Showing papers on "Bicubic interpolation published in 1985"
TL;DR: In this paper, the applicability of various proposed interpolation techniques for estimating annual precipitation at selected sites was compared using 30 years of annual precipitation data at 29 stations located in the Region II of the North Central continental United States.
Abstract: One of the problems which often arises in engineering hydrology is to estimate data at a given site because either the data are missing or the site is ungaged. Such estimates can be made by spatial interpolation of data available at other sites. A number of spatial interpolation techniques are available today with varying degrees of complexity. It is the intent of this paper to compare the applicability of various proposed interpolation techniques for estimating annual precipitation at selected sites.
The interpolation techniques analyzed include the commonly used Thiessen polygon, the classical polynomial interpolation by least-squares or Lagrange approach, the inverse distance technique, the multiquadric interpolation, the optimal interpolation and the Kriging technique. Thirty years of annual precipitation data at 29 stations located in the Region II of the North Central continental United States have been used for this study. The comparison is based on the error of estimates obtained at five selected sites. Results indicate that the Kriging and optimal interpolation techniques are superior to the other techniques. However, the multiquadric technique is almost as good as those two. The inverse distance interpolation and the Thiessen polygon gave fairly satisfactory results while the polynomial interpolation did not produce good results.
555 citations
TL;DR: In this article, an explicit representation of a piecewise rational cubic function is developed which can be used to solve the problem of shape preserving interpolation, and an error analysis of the interpolant is given.
Abstract: An explicit representation of a $C^1 $ piecewise rational cubic function is developed which can be used to solve the problem of shape preserving interpolation. It is shown that the interpolation method can be applied to convex and/or monotonic sets of data and an error analysis of the interpolant is given. The scheme includes, as a special case, the monotonic rational quadratic interpolant considered by the authors in [1] and [5]. However, the requirement of convexity necessitates the generalization to the rational cubic form employed here.
149 citations
01 Oct 1985
TL;DR: Hardy's multiquadric interpolation (MQI) scheme is a global, continuously differentiable interpolation method for solving scattered data interpolation problems as discussed by the authors, which is capable of producing monotonic, extremely accurate interpolating functions, integrals, and derivatives.
Abstract: Hardy's multiquadric interpolation (MQI) scheme is a global, continuously differentiable interpolation method for solving scattered data interpolation problems. It is capable of producing monotonic, extremely accurate interpolating functions, integrals, and derivatives. Derivative estimates for a variety of one and two-dimensional surfaces were obtained. MQI was then applied to the spherical blast wave problem of von Neumann. The numerical solution agreed extremely well with the exact solution. 17 refs., 3 figs., 2 tabs.
51 citations
TL;DR: It is shown that for cubic splines or bicubic patches, accurate tool motion can be generated by a fast DDA-like (Digital Differential Analyser) digital integration scheme, which reduces integration to successive additions.
Abstract: A method is proposed for generating tool centre paths with an accuracy equal to the resolution of the machine tool. The method exploits the properties of Bertrand curves and surfaces, to generate tool motion which secures the desired degree of continuity in the machined contour or surface. It is shown that for cubic splines or bicubic patches, accurate tool motion can be generated by a fast DDA-like (Digital Differential Analyser) digital integration scheme, which reduces integration to successive additions. Accuracy is achieved by eliminating the integration error inherent in DDA.
43 citations
TL;DR: The two algorithms INTCOL and HERMCOL presented here implement the collocation method as described in [l] for the special case of rectangular domains as descendents of the module P3Cl COLLOCATION contained in early versions of ELLPACK.
Abstract: The two algorithms INTCOL and HERMCOL presented here implement the collocation method as described in [l] for the special case of rectangular domains. These algorithms are descendents of the module P3Cl COLLOCATION contained in early versions of ELLPACK [2]. One of the strengths of the collocation method is the ease of its implementation for rectangular domains. The distinguishing features of these two algorithms are (1) they are tailored to take maximum advantage of the two dimensional rectangular domain, (2) they are compatible in structure and design with the algorithm GENCOL, and (3) they provide for the easy use of various software for solving the linear system of equations generated. The driver provided includes a program to format the linear system as a band matrix (with minimal bandwidth) and a program to solve the system by Gauss elimination with scaled partial pivoting. These two algorithms are included in the ELLPACK system [3] as INTERIOR COLLOCATION and HERMITE COLLOCATION.
24 citations
TL;DR: In the present paper, a smoothing algorithm for bicubic spline surfaces is presented, having the piecewise cubic boundaries of the patches fixed, the algorithm chooses adequate twists factors in order to increase the smoothness.
