Shape Preserving Spline Interpolation

A rational spline solution to the problem of shape preserving interpolation is discussed. The rational spline is represented in terms of first derivative values at the knots and provides an alternative to the spline-under-tension. The idea of making the shape control parameters dependent on the first derivative unknowns is then explored. The monotonic or convex shape of the interpolation data can then be preserved automatically through the solution of the resulting non-linear consistency equations of the spline

Data visualization using rational spline interpolation

AbstractA smooth curve interpolation scheme for positive, monotonic, and convex data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The rational spline scheme has a unique representation. The degree of smoothness attained is C1

Shape-preserving algorithms for curve and surface design

PhD ThesisThis thesis investigates, develops and implements algorithms for shape- preserving curve and surface design that aim to reflect the shape characteristics of the underlying geometry by achieving a visually pleasing interpolant to a set of data points in one or two dimensions. All considered algorithms are local and useful in computer graphics applications. The thesis begins with an introduction to existing methods which attempt to solve the shape-preserving 1 curve interpolation problem using C cubic and quadratic splines. Next, a new generalized slope estimation method involving a parameter t, which is used to control the size of the estimated slope and, in turn, produces a more visually pleasing shape of the resulting curve, is proposed. Based on this slope generation formula, new automatic and interactive algorithms for constructing 1 shape-preserving curves from C quadratic and cubic splines are developed and demonstrated on a number of data sets. The results of these numerical experiments are also presented. Finally, a method suggested by Roulier which 1 generates C surfaces interpolating arbitrary sets of convex data on rectangular grids is considered in detail and modified to achieve more visually pleasing surfaces. Some numerical examples are given to demonstrate the performance of the method.Ministry of Education, Government of Pakista

Splainidega lahendamine

Dissertatsioonis on käsitletud kolme probleemi splainide teooriast. Esiteks vaadeldakse kuupsplainidega histopoleerimist. Antud on suvaliselt paiknevate sõlmedega ja suvaliste kõrgustega histogramm. Näidatakse, et alati on olemas histopoleeriv kuupsplain üldka-su¬tatavate rajatingimuste korral. Splaini leidmisel kasutatakse erinevaid esitusi. Teiseks uuritakse suvalise võrgu korral histopoleeriva polünomiaalse perioodilise splaini ole¬mas-olu. Saadud tulemustest järelduvad kirjandusest varem teadaolevad tulemused ühtla¬se võrgu korral. Kolmas probleem, mida dissertatsioonis lahendatakse, on ruut/lineaar ratsionaalsplainidega histopoleerimine. Taolised splainid säilitavad lähteandmete kume-ru¬se, sest nad on ise alati selle omadusega. Et vabadus on siin splaini sõlmede valikus, on loomulik küsida, kas suvalise rangelt kumera histogrammi korral on võimalik valida splaini sõlmed nii, et eksisteeriks histopoleeriv ruut/lineaar ratsionaalsplain. Vastus on siin eitav. Uurimisel leitakse splaini parameetreid määrav baasvõrrandite süsteem ning splaini eksisteerimine on samaväärne selle mittelineaarse süsteemi lahendi olemasoluga. Probleemi lahendamiseks on leitud splaini sobiv esitus.The dissertation treats three kinds of problems from the theory of splines. Firstly, a par¬ti-cular interpolation problem about the cubic spline histopolation with arbitrary place¬ment of histogram knots and spline knots between them is discussed. A cubic spline is studied provided that its integral on a prescribed interval equals the area of the corres¬pon¬ding histogram rectangle. It is considered the most common boundary value con¬di¬tions like given values of the spline and its first and second derivatives in endpoints of given interval and then solved the problem of existence and uniqueness of the solution for such histopolation problem. Secondly, the periodic polynomial spline histopolation problem with the arbitrary placement of histogram knots and coinciding histogram knots is considered. Several results about the existence and uniqueness of solution are obtained and they imply known results in the case of uniform grid. In the last problem, the rational spline histopolation of convex data is studied. For the concern about the con¬vexity, an appropriate tool is interpolation or histopolation with quadratic/linear rational splines because these splines keep the sign of its second derivative on the whole interval. For this reason the given histogram is assumed to be strictly convex. The main task is at the study of existence of solution for a nonlinear system of basic equations to determine the values of second derivatives in spline knots. The other parameters in the representation of spline are determined from a linear system with regular matrix. It is shown that there is a strictly convex histogram without the solution of histopolation problem for any choice of spline knots.https://www.ester.ee/record=b524283

Shape preserving piecewise rational interpolation

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.For abstract see full text