Topological Methods in Nonlinear Analysis

On the nonlinear analysis of optical flow

Shengxiang Xia and Yanmin Yin

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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


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