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
Full-text PDF Free Access
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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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R. Bryant and P. Griffiths, Reduction of order for the constrained variational problem and $\smallint {k^2}/2 ds$, Amer. J. Math. 108 (1986), 525–570.
G. Dziuk, E. Kuwert, and R. Schätzle, Evolution of elastic curves in ${\mathbb {R}^{n}}$: existence and computation, SIAM J. Math. Anal. 33 (2002), 1228–1245.
G. Guy and G. Medioni, Inferring global perceptual contours from local features, Int’l J. Computer Vision 20 (1996), 113–133.
L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattisimo sensu accepti, Bousquet, Lausannae e Genevae 24 (1744) E65A. O. O. Ser. I.
M. W. Jones and M. Chen, A new approach to the construction of surfaces from contour data, Computer Graphics Forum 13 (1994), C-85–C-84.
V. Jurdjevic, Non-Euclidean elastica, Amer. J. Math. 117 (1995), 93–124.
N. Koiso, On the motion of a curve towards elastica, in: Actes de la Table Ronde de Géometrie Différentielle (Luminy 1992) Sémin. Congr. 1 (Soc. Math. France, Paris, 1996), 403–436.
J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differential Geometry 20 (1984), 1–22.
J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology 24 (1985), 75–88.
J. Langer and D. A. Singer, Curve straightening in Riemannian manifolds, Ann. Global Anal. Geom. 5 (1987), 133–150.
D. Mumford, Elastica and computer vision, in: Algebraic Geometry and its Applications (West Lafayette, IN, 1990) (Springer, New York, 1994), 491–506.
R. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299–340.
R. Palais, Critical point theory and the minimax principle, Global Analysis, Proc. Symp. Pure Math. XV (1970), 185–212.
S. T. Roweis and L. K. Saul, Nonlinear dimensionality reduction by locally linear embedding, Science 290 (2000), 2323–2326.
E. Sharon, A. Brandt, and R. Basri, Completion energies and scale, IEEE Trans. Pattern Analysis and Machine Intelligence 22 (2000), 1117–1131.
C. Truesdell, The influence of elasticity on analysis: the classical heritage, Bull. Amer. Math. Soc. 9 (1983), 293–310.
I. Weiss, 3D Shape representation by contours, Computer Vision, Graphics, and Image Processing 41 (1988), 80–100.
L. R. Williams and D. W. Jacobs, Stochastic completion fields: a neural model of illusory contour shape and salience, Neural Computation 9 (1997), 837–858.
Y. Wen, Curve straightening flow deforms closed plane curves with nonzero rotation number to circles, J. Differential Equations 120 (1995), 89–107.
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© Copyright 2004
American Mathematical Society
