Decomposition of optical flow on the sphere

  • Clemens Kirisits
  • Lukas F. LangEmail author
  • Otmar Scherzer
Original Paper


We propose a number of variational regularisation methods for the estimation and decomposition of motion fields on the \(2\)-sphere. While motion estimation is based on the optical flow equation, the presented decomposition models are motivated by recent trends in image analysis. In particular we treat \(u+v\) decomposition as well as hierarchical decomposition. Helmholtz decomposition of motion fields is obtained as a natural by-product of the chosen numerical method based on vector spherical harmonics. All models are tested on time-lapse microscopy data depicting fluorescently labelled endodermal cells of a zebrafish embryo.


Optical flow Vector spherical harmonics Biomedical imaging Computer vision Variational methods  Vector field decomposition 

Mathematics Subject Classification

92C55 92C37 92C17 35A15 68U10 33C55 



We thank Pia Aanstad from the University of Innsbruck for sharing her biological insight and for kindly providing the microscopy data. This work has been supported by the Vienna Graduate School in Computational Science (IK I059-N) funded by the University of Vienna. In addition, we acknowledge the support by the Austrian Science Fund (FWF) within the national research networks “Photoacoustic Imaging in Biology and Medicine” (project S10505-N20, Reconstruction Algorithms for PAI) and “Geometry + Simulation” (project S11704, Variational Methods for Imaging on Manifolds).


