Additive Operator Splittings

An efficient numerical approach for the acceleration of algorithms is the use of so called splitting schemes. They allow for a decomposition of a single problem into multiple problems that offer certain advantages compared to the original one. In general, the resulting problems can be solved very efficiently with standard numerical methods. Some of them even offer advantages regarding a possible parallelisation.

Our research mainly focuses on additive operator splitting (AOS) schemes (Lu et al. 1991), which have the advantage over multiplicative splittings that the result does not depend on the order in which the operators are applied. In [1] such an AOS scheme has been introduced to the image processing and computer vision community. Apart from nonlinear diffusion filtering, variants for regularisation methods [2] and optic flow computation [3] have been developed. We have also used AOS schemes successfully in the context of active contour models [4].

  1. J. Weickert, B.M. ter Haar Romeny, M.A. Viergever:
    Efficient and reliable schemes for nonlinear diffusion filtering.
    IEEE Transactions on Image Processing, Vol. 7, No. 3, 398-410, March 1998.

  2. J. Weickert:
    Efficient image segmentation using partial differential equations and morphology
    Pattern Recognition, Vol. 34, No. 9, 1813-1824, September 2001.
    Also available as Technical Report No. 3/2000. Computer Science Series, University of Mannheim, Germany, February 2000.

  3. J. Weickert, C. Schnörr:
    Variational optic flow computation with a spatio-temporal smoothness constraint.
    Journal of Mathematical Imaging and Vision, Vol. 14, No. 3, 245-255, May 2001.
    Revised version of
    Technical Report No. 15/2000. Computer Science Series, University of Mannheim, Germany, July 2000.

  4. J. Weickert, G. Kühne:
    Fast methods for implicit active contour models.
    In S. Osher, N. Paragios (Eds.): Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, 2003.

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