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

Abstract

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.

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Weickert, J., Schnörr, C. A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion. International Journal of Computer Vision 45, 245–264 (2001). https://doi.org/10.1023/A:1013614317973

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  • DOI: https://doi.org/10.1023/A:1013614317973

  • optic flow
  • differential methods
  • regularization
  • diffusion filtering
  • well-posedness