  1. Abhau, J., Belhachmi, Z., Scherzer, O.: On a decomposition model for optical flow. Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, pp. 126–139. Springer, Berlin (2009)CrossRefGoogle Scholar
  2. Amat, F., Myers, E.W., Keller, P.J.: Fast and robust optical flow for time-lapse microscopy using super-voxels. Bioinformatics 29(3), 373–380 (2013)CrossRefGoogle Scholar
  3. Aubert, G., Kornprobst, P.: Mathematical problems in image processing. In: Partial Differential Equations and the Calculus of Variations, With a Foreword by Olivier Faugeras, 2nd edn. Applied Mathematical Sciences, vol. 147. Springer, New York (2006)Google Scholar
  4. Baker, S., Scharstein, D., Lewis, J.P., Roth, S., Black, M.J., Szeliski, R.: A database and evaluation methodology for optical flow. Int. J. Comput. Vis. 92(1), 1–31 (2011)CrossRefGoogle Scholar
  5. Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., Lévy, B.: Polygon Mesh Processing. A K Peters, Wellesley (2010)Google Scholar
  6. Freeden, W., Schreiner, M.: Spherical Functions of Mathematical Geosciences. A Scalar, Vectorial, and Tensorial Setup. Springer, Berlin (2009)zbMATHGoogle Scholar
  7. Frühauf, F., Pontow, C., Scherzer, O.: Texture enhancing based on variational image decomposition. In: Bergounioux, M. (ed.) Mathematical Image Processing. Springer Proceedings in Mathematics, vol. 5, pp. 127–140. Springer, Berlin (2011)CrossRefGoogle Scholar
  8. Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artif. Intell. 17, 185–203 (1981)CrossRefGoogle Scholar
  9. Imiya, A., Sugaya, H., Torii, A., Mochizuki, Y.: Variational analysis of spherical images. In: Gagalowicz, A., Philips, W. (eds.) Computer Analysis of Images and Patterns. Lecture Notes in Computer Science, vol. 3691, pp. 104–111. Springer, Berlin (2005)CrossRefGoogle Scholar
  10. Khan, S., Lefèvre, J., Ammari, H., Baillet, S.: Feature detection and tracking in optical flow on non-flat manifolds. Pattern Recognit. Lett. 32(15), 2047–2052 (2011)CrossRefGoogle Scholar
  11. Kimmel, C.B., Ballard, W.W., Kimmel, S.R., Ullmann, B., Schilling, T.F.: Stages of embryonic development of the zebrafish. Dev. Dyn. 203(3), 253–310 (1995)CrossRefGoogle Scholar
  12. Kirisits, C., Lang, L.F., Scherzer, O.: Optical flow on evolving surfaces with an application to the analysis of 4D microscopy data. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds.) SSVM’13: Proceedings of the Fourth International Conference on Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 7893, pp. 246–257. Springer, Berlin (2013a)CrossRefGoogle Scholar
  13. Kirisits, C., Lang, L.F., Scherzer, O.: Optical Flow on Evolving Surfaces with Space and Time Regularisation. University of Vienna, Austria (2013b). Preprint on ArXiv arXiv:1301.0322
  14. Kohlberger, T., Memin, E., Schnörr, C.: Variational dense motion estimation using the Helmholtz decomposition. In: Griffin, L.D., Lillholm, M. (eds.) Scale Space Methods in Computer Vision, vol. 2695. Lecture Notes in Computer Science, pp. 432–448. Springer, Berlin (2003)Google Scholar
  15. Lefèvre, J., Baillet, S.: Optical flow and advection on 2-Riemannian manifolds: a common framework. IEEE Trans. Pattern Anal. Mach. Intell. 30(6), 1081–1092 (2008)CrossRefGoogle Scholar
  16. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Die Grundlehren der Mathematischen Wissenschaften. Springer, New York (1972)CrossRefGoogle Scholar
  17. Megason, S.G., Fraser, S.E.: Digitizing life at the level of the cell: high-performance laser-scanning microscopy and image analysis for in toto imaging of development. Mech. Dev. 120(11), 1407–1420 (2003)CrossRefGoogle Scholar
  18. Melani, C., Campana, M., Lombardot, B., Rizzi, B., Veronesi, F., Zanella, C., Bourgine, P., Mikula, K., Peyriéras, N., Sarti, A.: Cells tracking in a live zebrafish embryo. In: Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS 2007), pp. 1631–1634 (2007)Google Scholar
  19. Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations, vol. 22. University Lecture Series. American Mathematical Society, Providence, RI (2001). The fifteenth Dean Jacqueline B. Lewis memorial lecturesGoogle Scholar
  20. Michel, V.: Lectures on Constructive Approximation. Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball. Birkhäuser, New York (2013)Google Scholar
  21. Mizoguchi, T., Verkade, H., Heath, J.K., Kuroiwa, A., Kikuchi, Y.: Sdf1/Cxcr4 signaling controls the dorsal migration of endodermal cells during zebrafish gastrulation. Development 135(15), 2521–2529 (2008)CrossRefGoogle Scholar
  22. Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(H^{-1}\)-norm. Multiscale Model. Simul. 1(3), 349–370 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  23. Quelhas, P., Mendonça, A.M., Campilho, A.: Optical flow based Arabidopsis thaliana root meristem cell division detection. In: Campilho, A., Kamel, M. (eds.) Image Analysis and Recognition, vol. 6112. Lecture Notes in Computer Science, pp. 217–226. Springer, Berlin (2010)Google Scholar
  24. Schmid, B., Shah, G., Scherf, N., Weber, M., Thierbach, K., Campos Pérez, C., Roeder, I., Aanstad, P., Huisken, J.: High-speed panoramic light-sheet microscopy reveals global endodermal cell dynamics. Nat. Commun. 4, 2207 (2013)CrossRefGoogle Scholar
  25. Schnörr, C.: Determining optical flow for irregular domains by minimizing quadratic functionals of a certain class. Int. J. Comput. Vis. 6, 25–38 (1991)CrossRefGoogle Scholar
  26. Schuster, T., Weickert, J.: On the application of projection methods for computing optical flow fields. Inverse Probl. Imaging 1(4), 673–690 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  27. Tadmor, E., Nezzar, S., Vese, L.: A multiscale image representation using hierarchical \((BV, L^2)\) decompositions. Multiscale Model. Simul. 2(4), 554–579 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  28. Torii, A., Imiya, A., Sugaya, H., Mochizuki, Y.: Optical flow computation for compound eyes: variational analysis of omni-directional views. In: De Gregorio, M., Di Maio, V., Frucci, M., Musio, C. (eds.) Vision, and Artificial Intelligence, vol. 3704. Lecture Notes in Computer Science, pp. 527–536. Springer, Berlin (2005)Google Scholar
  29. Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput 19(1–3), 553–572 (2003). Special issue in honor of the sixtieth birthday of Stanley OsherGoogle Scholar
  30. Warga, R.M., Nüsslein-Volhard, C.: Origin and development of the zebrafish endoderm. Development 126(4), 827–838 (1999)Google Scholar
  31. Weickert, J., Schnörr, C.: Variational optic flow computation with a spatio-temporal smoothness constraint. J. Math. Imaging Vis. 14, 245–255 (2001)CrossRefzbMATHGoogle Scholar
  32. Weickert, J., Bruhn, A., Brox, T., Papenberg, N.: A survey on variational optic flow methods for small displacements. In: Scherzer, O. (ed.) Mathematical Models for Registration and Applications to Medical Imaging, vol. 10. Mathematics in Industry, pp. 103–136. Springer, Berlin (2006)Google Scholar
  33. Weiskopf, D., Erlebacher, G.: Overview of flow visualization. In: Hansen, C.D., Johnson, C.R. (eds.) The Visualization Handbook, pp. 261–278. Elsevier, Amsterdam (2005)CrossRefGoogle Scholar
  34. Yuan, J., Schnörr, C., Steidl, G.: Simultaneous higher-order optical flow estimation and decomposition. SIAM J. Sci. Comput. 29(6), 2283–2304 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  35. Yuan, J., Schnörr, C., Steidl, G.: Convex Hodge decomposition and regularization of image flows. J. Math. Imaging Vis. 33(2), 169–177 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Clemens Kirisits
    • 1
  • Lukas F. Lang
    • 1
    Email author
  • Otmar Scherzer
    • 1
    • 2
  1. 1.Computational Science CenterUniversity of ViennaViennaAustria
  2. 2.Radon Institute of Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

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