision 54 (2003), 83–103.
• S. Roth, and M. J. Black, On the spatial statistics of optical flow, Internat. J. Comput. Vision 74 (2007), 33–50.
• D. Sun, S. Roth and M. J. Black, A quantitative analysis of current practices in optical flow estimation and the principles behind Them, Internat. J. Comput. Vision 106 (2014), 115–137.
• J.H. van Hateren, Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation, J. Comput. Physiology A 171 (1992), 157–170.
• D.H. Warren and E.R. Strelow, Electronic Spatial Sensing for the Blind: Contributions from Perception, Springer, 1985.
• S. Xia, On the local behavior of spaces of range image patches, to appear.
• A. Zomorodian and G. Carlsson, Computing persistent homology, Discr. Comput. Geom. 33 (2005), 249–274.