A general framework for the construction of piecewise-polynomial local interpolants of minimum degree

Antonelli, Michele; Beccari, Carolina Vittoria; Casciola, Giulio

doi:10.1007/s10444-013-9335-y

Cited by 16 publications

(33 citation statements)

References 18 publications

(40 reference statements)

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“…where the coefficients r[k] = (r x [k], r y [k], r z [k]) with k ∈ Z are the control points. The curve (29) can be locally modified by changing the position of a single control point. The shapes that r can adopt (e.g., polynomial, circular, elliptic) depend on the properties of the generator.…”

Section: Reproduction Of Idealized Shapesmentioning

confidence: 99%

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Compactly-supported smooth interpolators for shape modeling with varying resolution

et al. 2017

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A B S T R A C TIn applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.

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Section: Reproduction Of Idealized Shapesmentioning

confidence: 99%

“…The shapes that r can adopt (e.g., polynomial, circular, elliptic) depend on the properties of the generator. One can also extend the curve model (29) to represent separable tensorproduct surfaces. In this case, a surface σ is parameterized by u, v ∈ R as…”

Section: Reproduction Of Idealized Shapesmentioning

confidence: 99%

Compactly-supported smooth interpolators for shape modeling with varying resolution

et al. 2017

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“…Furthermore, from [23] we know that, for α ∈ α n 0 of multiplicity q + 1, there exist sequences p n such that (12) for n = 0, . .…”

Section: Reproduction Of Exponential Polynomialsmentioning

confidence: 99%

“…Especially in 3D applications, this can be inconvenient because it is no longer intuitive to interactively modify complex shapes. More recently a method to construct piecewise polynomial interpolators has been presented in [11,12].…”

Section: Introductionmentioning

confidence: 99%

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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“…The primary contributions of this work are: 1) a new geometrical representation based on subdivision. A crucial aspect is the choice of the subdivision mask that determines important properties of the model such as its approximation properties, the capability of reproducing circular, elliptical, or polynomial shapes [19], as well as the possibility of being interpolatory [20], [21] or not; 2) the derivation of associated energy functions such as regionand edge-based terms; 3) the presentation of an integrated strategy where the snake is optimized in a coarse-to-fine fashion. This multiscale approach is algorithmic and inherently recursive: We increase the number of points describing the curve as the algorithm progresses to the solution; at each step, the scale of the image feature (on which the optimization is performed) is matched to the density of the point cloud.…”

Section: Introductionmentioning

confidence: 99%

Multiresolution Subdivision Snakes

Badoual

Schmitter

Uhlmann

et al. 2017

IEEE Trans. on Image Process.

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Abstract-We present a new family of snakes that satisfy the property of multiresolution by exploiting subdivision schemes. We show in a generic way how to construct such snakes based on an admissible subdivision mask. We derive the necessary energy formulations and provide the formulas for their efficient computation. Depending on the choice of the mask, such models have the ability to reproduce trigonometric or polynomial curves. They can also be designed to be interpolating, a property that is useful in user-interactive applications. We provide explicit examples of subdivision snakes and illustrate their use for the segmentation of bioimages. We show that they are robust in the presence of noise and provide a multiresolution algorithm to enlarge their basin of attraction, which decreases their dependence on initialization compared to singleresolution snakes. We show the advantages of the proposed model in terms of computation and segmentation of structures with different sizes.

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A general framework for the construction of piecewise-polynomial local interpolants of minimum degree

Cited by 16 publications

References 18 publications

Compactly-supported smooth interpolators for shape modeling with varying resolution

Compactly-supported smooth interpolators for shape modeling with varying resolution

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

Multiresolution Subdivision Snakes

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