22 citations
01 Oct 1985
TL;DR: In this article, it was shown that cubic spline interpolation with the not-a-knot side condition converges to any C squared without any mesh-ratio restriction as the mesh size goes to zero.
Abstract: : It is shown that cubic spline interpolation with the not-a-knot side condition converges to any C squared without any mesh-ratio restriction as the mesh size goes to zero Keywords: Cubic spline; Interpolation; Not-a-knot; Convergence; Total positivity
19 citations
TL;DR: A new algorithm for piecewise bicubic interpolation to bivariate data on a rectangular mesh is described and the Hermite form is used to represent the resulting surface.
16 citations
TL;DR: This paper presents an algorithm for the collocation method described in the companion paper Collocation Software for Second Order Elliptic Partial Differential---Equations -~ Houstis, Mitchell and Rice.
Abstract: This paper presents an algorithm for the collocation method described in the companion paper Collocation Software for Second Order Elliptic Partial Differential---Equations -~ Houstis, Mitchell and Rice. The problem solved has a general elliptic linear operator with variable coefficients, general linear boundary conditions and a general two dimensional domain. This algorithm uses the output of the domain processor [Rice, 1982] (which can be generated "by hand" for simple domains). The basis functions used are bicubic Hermite polynomials defined on a tensor product grid covering the problem domain. This paper describes the driver, use of test problems and the files of the algorithm. The input and output of the algorithms themselves are documented in the initial comments of the algorithm. This documentation is reproduced here.
14 citations
TL;DR: In this paper, necessary and sufficient conditions for the convergence of cardinal interpolation with bivariate box splines as the degree tends to infinity are given. But they do not consider the case where the spline is fixed.
Abstract: We give necessary and sufficient conditions for the convergence of cardinal interpolation with bivariate box splines as the degree tends to infinity.
11 citations
TL;DR: In this paper, the existence of weight functions for which the Lagrange interpolating polynomials associated with the zeros of the corresponding orthogonal polynomorphials diverge in every L p space with p > 2 for some continuous function is proved.
TL;DR: The final author version and the galley proof are versions of the publication after peer review that features the final layout of the paper including the volume, issue and page numbers.
TL;DR: In this article, surface splines are obtained while minimizing a rotation invariant inner product in an Hilbert space, which is a very good technique to fit a surface to a noisy scattered data set.
01 Jun 1985
TL;DR: In this paper, a bumpless monotonic bicubic interpolation (MBI) technique is proposed for guessing at a very smooth interpolated curved surface, which can be used for very small number micron and/or submicron VLSI MOSFET device modelling.
Abstract: A bumpless monotonic bicubic interpolation (MBI) technique is proposed. The method is applied to MOSFET device modelling for guessing at a very smooth interpolated curved surface. Monotonic increase in a two-dimensional surface can be held, even if the actual device characteristics show steepest change, like punchthrough characteristics. The technique can be utilised for very small number micron and/or submicron VLSI MOSFET device modelling.
DOI•
01 Sep 1985TL;DR: In this paper, a grid generation algorithm using a mapping composed of bicubic B-splines is presented. But the concept of the degree of a mapping and how it can be used to determine what requirements are necessary if a mapping is to produce a suitable grid is examined.
Abstract: Finite difference methods are more successful if the accompanying grid has lines which are smooth and nearly orthogonal. The development of an algorithm which produces such a grid when given the boundary description. Topological considerations in structuring the grid generation mapping are discussed. The concept of the degree of a mapping and how it can be used to determine what requirements are necessary if a mapping is to produce a suitable grid is examined. The grid generation algorithm uses a mapping composed of bicubic B-splines. Boundary coefficients are chosen so that the splines produce Schoenberg's variation diminishing spline approximation to the boundary. Interior coefficients are initially chosen to give a variation diminishing approximation to the transfinite bilinear interpolant of the function mapping the boundary of the unit square onto the boundary grid. The practicality of optimizing the grid by minimizing a functional involving the Jacobian of the grid generation mapping at each interior grid point and the dot product of vectors tangent to the grid lines is investigated. Grids generated by using the algorithm are presented.
01 Jan 1985
TL;DR: An algorithm is introduced for the solution of the problem to interpolate the shape of the object between the “known” planar slices and used for the three-dimensional display of theobject and to provide cross-sectional views from specified perspectives.
Abstract: The general problem is introduced by way of a particular application. An unknown, three-dimensional object is sampled discretely on parallel planar slices. The problem is to interpolate the shape of the object between the “known” planar slices. An algorithm is introduced for the solution of the problem and is illustrated by examples. Other possible approaches are discussed. The algorithm is used for the three-dimensional display of the object and to provide cross-sectional views from specified perspectives.
