Construction and characterization of non-uniform local interpolating polynomial splines

Beccari, Carolina Vittoria; Casciola, Giulio; Romani, Lucia

doi:10.1016/j.cam.2012.06.025

Cited by 16 publications

(36 citation statements)

References 14 publications

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“…Other variants have been presented in Refs. [40][41][42]. An interpolating subdivision scheme was originally introduced by Deslaurier and Dubuc [43].…”

Section: Interpolationmentioning

confidence: 99%

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

2017

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Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface.Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.

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“…Other variants have been presented in Refs. [40][41][42]. An interpolating subdivision scheme was originally introduced by Deslaurier and Dubuc [43].…”

Section: Interpolationmentioning

confidence: 99%

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

2017

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“…The shifted exponential B-splines in (9) also have the same reproduction property. By combining (11) and (12) and considering an arbitrary shift m, we see that (13) which is a linear combination of polynomials in t of degree up to n that are multiplied by e αt . Thus, we can collect all the factors multiplying t k and rewrite them as b k to express (13) as (14) 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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Construction and characterization of non-uniform local interpolating polynomial splines

Cited by 16 publications

References 14 publications

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

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

Multiresolution Subdivision Snakes

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