Skip to main content
SpringerLink
Log in

On the Nonlinear Statistics of Optical Flow

  • Conference paper
  • First Online:
Computational Topology in Image Context (CTIC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11382))

Included in the following conference series:

  • 514 Accesses

Abstract

In A naturalistic open source movie for optical flow evaluation, Butler et al. create a database of ground-truth optical flow from the computer-generated video Sintel. We study the high-contrast \(3\times 3\) patches from this video, and provide evidence that this dataset is well-modeled by a torus (a nonlinear 2-dimensional manifold). Our main tools are persistent homology and zigzag persistence, which are popular techniques from the field of computational topology. We show that the optical flow torus model is naturally equipped with the structure of a fiber bundle, which is furthermore related to the statistics of range images.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A range image contains a distance at each pixel.

References

  1. Adams, H., Atanasov, A., Carlsson, G.: Nudged elastic band in topological data analysis. Topological Methods Nonlinear Anal. 45(1), 247–272 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, H., Carlsson, G.: On the nonlinear statistics of range image patches. SIAM J. Imaging Sci. 2(1), 110–117 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adams, H., et al.: Persistence images: a vector representation of persistent homology. J. Mach. Learn. Res. 18(8), 1–35 (2017)

    MathSciNet  Google Scholar 

  4. Armstrong, M.A.: Basic Topology. Springer, Heidelberg (2013). https://doi.org/10.1007/978-1-4757-1793-8

    Book  MATH  Google Scholar 

  5. Baker, S., Scharstein, D., Lewis, J., Roth, S., Black, M.J., Szeliski, R.: A database and evaluation methodology for optical flow. Int. J. Comput. Vis. 92(1), 1–31 (2011)

    Article  Google Scholar 

  6. Bao, W., Li, H., Li, N., Jiang, W.: A liveness detection method for face recognition based on optical flow field. In: 2009 International Conference on Image Analysis and Signal Processing, IASP 2009, pp. 233–236. IEEE (2009)

    Google Scholar 

  7. Barron, J.L., Fleet, D.J., Beauchemin, S.S.: Performance of optical flow techniques. Int. J. Comput. Vis. 12(1), 43–77 (1994)

    Article  Google Scholar 

  8. Baryshnikov, Y., Ghrist, R.: Target enumeration via euler characteristic integrals. SIAM J. Appl. Math. 70(3), 825–844 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bauer, U.: Ripser: a lean C++ code for the computation of Vietoris-Rips persistence barcodes. Software (2017). https://github.com/Ripser/ripser

  10. Beauchemin, S.S., Barron, J.L.: The computation of optical flow. ACM Comput. Surv. (CSUR) 27(3), 433–466 (1995)

    Article  Google Scholar 

  11. Bendich, P., Marron, J.S., Miller, E., Pieloch, A., Skwerer, S.: Persistent homology analysis of brain artery trees. Ann. Appl. Stat. 10(1), 198 (2016)

    Article  MathSciNet  Google Scholar 

  12. Bubenik, P.: Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16(1), 77–102 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Burghelea, D., Dey, T.K.: Topological persistence for circle-valued maps. Discrete Comput. Geom. 50(1), 69–98 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Butler, D.J., Wulff, J., Stanley, G.B., Black, M.J.: A naturalistic open source movie for optical flow evaluation. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012. LNCS, vol. 7577, pp. 611–625. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33783-3_44

    Chapter  Google Scholar 

  15. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carlsson, G., De Silva, V., Morozov, D.: Zigzag persistent homologyand real-valued functions. In: Proceedings of the Twenty-Fifth annual Symposium on Computational Geometry, pp. 247–256. ACM (2009)

    Google Scholar 

  17. Carlsson, G., Ishkhanov, T., De Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76(1), 1–12 (2008)

    Article  MathSciNet  Google Scholar 

  18. Carlsson, G., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chung, M.K., Bubenik, P., Kim, P.T.: Persistence diagrams of cortical surface data. In: Prince, J.L., Pham, D.L., Myers, K.J. (eds.) IPMI 2009. LNCS, vol. 5636, pp. 386–397. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02498-6_32

