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The ubiquitous ellipse

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Abstract

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.

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Authors and Affiliations

  1. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Guillermo Sapiro

  2. Department of Computer Science, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Alfred M. Bruckstein

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  1. Guillermo Sapiro
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  2. Alfred M. Bruckstein
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Sapiro, G., Bruckstein, A.M. The ubiquitous ellipse. Acta Appl Math 38, 149–161 (1995). https://doi.org/10.1007/BF00992844

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