Acta Applicandae Mathematica

, Volume 38, Issue 2, pp 149–161

The ubiquitous ellipse

  • Guillermo Sapiro
  • Alfred M. Bruckstein


We discuss three different affine invariant evolution processes for smoothing planar curves. The first one is derived from ageometric heat-type flow, both the initial and the smoothed curves being differentiable. The second smoothing process is obtained from a discretization of this affine heat equation. In this case, the curves are represented by planarpolygons. The third process is based onB-spline approximations. For this process, the initial curve is a planar polygon, and the smoothed curves are differentiable and even analytic. We show that, in the limit, all three affine invariant smoothing processes collapse any initial curve into anelliptic point.

Mathematics subject classifications (1991)

35Q80 41A15 52B99 53A15 

Key words

affine invariant multi-scale smoothing geometric heat flows polygons B-splines ellipses 


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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Guillermo Sapiro
    • 1
  • Alfred M. Bruckstein
    • 2
  1. 1.Department of Electrical EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Computer ScienceTechnion-Israel Institute of TechnologyHaifaIsrael
  3. 3.LIDS, MITCambridgeUSA

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