The variational approach to shape preservation
Curves and surfaces
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
Two dimensional spline interpolation algorithms
Two dimensional spline interpolation algorithms
Computer Aided Geometric Design
Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design
Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design
Theory and algorithms for cubic l(1) splines
Theory and algorithms for cubic l(1) splines
Shape-preserving, multiscale interpolation by bi- and multivariate cubic L1 splines
Computer Aided Geometric Design
An Efficient Algorithm for Generating Univariate Cubic L1 Splines
Computational Optimization and Applications
Shape-preserving, first-derivative-based parametric and nonparametric cubic L1 spline curves
Computer Aided Geometric Design
A compressed primal-dual method for generating bivariate cubic L1 splines
Journal of Computational and Applied Mathematics
C1 and C2-continuous polynomial parametric Lp splines (p≥1)
Computer Aided Geometric Design
Geometric dual formulation for first-derivative-based univariate cubic L1 splines
Journal of Global Optimization
Computer Aided Geometric Design
Shape-preserving, first-derivative-based parametric and nonparametric cubic L1 spline curves
Computer Aided Geometric Design
A geometric programming approach for bivariate cubic L1 splines
Computers & Mathematics with Applications
Sample-based polynomial approximation of rational Bézier curves
Journal of Computational and Applied Mathematics
Fast L1kCk polynomial spline interpolation algorithm with shape-preserving properties
Computer Aided Geometric Design
Nonlinear L1C1 interpolation: application to Images
Proceedings of the 7th international conference on Curves and Surfaces
Computational Optimization and Applications
Robust univariate spline models for interpolating interval data
Operations Research Letters
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The results in this paper quantify the ability of cubic L2 splines to preserve the shape of nonparametric data. The data under consideration include multiscale data, that is, data with abrupt changes in spacing and magnitude. A simplified dual-to-primal transformation for a geometric programming model for cubic L1 splines is developed. This transformation allows one to establish in a transparent manner relationships between the shape-preserving properties of a cubic L1 spline and the solution of the dual geometric-programming problem. Properties that have often been associated with shape preservation in the past include preservation of linearity and convexity/concavity. Under various circumstances, cubic L1 splines preserve linearity and convexity/concavity of data. When four consecutive data points lie on a straight line, the cubic L1 spline is linear in the interval between the second and third data points. Cubic L1 splines of convex/concave data preserve convexity/concavity if the first divided differences of the data do not increase/decrease too rapidly. When cubic L2 splines do not preserve convexity/concavity, they still do not cross the piecewise linear interpolant and, therefore, they do not have extraneous oscillation.