Ellipse-preserving Hermite interpolation and subdivision

Conti, Costanza; Romani, Lucia; Unser, Michaël

doi:10.1016/j.jmaa.2015.01.017

Cited by 46 publications

(27 citation statements)

References 27 publications

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“…Here, we focus on the cubic polynomial case. Note that a variety of Hermite schemes can be constructed, for instance based on exponential functions, as in [24] and [25].…”

Section: A Characterization Of Cubic Hermite Splinesmentioning

confidence: 99%

Hermite Snakes With Control of Tangents

Uhlmann

Fageot

Unser

2016

IEEE Trans. on Image Process.

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Abstract-We introduce a new model of parametric contours defined in a continuous fashion. Our curve model relies on Hermite spline interpolation and can easily generate curves with sharp discontinuities; it also grants direct access to the tangent at each location. With these two features, the Hermite snake distinguishes itself from classical spline-snake models and allows one to address certain bioimaging problems in a more efficient way. More precisely, the Hermite snake construction allows introducing sharp corners in the snake curve and designing directional energy functionals relying on local orientation information in the input image. Using the formalism of spline theory, the model is shown to meet practical requirements such as invariance to affine transformations and good approximation properties. Finally, the dependence on initial conditions and the robustness to the noise is studied on synthetic data in order to validate our Hermite snake model, and its usefulness is illustrated on real biological images acquired using brightfield, phase-contrast, differentialinterference-contrast, and scanning-electron microscopy.

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“…Here, we focus on the cubic polynomial case. Note that a variety of Hermite schemes can be constructed, for instance based on exponential functions, as in [24] and [25].…”

Section: A Characterization Of Cubic Hermite Splinesmentioning

confidence: 99%

Hermite Snakes With Control of Tangents

Uhlmann

Fageot

Unser

2016

IEEE Trans. on Image Process.

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“…We can also reproduce spheres or ellipsoids by using the basis function defined in (18). Similar to [9], a possible parameterization of the sphere is given by …”

Section: Spherementioning

confidence: 99%

“…Again, using the same basis function of our working example (18) . The basis functions that were used to construct the noninterpolatory surfaces correspond to the ones presented in [13].…”

Section: Torusmentioning

confidence: 99%

“…This convenient property is particularly relevant for the exact rendering of conic sections such as circles, ellipses, or parabolas, as well as other trigonometric and hyperbolic curves and surfaces [13]. In its absence, one must resort to subdivision to tackle this aspect [14][15][16][17][18]. However, existing comparable subdivision schemes usually rely on basis functions that are defined as a limit process and do not have a closed-form expression [19].…”

Section: Introductionmentioning

confidence: 99%

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A family of smooth and interpolatory basis functions for parametric curve and surface representation

Schmitter

Delgado-Gonzalo

Unser

2016

Applied Mathematics and Computation

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a b s t r a c tInterpolatory basis functions are helpful to specify parametric curves or surfaces that can be modified by simple user-interaction. Their main advantage is a characterization of the object by a set of control points that lie on the shape itself (i.e., curve or surface). In this paper, we characterize a new family of compactly supported piecewise-exponential basis functions that are smooth and satisfy the interpolation property. They can be seen as a generalization and extension of the Keys interpolation kernel using cardinal exponential B-splines. The proposed interpolators can be designed to reproduce trigonometric, hyperbolic, and polynomial functions or combinations of them. We illustrate the construction and give concrete examples on how to use such functions to construct parametric curves and surfaces.

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“…in the sense that SaV [2,4,6,21,22]. Note that one really has to consider di↵erent spaces here, because the result of the subdivision operator corresponds to a sequence on the finer grid Z/2 due to the upsampling incorporated into the subdivision operator.…”

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confidence: 99%

Factorization of Hermite subdivision operators preserving exponentials and polynomials

2016

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In this paper we focus on Hermite subdivision operators that act on vector valued data interpreting their components as function values and associated consecutive derivatives. We are mainly interested in studying the exponential and polynomial preservation capability of such kind of operators, which can be expressed in terms of a generalization of the spectral condition property in the spaces generated by polynomials and exponential functions. The main tool for our investigation are convolution operators that annihilate the aforementioned spaces, which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator.

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Ellipse-preserving Hermite interpolation and subdivision

Cited by 46 publications

References 27 publications

Hermite Snakes With Control of Tangents

Hermite Snakes With Control of Tangents

A family of smooth and interpolatory basis functions for parametric curve and surface representation

Factorization of Hermite subdivision operators preserving exponentials and polynomials

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