A smoothness criterion for monotonicity-preserving subdivision

Floater, Michael S.; Beccari, Carolina Vittoria; Cashman, Thomas J.; Romani, Lucia

doi:10.1007/s10444-012-9275-y

Cited by 9 publications

(3 citation statements)

References 18 publications

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“…Mustafa et al [7] introduced n-ary interpolating SS in which convexity preservation of the scheme was also examined. Floater et al [8] offered monotonicity preservation of input data having conditions that assure differentiability to the limit function of the SS.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

et al. 2020

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Shape preservation has been the heart of subdivision schemes (SSs) almost from its origin, and several analyses of SSs have been established. Shape preservation properties are commonly used in SSs and various ways have been discovered to connect smooth curves/surfaces generated by SSs to applied geometry. With an eye on connecting the link between SSs and applied geometry, this paper analyzes the geometric properties of a ternary four-point rational interpolating subdivision scheme. These geometric properties include monotonicity-preservation, convexity-preservation, and curvature of the limit curve. Necessary conditions are derived on parameter and initial control points to ensure monotonicity and convexity preservation of the limit curve of the scheme. Furthermore, we analyze the curvature of the limit curve of the scheme for various choices of the parameter. To support our findings, we also present some examples and their graphical representation.

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Section: Introductionmentioning

confidence: 99%

Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

et al. 2020

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“…For instance, in [GSV08], arithmetic means are replaced by geometric means. Other examples include schemes based on median-interpolating polynomials [DY00, Osw04,XY05], interpolating rational functions [KvD99], or interpolating circles [FBCR13]. For all these schemes, some specialized smoothness analysis is available.…”

Section: Introductionmentioning

confidence: 99%

Hölder Regularity of Geometric Subdivision Schemes

2015

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We present a framework for analyzing non-linear R d -valued subdivision schemes which are geometric in the sense that they commute with similarities in R d . It admits to establish C 1,α -regularity for arbitrary schemes of this type, and C 2,α -regularity for an important subset thereof, which includes all real-valued schemes. Our results are constructive in the sense that they can be verified explicitly for any scheme and any given set of initial data by a universal procedure. This procedure can be executed automatically and rigorously by a computer when using interval arithmetics.

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“…Let us finally mention that monotonicity preservation has also been investigated for scattered data approximation [19], for subdivision curves [9,14], convolution of B-splines [18], for rational surfaces [5] and for non-polynomial functions [15].…”

Section: Related Workmentioning

confidence: 99%

Piecewise Polynomial Monotonic Interpolation of 2D Gridded Data

Allemand-Giorgis

Bonneau

Hahmann

et al. 2014

Mathematics and Visualization

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A method for interpolating monotone increasing 2D scalar data with a monotone piecewise cubic C 1 -continuous surface is presented. Monotonicity is a sufficient condition for a function to be free of critical points inside its domain. The standard axial monotonicity for tensor-product surfaces is however too restrictive. We therefore introduce a more relaxed monotonicity constraint. We derive sufficient conditions on the partial derivatives of the interpolating function to ensure its monotonicity. We then develop two algorithms to effectively construct a monotone C 1 surface composed of cubic triangular Bézier surfaces interpolating a monotone gridded data set. Our method enables to interpolate given topological data such as minima, maxima and saddle points at the corners of a rectangular domain without adding spurious extrema inside the function domain. Numerical examples are given to illustrate the performance of the algorithm.

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A smoothness criterion for monotonicity-preserving subdivision

Cited by 9 publications

References 18 publications

Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

Hölder Regularity of Geometric Subdivision Schemes

Piecewise Polynomial Monotonic Interpolation of 2D Gridded Data

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