Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix

Charina, Maria; Conti, Costanza; Romani, Lucia

doi:10.1007/s00211-013-0587-8

Cited by 43 publications

(45 citation statements)

References 36 publications

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“…Indeed, we provide the expressions of the inverse matrices of the linear systems arising by imposing the algebraic conditions for exponential polynomial reproduction. Such conditions were first given in [17] and successively extended to any arbitrary arity in [7]. We also show that, under the symmetry assumption on Γ (or on a subset of it), the symbol a (k) M,N,Γ (z) has the same symmetry as B (k) N,Γ (z).…”

Section: Introductionmentioning

confidence: 78%

“…The most popular examples of subdivision schemes are B-spline subdivision schemes and their nonstationary counterparts, namely exponential B-spline subdivision schemes (see, e.g., [16,43]), characterized by the property of representing polynomials and exponential polynomials, respectively. Since in many applicative areas the capability of representing shapes described by exponential polynomial functions is fundamental, interpolating and approximating subdivision schemes based on exponential B-splines and inheriting their generation properties, have been recently introduced (see, e.g., [2,3,5,7,13,14,16,17,28,35,39]). We recall that, while the term generation usually refers to the subdivision scheme capability of providing specific types of limit functions, with reproduction we mean the capability of a subdivision scheme to reproduce in the limit exactly the same function from which the data are sampled.…”

Section: Introductionmentioning

confidence: 99%

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Exponential pseudo-splines: Looking beyond exponential B-splines

Conti

Gemignani

Romani

2016

Journal of Mathematical Analysis and Applications

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Pseudo-splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, B-spline functions, universally known for their usefulness in different fields of application. When replacing polynomial reproduction by exponential polynomial reproduction, a new family of functions is obtained. This new family is here constructed and called the family of exponential pseudo-splines. It is the nonstationary counterpart of (polynomial) pseudo-splines and includes exponential B-splines as a special subclass. In this work we provide a computational strategy for deriving the explicit expression of the Laurent polynomial sequence that identifies the family of exponential pseudo-spline nonstationary subdivision schemes. For this family we study its symmetry properties and perform its convergence and regularity analysis. Finally, we also show that the family of primal exponential pseudo-splines fills in the gap between exponential B-splines and interpolatory cardinal functions. This extends the analogous property of primal pseudo-spline stationary subdivision schemes.

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Exponential pseudo-splines: Looking beyond exponential B-splines

Conti

Gemignani

Romani

2016

Journal of Mathematical Analysis and Applications

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“…Using this proposition, we deduce that, for n ∈ [n 0 , N], the convolution product β αn\αn 0 * β αn 0 (16) preserves the exponential reproduction properties of β αn 0 . Note that in (16), the term β αn 0 \αn 0 = 1.…”

Section: Proposition (Unser and Blumentioning

confidence: 81%

“…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%

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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“…As opposed to polygon mesh models, subdivision methods do not necessarily have interpolating control points. Different methods based on nonstationary refinement rules have been proposed to approximate spheres using subdivision [27][28][29]. One drawback of polygon and subdivision methods is that they require a large number of parameters which can be a challenge when computational speed is important (e.g., in finite element models [30]).…”

Section: Discrete Closed Surfacesmentioning

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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Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix

Cited by 43 publications

References 36 publications

Exponential pseudo-splines: Looking beyond exponential B-splines

Exponential pseudo-splines: Looking beyond exponential B-splines

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

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

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