On the Coupling of Decimation Operator with Subdivision Schemes for Multi-scale Analysis

Kui, Zhiqing; Baccou, Jean; Liandrat, Jacques

doi:10.1007/978-3-319-67885-6_9

Cited by 3 publications

(5 citation statements)

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“…At edges, the coefficients involved for S and D must be adapted as described in [13]. For the schemes used in this paper, edges adaptations are precised in Appendix D. However, with this redefinition at edges, the uniformity of the schemes is lost, which is essential for the multiresolution analysis.…”

Section: Application: Wellbore Trajectoriesmentioning

confidence: 99%

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Enforcing convergence of derivatives for L∞ approximation of a regular curve

Garcia

Liandrat

2021

Computer Aided Geometric Design

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Converging approximation of a regular curve by polygonal lines in the uniform norm does not imply the convergence of the discrete differentials to their smooth counterpart. In this paper, we provide a constructive approach that, given a converging polygon sequence and an approximation of its distance to the objective curve, provides another sequence of polygons for which convergence of discrete differentials occurs as well. This approach is based on the notion of local scale of a polygon and uses multi-resolution decomposition as well as a non linear smoothing process. We provide the proof of the convergence and some numerical evidence of it, with application to the evaluation of solid friction in a pipe.

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Section: Application: Wellbore Trajectoriesmentioning

confidence: 99%

“…11 9X j 0 − 105X j 1 + 945X j 2 + 1575X j 3 − 525X j 4 + 189X j 5 − 45X j 6 + 5X j 7 DX j 2 = 1 2 11 5X j 0 − 45X j 1 + 189X j 2 − 525X j 3 + 1575X j 4 + 945X j 5 − 105X j 6 + 9X j 7 DX j 3 = 1 2 11 33X j 0 − 273X j 1 + 1001X j 2 − 2145X j 3 + 3003X j 4 − 3003X j 5 + 3003X j 6 + 429X j 7 (13)…”

Section: Subdivisionunclassified

Enforcing convergence of derivatives for L∞ approximation of a regular curve

Garcia

Liandrat

2021

Computer Aided Geometric Design

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“…Linear case. A generic method to construct the mask of any decimation consistent with a given linear uniform subdivision has been proposed in [15]. The main results are recalled in the two following propositions.…”

Section: Construction Of Consistent Decimation Operatorsmentioning

confidence: 99%

“…In the linear case, the decay of the prediction error for consistent scheme (h, h) was studied in [15] and is recalled in the following proposition, Proposition 9. Let h be a linear subdivision scheme such that ∀n = 0, 1, 2, .…”

Section: Decay Of the Prediction Errormentioning

confidence: 99%

“…Two examples of schemes for which no associated multiresolution existed are also given to emphasize the main motivation of our work. Then starting from first results presented in [15], we focus on the construction of decimation operators consistent with general subdivision schemes (Section 3). Section 4 is devoted to the construction of detail operators while the analysis of linear and non-linear multiresolutions is provided in section 5.…”

Section: Introductionmentioning

confidence: 99%

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On the construction of multiresolution analyses associated to general subdivision schemes

Kui¹,

Baccou²,

Liandrat³

2021

Math. Comp.

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Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory. This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.

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