A formula for estimating the deviation of a binary interpolatory subdivision curve from its data polygon

Deng, Chongyang; Jin, Wenbiao; Li, Yajuan; Xu, Huirong

doi:10.1016/j.amc.2017.01.035

Cited by 3 publications

(3 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 4. If P 0 = {p 0 i,j , i ∈ Z} is the initial polygon and P k = {p k i,j , i ∈ Z} is the polygon obtained by (17). Then the error bound between two successive levels is…”

Section: The Error Bounds and Subdivision Depth For Surface Modelsmentioning

confidence: 99%

See 1 more Smart Citation

A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes

et al. 2021

View full text Add to dashboard Cite

In this paper, an advanced computational technique has been presented to compute the error bounds and subdivision depth of quaternary subdivision schemes. First, the estimation is computed of the error bound between quaternary subdivision limit curves/surfaces and their polygons after kth-level subdivision by using l0 order of convolution. Secondly, by using the error bounds, the subdivision depth of the quaternary schemes has been computed. Moreover, this technique needs fewer iterations (subdivision depth) to get the optimal error bounds of quaternary subdivision schemes as compared to the existing techniques.

show abstract

“…Theorem 4. If P 0 = {p 0 i,j , i ∈ Z} is the initial polygon and P k = {p k i,j , i ∈ Z} is the polygon obtained by (17). Then the error bound between two successive levels is…”

Section: The Error Bounds and Subdivision Depth For Surface Modelsmentioning

confidence: 99%

“…We also mention the drawback of this technique in this paper. The second technique is introduced by Deng et al [17]. It is not mature enough.…”

Section: Introductionmentioning

confidence: 99%

A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes

et al. 2021

View full text Add to dashboard Cite

show abstract

“…Wang et al [22][23][24], Zhou and Zeng [25], Zeng and Chen [26], and Huang and Wang [27][28][29][30][31] presented different versions of the bounds and subdivision depth for the Doo-Sabin [32], Loop [33], and Catmull-Clark [34] schemes based on the 1-norm, 2-norm, and infinity-norm. Further, Mustafa et al [35] and Deng et al [36] presented error bounds and subdivision depth based on the infinity-norm of stationary schemes of different arity (i.e., binary, ternary, etc.) and their generalizations.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes

et al. 2023

View full text Add to dashboard Cite

Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations.

show abstract

A Computational Method for Subdivision Depth of Ternary Schemes

et al. 2020

View full text Add to dashboard Cite

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.

show abstract

A formula for estimating the deviation of a binary interpolatory subdivision curve from its data polygon

Cited by 3 publications

References 22 publications

A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes

A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes

An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes

A Computational Method for Subdivision Depth of Ternary Schemes

Contact Info

Product

Resources

About