Numerical Integration of Riemannian Gradient Flows for Image Labeling

  • Fabrizio SavarinoEmail author
  • Ruben Hühnerbein
  • Freddie Åström
  • Judit Recknagel
  • Christoph Schnörr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)


The image labeling problem can be described as assigning to each pixel a single element from a finite set of predefined labels. Recently, a smooth geometric approach was proposed [2] by following the Riemannian gradient flow of a given objective function on the so-called assignment manifold. In this paper, we adopt an approach from the literature on uncoupled replicator dynamics and extend it to the geometric labeling flow, that couples the dynamics through Riemannian averaging over spatial neighborhoods. As a result, the gradient flow on the assignment manifold transforms to a flow on a vector space of matrices, such that parallel numerical update schemes can be derived by established numerical integration. A quantitative comparison of various schemes reveals a superior performance of the adaptive scheme originally proposed, regarding both the number of iterations and labeling accuracy.


Image labeling Assignment manifold Riemannian gradient flow Replicator equation Multiplicative updates 


  1. 1.
    Absil, P.-A., Mathony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, Woodstock (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Åström, F., Petra, S., Schmitzer, B., Schnörr, C.: Image labeling by assignment. J. Math. Imaging Vis. 58(2), 211–238 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ay, N., Erb, I.: On a notion of linear replicator equations. J. Dyn. Differ. Equ. 17(2), 427–451 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bergmann, R., Fitschen, J.H., Persch, J., Steidl, G.: Iterative multiplicative filters for data labeling. Int. J. Comput. Vis. 1–19 (2017).
  5. 5.
    Burbea, J., Rao, C.R.: Entropy differential metric, distance and divergence measures in probability spaces: a unified approach. J. Multivar. Anal. 12, 575–596 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics, vol. 8, 2nd edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  7. 7.
    Kappes, J.H., Andres, B., Hamprecht, F.A., Schnörr, C., Nowozin, S., Batra, D., Kim, S., Kausler, B.X., Kröger, T., Lellmann, J., Komodakis, N., Savchynskyy, B., Rother, C.: A comparative study of modern inference techniques for structured discrete energy minimization problems. IJCV 155(2), 155–184 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Fabrizio Savarino
    • 1
    Email author
  • Ruben Hühnerbein
    • 1
  • Freddie Åström
    • 1
  • Judit Recknagel
    • 1
  • Christoph Schnörr
    • 1
  1. 1.Image and Pattern Analysis Group, RTG 1653Heidelberg UniversityHeidelbergGermany

Personalised recommendations