01 Jan 1985
TL;DR: In this article, the authors investigate the question of positivity and convergence for higher Hermite-Fejer interpolation with boundary conditions and show that it is NP-hard to achieve convergence.
Abstract: In this paper we investigate the question of positivity and convergence for Hermite-Fejer and higher Hermite-Fejer interpolation with boundary conditions.
TL;DR: In this article, the authors proposed a more regular approximation, which can be analytically differentiated, obtained by bicubic splines, based on bilinear finite elements for 3D wind field analysis.
Abstract: An objective representation of an observed meteorological field is obtained by minimizing a quadratic functional that measures both the smoothness (and regularity) of the objective field and the closeness to the observed data. This is a particular form of the general model for numerical variational analysis suggested by Wahba and Wendelberger to generalize the idea introduced by Sasaki. The solution of this minimization problem can be obtained by using homogeneous splines and cross validation or by using the finite element method to determine an approximate solution. Finite elements can provide data compression for large N. Testud and Chong proposed such a method, based on bilinear finite elements for 3-dimensional wind field analysis. We go further with the same principle and study a more regular approximation, which can be analytically differentiated, obtained by bicubic splines. The numerical simulations proposed by Chong and Testud are used to compare the capability of all these methods (spli...
01 Jan 1985
TL;DR: In this paper, a grid generation algorithm using a mapping composed of bicubic B-splines is presented. But it is not shown how to generate a grid when given the boundary description, only the boundary coefficients are chosen so that the splines produce a variation diminishing spline approximation to the boundary.
Abstract: In general, finite difference methods are more successful if the accompanying grid has lines which are smooth and nearly orthogonal. This thesis discusses the development of an algorithm which produces such a grid when given the boundary description.
Topological considerations in structuring the grid generation mapping are discussed. In particular, this thesis examines the concept of the degree of a mapping and how it can be used to determine what requirements are necessary if a mapping is to produce a suitable grid.
The grid generation algorithm uses a mapping composed of bicubic B-splines. Boundary coefficients are chosen so that the splines produce Schoenberg's variation diminishing spline approximation to the boundary. Interior coefficients are initially chosen to give a variation diminishing approximation to the transfinite bilinear interpolant of the function mapping the boundary of the unit square onto the boundary of the grid.
The practicality of optimizing the grid by minimizing a functional involving the Jacobian of the grid generation mapping at each interior grid point and the dot product of vectors tangent to the grid lines is investigated.
Grids generated by using the algorithm are presented.
01 Jan 1985
TL;DR: In many application fields, such as geodesy, cartography, and surveying large surfaces are represented by triangular networks, which must be manipulated and modified frequently as they are generated.
Abstract: In many application fields, such as geodesy, cartography, and surveying large surfaces are represented by triangular networks (3, 11, 12, 17). These networks must be manipulated and modified frequently as they are generated. Afterwards, the graphical presentation of these surfaces should be pleasing to the eye of a critical observer.
TL;DR: A computational method for fitting smoothed natural or periodic bicubic splines to data given at the grid points of a rectangular network is proposed and the defining equations are presented.
Abstract: A computational method for fitting smoothed natural or periodic bicubic splines to data given at the grid points of a rectangular network is proposed. The one-dimensional smoothed spline fit, introduced by Reinsch, defines the smoothness properties well. These are generalized for a two-dimensional approximation by solving the corresponding variational problem. The defining equations are presented here together with an efficient method of determining the necessary parameters and computing the resultant spline.
01 Jan 1985
TL;DR: Parts of this program that allows this program to generate realistic scenes and special effects that are characteristic of B-Splines are described.
Abstract: In our program RODIN, objects are formed through the use of bicubic B-Splines. We will describe aspects of this program that allows us to generate realistic scenes and special effects that are characteristic of B-Splines.
TL;DR: In this paper, a simple, inexpensive and easily integrable circuit is described which reconstructs an analogue signal by linear interpolation between its sample values, which can be used to construct a simple and easily integrated circuit.
Abstract: A simple, inexpensive and easily integrable circuit is described which reconstructs an analogue signal by linear interpolation between its sample values.
01 Jan 1985
TL;DR: In this article, the objective is to construct new Boolean interpolation schemes which have functions of (N-m) independent variables as interpolation parameters with 1£m£N.
Abstract: N-variate blending interpolation involves functions of (N-1) independent variables as data functions in contrast to N-variate product interpolation which depends on scalars only. It is the objective of this paper to construct new Boolean interpolation schemes which have functions of (N-m) independent variables as interpolation parameters with 1£m£N.