Topic

# Monotone cubic interpolation

About: Monotone cubic interpolation is a(n) research topic. Over the lifetime, 1740 publication(s) have been published within this topic receiving 38111 citation(s).

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TL;DR: It can be shown that the order of accuracy of the cubic convolution method is between that of linear interpolation and that of cubic splines.

Abstract: Cubic convolution interpolation is a new technique for resampling discrete data. It has a number of desirable features which make it useful for image processing. The technique can be performed efficiently on a digital computer. The cubic convolution interpolation function converges uniformly to the function being interpolated as the sampling increment approaches zero. With the appropriate boundary conditions and constraints on the interpolation kernel, it can be shown that the order of accuracy of the cubic convolution method is between that of linear interpolation and that of cubic splines. A one-dimensional interpolation function is derived in this paper. A separable extension of this algorithm to two dimensions is applied to image data.

2,789 citations

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TL;DR: In this article, a monotone piecewise bicubic interpolation algorithm was proposed for data on a rectangular mesh, where the first partial derivatives and first mixed partial derivatives are determined by the mesh points.

Abstract: In a 1980 paper [SIAM J. Numer. Anal., 17 (1980), pp. 238–246] the authors developed a univariate piecewise cubic interpolation algorithm which produces a monotone interpolant to monotone data. This paper is an extension of those results to monotone $\mathcal{C}^1 $ piecewise bicubic interpolation to data on a rectangular mesh. Such an interpolant is determined by the first partial derivatives and first mixed partial (twist) at the mesh points. Necessary and sufficient conditions on these derivatives are derived such that the resulting bicubic polynomial is monotone on a single rectangular element. These conditions are then simplified to a set of sufficient conditions for monotonicity. The latter are translated to a system of linear inequalities, which form the basis for a monotone piecewise bicubic interpolation algorithm.

1,942 citations

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TL;DR: In this article, the authors describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates, which can help prevent the problems that result from inappropriate linearity assumptions.

Abstract: We describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates. This simple method can help prevent the problems that result from inappropriate linearity assumptions. We compare restricted cubic spline regression to non-parametric procedures for characterizing the relationship between age and survival in the Stanford Heart Transplant data. We also provide an illustrative example in cancer therapeutics.

1,878 citations

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01 Nov 1987TL;DR: This paper presents an Explicity Formulation for Cubic Beta-splines, a simple Approximation technique for Uniform Cubic B-spline Surfaces, and discusses its applications in Rendering and Evaluation and simulation.

Abstract: 1 Introduction 2 Preliminaries 3 Hermite and Cubic Spline Interpolation 4 A Simple Approximation Technique - Uniform Cubic B-splines 5 Splines in a More General Setting 6 The One-Sided Basis 7 Divided Differences 8 General B-splines 9 B-spline Properties 10 Bezier Curves 11. Knot Insertion 12 The Oslo Algorithm 13 Parametric vs. Geometric Continuity 14 Uniformly-Shaped Beta-spline Surfaces 15 Geometric Continuity, Reparametrization, and the Chain Rule 16 Continuously-Shaped Beta-splines 17 An Explicity Formulation for Cubic Beta-splines 18 Discretely-Shaped Beta-splines 19 B-spline Representations for Beta-splines 20 Rendering and Evaluation 21 Selected Applications

1,286 citations

01 Jan 2010

TL;DR: In this article, a cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain is described.

Abstract: article i nfo We describe a novel cubic Hermite collocation scheme for the solution of the coupled integro-partial differential equations governing the propagation of a hydraulic fracture in a state of plane strain. Special blended cubic Hermite-power-law basis functions, with arbitrary index 0b αb1, are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. The implementation of blended infinite elements to model semi-infinite crack problems is also described. Explicit formulae for the integrated kernels associated with the cubic Hermite and blended basis functions are provided. The cubic Hermite collocation algorithm is used to solve a number of different test problems with two distinct propagation regimes and the results are shown to converge to published similarity and asymptotic solutions. The convergence rate of the cubic Hermite scheme is determined by the order of accuracy of the tip asymptotic expansion as well as the O(h 4 ) error due to the Hermite cubic interpolation. The errors due to these two approximations need to be matched in order to achieve optimal convergence. Backward Euler time-stepping yields a robust algorithm that, along with geometric increments in the time-step, can be used to explore the transition between propagation regimes over many orders of magnitude in time.

923 citations