Showing papers on "Monotone cubic interpolation published in 1981"
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.
3,280 citations
TL;DR: The boundary integral method is formulated and applied using cubic spline interpolation along the boundary for both the geometry and the primary variables as discussed by the authors, which allows the accurate calculation of derivative dependent functions (on the boundary) such as velocity in potential flow.
Abstract: The boundary integral method is formulated and applied using cubic spline interpolation along the boundary for both the geometry and the primary variables. The cubic spline interpolation has continuous first and second derivatives between elements, thus allowing the accurate calculation of derivative dependent functions (on the boundary) such as velocity in potential flow. The spline functions also smooth the geometry and can represent curved sections with fewer nodes.
The results of numerical experiments indicate that the accuracy of the boundary integral equation method is improved for a given number of elements by using cubic spline interpolation. It is, however, necessary to use numerical quadrature. The quadrature slows calculation and/or degrades the accuracy. The numerical experiments indicate that most problems run faster for a given accuracy using linear interpolation. There seems to be a class of problems, however, which requires higher order interpolation and/or continuous derivatives for which the cubic spline interpolation works much better than linear interpolation.
58 citations
TL;DR: In this paper, a cubic spline method is described for the numerical solution of a two-point boundary value problem involving a fourth order linear differential equation, which is closely related to a known fourth order finite difference scheme.
31 citations
TL;DR: In this paper, a variable mesh cubic spline technique is developed for the shock-like solution of the one-dimensional Burgers' equation so that the necessity of taking very fine mesh all over the computational region could be avoided.
26 citations
TL;DR: In this paper, the problem of deriving accurate end conditions for cubic spline interpolation at equally spaced knots was considered and a number of end conditions which lead to derivative approximations of high accuracy was derived.
8 citations
TL;DR: In this article, the authors consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points, specified by a finite incidence matrix E.
Abstract: We consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points, specified by a finite incidence matrix E . Similarly the allowable discontinuities of the piecewise polynomials are specified by a finite incidence matrix F . We first find necessary conditions on ( E , F ) for the problem to be poised, that is to have a unique solution for any given data. The main result gives sufficient conditions on ( E , F ) for the problem to be poised, generalising a well-known result of Atkinson and Sharma. To this end we prove some results involving estimates of the numbers of zeros of the relevant piecewise polynomials.
7 citations
TL;DR: In this article, the authors studied deficient cubic splines by making less restrictive continuity requirements at the joints and having two interpolatory conditions, one of which is the matching condition at appropriate points of the dividing intervals while the other is matching of the integral means.
6 citations
6 citations
TL;DR: In this article, a simple nine freedom finite element for plate bending is obtained by cubic interpolation of the slopes parallel and linear interpolation on the slopes normal to each side, giving twelve local freedoms.
5 citations
01 Mar 1981
TL;DR: In this paper, the interpolation problem of matching a cubic spline at one intermediate point and a spline with multiple knots at two intermediate points between the successive knots is studied when the interpolatory points are not necessarily equispaced.
Abstract: The interpolation problem of matching a cubic spline at one intermediate point and cubic spline with multiple knots at two intermediate points between the successive knots are studied when the interpolatory points are not necessarily equispaced.
01 Dec 1981
TL;DR: In this paper, modified B-spline interpolations are conceived and operated on these non-ideal samples, in addition to smooth interpolation, the derivation and experiment also show reduction of interpolation error due to finite width sampling.
Abstract: The image samples from most scanning systems are integral averages of the light intensity over resolution elements. New modified B-spline interpolations are conceived and operated on these nonideal samples. In addition to smooth interpolation, the derivation and experiment also show reduction of interpolation error due to finite width sampling.
TL;DR: This article describes an APL algorithm that does satisfy the energy condition, be it that greater programming complexity is to be accepted and the basic theory presented in [4] has been extended in order to allow for all kinds of boundary conditions.
Abstract: Introduction APL functions for §pline interpolation have been described in [I] and [2], the latter based on an algorithm presented in [3]. Although simple in their construction and useful for many applications, these methods suffer from some deficiencies that make them less applicable when the data-points are not approximately equidistant. Such curves have slope and curvature continuity at the junctions, but do not satisfy the condition for minimization of total energy. This may lead to spurious solutions, as we shall illustrate with a simple example. This article describes an APL algorithm that does satisfy the energy condition, be it that greater programming complexity is to be accepted. The basic theory presented in [4] has been extended in order to allow for all kinds of boundary conditions. The program distinguishes graphs and planar splines-a graph having y as a single-valued linear-cubic spline through (x,y)-pairs, and a planar spline having x and y as separate linear-cubic spline functions of the arc length of the polygon formed by the nodes. Basic theory An open cubic spline is a function y = f x defined on the entire interval X[0] s x X[N]. On each subinterval X[U-I] s x X[U], y = f x is a cubic in x. The coefficients in the cubics change from interval to interval, but in such a way that y and its first and second derivative are continuous. Then their splined ordinates F over interval U can be found by: where the coefficients AS are yet to be determined. Note that since W ranges from 0 to I and V from I to O, the first two terms in (4) represent simple linear interpolation. The third term is a cubic correction term used to obtain continuous first and second derivatives. It is easy to prove that (4) is a non-degenerate cubic. (6) In order to determine the vector AS, we require the two expressions for FI to have the same value on both sides of X[U]~ this provides the tridiagonal set of linear equations: (AA+.xAS):D (7) In the case of open splines, matrix AA is symmetric and tridiagonal, and has as main and lower diagonals respectively: B÷-I+2×(DX+I~DX) (8) A÷i+-i~DX (9) and the right side of (7) is: D+-I~(iCDY)-DY (10) The elements of matrix AA depend on the interval sizes and are different for each interval; this is the fundamental difference from other methods where the intervals are, so …
TL;DR: In this paper, the use of spline function interpolation in the evaluation of Fourier integrals has been proposed, and the numerical results of some common functions by various interpolation methods and a simplified method of construction of splines function for various boundary conditions are also presented.
Abstract: This paper suggests the use of spline function interpolation in the evaluation of Fourier integrals At the same time, the numerical results of some common functions by various interpolation methods and a simplified method of construction of spline function for various boundary conditions are also presented
TL;DR: In this article, a method for improving the computational accuracy of numerical derivatives based on cubic splines and computed from a given set of function values with possible errors is described, which incorporates smoothing, a derivative profile and/or numerical derivatives in the computational process by solving a pentadiagonal system of linear equations.
Abstract: A method for improving the computational accuracy of numerical derivatives based on cubic splines and computed from a given set of function values with possible errors is described. The method incorporates smoothing, a derivative profile and/or numerical derivatives in the computational process by solving a pentadiagonal system of linear equations. Results comparing the derivatives computed by the present method, cubic spline and finite differences are given.
TL;DR: In this article, a method of approximation, minimizing the number of knots and satisfying the smoothing and fitting properties of the cubic spline approximation, is suggested, and two modifications of the method are presented.
Abstract: A method of approximation, minimizing the number of knots and satisfying the smoothing and fitting properties of the cubic spline approximation, is suggested. Two modifications of the method are presented. The first approach is specified to provide an adequate simple approximation to the digitalized smooth curve (such as the velocity-depth function in seismology), while the second is more general and may be used to approximate geophysical measurements, which may include errors.