    Chapter  Google Scholar 

  20. De Silva, V., Carlsson, G.: Topological estimation using witness complexes. SPBG 4, 157–166 (2004)

    Google Scholar 

  21. Edelsbrunner, H., Harer, J.L.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  22. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: 2000 Proceedings of 41st Annual Symposium on Foundations of Computer Science, pp. 454–463. IEEE (2000)

    Google Scholar 

  23. Fleet, D., Weiss, Y.: Optical flow estimation. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision, pp. 237–257. Springer, Boston (2006). https://doi.org/10.1007/0-387-28831-7_15

    Chapter  Google Scholar 

  24. Geiger, A., Lenz, P., Stiller, C., Urtasun, R.: Vision meets robotics: the KITTI dataset. Int. J. Robot. Res. (IJRR) 32, 1231–1237 (2013)

    Article  Google Scholar 

  25. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  26. Horn, B.K., Schunck, B.G.: Determining optical flow. Artif. Intell. 17(1–3), 185–203 (1981)

    Article  Google Scholar 

  27. Huang, J., Lee, A.B., Mumford, D.B.: Statistics of range images. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 324–332 (2000)

    Google Scholar 

  28. Lee, A.B., Pedersen, K.S., Mumford, D.: The nonlinear statistics of high-contrast patches in natural images. Int. J. Comput. Vis. 54(1–3), 83–103 (2003)

    Article  MATH  Google Scholar 

  29. Lum, P., et al.: Extracting insights from the shape of complex data using topology. Sci. Rep. 3, 1236 (2013)

    Article  Google Scholar 

  30. Mac Aodha, O., Humayun, A., Pollefeys, M., Brostow, G.J.: Learning a confidence measure for optical flow. IEEE Trans. Pattern Anal. Mach. Intell. 35(5), 1107–1120 (2013)

    Article  Google Scholar 

  31. Morozov, D.: Dionysus. http://www.mrzv.org/software/dionysus2/

  32. Roosendaal, T.: Sintel. Blender Foundation, Durian Open Movie Project (2010). http://www.sintel.org/

  33. Roth, S., Black, M.J.: On the spatial statistics of optical flow. Int. J. Comput. Vis. 74(1), 33–50 (2007)

    Article  Google Scholar 

  34. de Silva, V., Ghrist, R.: Coordinate-free coverage in sensor networks with controlled boundaries via homology. Int. J. Robot. Res. 25(12), 1205–1222 (2006)

    Article  MATH  Google Scholar 

  35. Topaz, C.M., Ziegelmeier, L., Halverson, T.: Topological data analysis of biological aggregation models. PloS One 10(5), e0126383 (2015)

    Article  Google Scholar 

  36. Xia, K., Wei, G.W.: Persistent homology analysis of protein structure, flexibility, and folding. Int. J. Numer. Methods Biomed. Eng. 30(8), 814–844 (2014)

    Article  MathSciNet  Google Scholar 

  37. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Gunnar Carlsson, Bradley Nelson, Jose Perea, and Guillermo Sapiro for helpful conversations.

Author information

Authors and Affiliations

  1. Colorado State University, Fort Collins, CO, 80523, USA

    Henry Adams, Johnathan Bush, Brittany Carr, Lara Kassab & Joshua Mirth

Authors
  1. Henry Adams
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johnathan Bush
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Brittany Carr
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lara Kassab
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Joshua Mirth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henry Adams .

Editor information

Editors and Affiliations

  1. Universidad de Málaga, Málaga, Spain

    Rebeca Marfil

  2. Universidad de Málaga, Málaga, Spain

    Mariletty Calderón

  3. University of Seville, Sevilla, Spain

    Fernando Díaz del Río

  4. University of Seville, Sevilla, Spain

    Pedro Real

  5. Universidad de Málaga, Málaga, Spain

    Antonio Bandera

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Adams, H., Bush, J., Carr, B., Kassab, L., Mirth, J. (2019). On the Nonlinear Statistics of Optical Flow. In: Marfil, R., Calderón, M., Díaz del Río, F., Real, P., Bandera, A. (eds) Computational Topology in Image Context. CTIC 2019. Lecture Notes in Computer Science(), vol 11382. Springer, Cham. https://doi.org/10.1007/978-3-030-10828-1_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-10828-1_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10827-4

  • Online ISBN: 978-3-030-10828-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation