Temporal scale spaces

  • Authors:
  • Daniel Fagerström

  • Affiliations:
  • Computational Vision and Active Perception Laboratory, Department of Numerical Analysis and Computing Science, KTH Royal Institute of Technology, Stockholm, Sweden

  • Venue:
  • Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
  • Year:
  • 2003

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Abstract

In this paper we discuss how to define a scale space suitable for temporal measurements. We argue that such a temporal scale space should possess the properties of: temporal causality, linearity, continuity, positivity, recursitivity as well as translational and scaling covariance. It is shown that these requirements imply a one parameter family of convolution kernels. Furthermore it is shown that these measurements can be realized in a time recursive way, with the current data as input and the temporal scale space as state, i.e. there is no need for storing earlier input. This family of measurement processes contains the diffusion equation on the half line (that represents the temporal scale) with the input signal as boundary condition on the temporal axis. The diffusion equation is unique among the measurement processes in the sense that it is preserves positivity (in the scale domain) and is infinitesimally generated. A numerical scheme is developed and relations to other approaches are discussed.