EPFL | Biomedical Imaging Group | Hermite Interpolation

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.

@ARTICLE(http://bigwww.epfl.ch/publications/conti1501.html,
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="")

© 2015 Elsevier. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Elsevier. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.