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 behavior is the same as the classical cubic Hermite spline algorithm. The same convergence properties—i.e., fourth order of approximation—are hence ensured.

Ellipse-preserving Hermite interpolation and subdivision / Costanza, Conti; Lucia, Romani; Michael, Unser. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - STAMPA. - 426:(2015), pp. 211-227. [10.1016/j.jmaa.2015.01.017]

Ellipse-preserving Hermite interpolation and subdivision

CONTI, COSTANZA;
2015

Abstract

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 behavior is the same as the classical cubic Hermite spline algorithm. The same convergence properties—i.e., fourth order of approximation—are hence ensured.
2015
426
211
227
Goal 9: Industry, Innovation, and Infrastructure
Costanza, Conti; Lucia, Romani; Michael, Unser
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1014664
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