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Ellipse-Preserving Hermite Interpolation and Subdivision

C. Conti, L. Romani, M. Unser

Journal of Mathematical Analysis and Applications, vol. 426, no. 1, pp. 211-227, June 1, 2015.

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behavioris the same as the classical cubic Hermite spline algorithm. The same convergence properties—i.e., fourth order of approximation—are hence ensured. 

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AUTHOR="Conti, C. and Romani, L. and Unser, M.",
TITLE="Ellipse-Preserving {H}ermite Interpolation and Subdivision",
JOURNAL="Journal of Mathematical Analysis and Applications",
YEAR="2015",
volume="426",
number="1",
pages="211--227",
month="June 1,",
note="")

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