Bézier Curves in the Space of Images

  • Alexander Effland
  • Martin Rumpf
  • Stefan Simon
  • Kirsten Stahn
  • Benedikt Wirth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)


Bézier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of Bézier curves to the infinite-dimensional space of images. To this end the space of images is equipped with a Riemannian metric which measures the cost of image transport and intensity variation in the sense of the metamorphosis model [MY01]. Bézier curves are then computed via the Riemannian version of de Casteljau’s algorithm, which is based on a hierarchical scheme of convex combination along geodesic curves. Geodesics are approximated using a variational discretization of the Riemannian path energy. This leads to a generalized de Casteljau method to compute suitable discrete Bézier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a Bézier approach for the modulation of face interpolation and shape animation via image sketches is presented.


De Casteljau algorithm Shape manifolds Metamorphosis 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander Effland
    • 1
  • Martin Rumpf
    • 1
  • Stefan Simon
    • 1
  • Kirsten Stahn
    • 1
  • Benedikt Wirth
    • 2
  1. 1.Institute for Numerical SimulationUniversität BonnBonnGermany
  2. 2.Institute for Computational and Applied MathematicsUniversity of MuensterMuensterGermany

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