International Journal of Computer Vision

, Volume 45, Issue 3, pp 245–264

A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion

  • Joachim Weickert
  • Christoph Schnörr

DOI: 10.1023/A:1013614317973

Cite this article as:
Weickert, J. & Schnörr, C. International Journal of Computer Vision (2001) 45: 245. doi:10.1023/A:1013614317973


Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for data-driven and flow-driven, isotropic and anisotropic, as well as spatial and spatio-temporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are well-posed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flow-driven regularizers is identified, and a design criterion is proposed for constructing anisotropic flow-driven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.

optic flow differential methods regularization diffusion filtering well-posedness 

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Joachim Weickert
    • 1
  • Christoph Schnörr
    • 1
  1. 1.Computer Vision, Graphics, and Pattern Recognition Group, Department of Mathematics and Computer ScienceUniversity of MannheimMannheimGermany

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