Interpolatory subdivision schemes with the optimal approximation order

Zhang, Baoxing; Zheng, Haibin; Song, Weijie; Lin, Zehao; Zhou, Jie

doi:10.1016/j.amc.2018.10.078

Cited by 5 publications

(2 citation statements)

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“…Specifically speaking, with a suitably chosen iteration and a function of this iteration, we properly modify the Loop subdivision rules in the regular part of the mesh and design the subdivision rules in the neighborhoods of extraordinary points in order to obtain the desired symmetric non-stationary Loop subdivision. More interestingly, in the spirit of the push-back operation [15], we also derive the limit positions of the initial points, which generalizes the existing result and can be used to interpolate the initial points with certain valence. Furthermore, we also present a nonuniform generalization which can locally adjust the shape of the limit surface and also derive the corresponding limit positions of the initial points.…”

Section: Introductionmentioning

confidence: 75%

A Symmetric Non-Stationary Loop Subdivision with Applications in Initial Point Interpolation

Zhang,

Zheng

2024

Symmetry

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Loop subdivision is a significant surface scheme with wide applications in fields like computer graphics and wavelet. As a type of stationary scheme, Loop subdivision cannot adjust the limit surface directly. In this paper, we present a new way to solve this problem by proposing a symmetric non-stationary Loop subdivision based on a suitable iteration. This new scheme can be used to adjust the limit surfaces freely and thus can generate surfaces with different shapes. For this new scheme, we show that it is C2 convergent in the regular part of mesh and is at least tangent plane continuous at the limit positions of the extraordinary points. Additionally, we present a non-uniform generalization of this new symmetric non-stationary subdivision so as to locally control the shape of the limit surfaces. More interestingly, we present the limit positions of the initial points, both for the symmetric non-stationary Loop subdivision and its non-uniform generalization. Such limit positions can be used to interpolate the initial points with different valences, generalizing the existing result. Several numerical examples are given to illustrate the performance of the new schemes.

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Section: Introductionmentioning

confidence: 75%

A Symmetric Non-Stationary Loop Subdivision with Applications in Initial Point Interpolation

Zhang,

Zheng

2024

Symmetry

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“…a j z j , z ∈ C \ {0} (1.5) as τ a = A (1) m (see, e.g., [5]) where A (z) denotes the first derivative of A(z). In particular, in [5] it is proven that, in case of a subdivision scheme with odd-symmetry (see Definition 2.2), τ a ∈ Z; conversely, in case of even-symmetry (see Definition 2.3), τ a ∈ Z 2 \ Z. Symmetric, compactly supported interpolating m-refinable functions that we can find in the literature always fulfill equation (1.4) in the case τ a ∈ Z, i.e., are associated with an odd-symmetric subdivision mask a satisfying (1.3) for an even integer M (see, e.g., [1,2,8,11,15,16,18,23,24]). In addition, for all of them the odd-symmetric subdivision mask a is such that…”

Section: Introduction and Purpose Of The Workmentioning

confidence: 99%

Interpolating m-refinable functions with compact support: The second generation class

Romani

2019

Applied Mathematics and Computation

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We present an algorithm for the construction of a new class of compactly supported interpolating refinable functions that we call the second generation class since, contrary to the existing class, is associated to subdivision schemes with an even-symmetric mask that does not contain the submask {0..., 0, 1, 0, ...0}.As application examples of the proposed algorithm we present interpolating 4-refinable functions that are generated by parameter-dependent, even-symmetric quaternary schemes never considered in the literature so far.

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