Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Interpolations with elasticae in Euclidean spaces


Authors: W. Mio, A. Srivastava and E. Klassen
Journal: Quart. Appl. Math. 62 (2004), 359-378
MSC: Primary 41A05; Secondary 58E10, 65D05, 68U10, 94A08
DOI: https://doi.org/10.1090/qam/2054604
MathSciNet review: MR2054604
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Abstract: Motivated by interpolation problems arising in image analysis, computer vision, shape reconstruction, and signal processing, we develop an algorithm to simulate curve straightening flows under which curves in $ {\mathbb{R}^{n}}$ of fixed length and prescribed boundary conditions to first order evolve to elasticae, i.e., to (stable) critical points of the elastic energy $ E$ given by the integral of the square of the curvature function. We also consider variations in which the length $ L$ is allowed to vary and the flows seek to minimize the scale-invariant elastic energy $ {E_{inv}}$, or the free elastic energy $ {E_\lambda }$. $ {E_{inv}}$ is given by the product of $ L$ and the elastic energy $ E$, and $ {E_\lambda }$ is the energy functional obtained by adding a term $ \lambda $-proportional to the length of the curve to $ E$. Details of the implementations, experimental results, and applications to edge completion problems are also discussed.


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DOI: https://doi.org/10.1090/qam/2054604
Article copyright: © Copyright 2004 American Mathematical Society


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