Visualization of positive and convex data by a rational cubic spline interpolation

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Abstract

A curve interpolation scheme for the visualization of scientific data has been developed. This scheme uses piecewise rational cubic functions and is meant for positive and convex data. 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. In addition to preserve the shape of positive and/or convex data sets, it also possesses extra features to modify the shape of the design curve as and when desired. The degree of smoothness attained is C1.

Introduction

Smooth curve representation, to visualize the scientific data, is of great significance in the area of information sciences, computer graphics and in particular data visualization. Particularly, when the data are obtained from some complex function or from some scientific phenomena, it becomes crucial to incorporate the inherited features of the data and to gain the hidden information inside. Moreover, smoothness is also one of the very important requirements for pleasing visual display. Ordinary spline schemes, although smoother, are not helpful for the interpolation of the shaped data. Extremely misguided results, violating the inherited features of the data, can be seen when undesired oscillations occur. Thus, unwanted oscillations, which completely destroy the data features, are needed to be controlled.

This paper examines the problem of shape preservation of positive as well as convex data sets. Various authors have worked in the area of shape preservation [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. In this paper, the shape preserving interpolation has been discussed for positive and convex data, using rational cubic spline. The motivation to this work is due to the past work of many authors, e.g. quadratic interpolation methodology has been adopted in [1], [14] for the shape preserving curves. Fritsch and Carlson [3] and Fritsch and Butland [5] have discussed the piecewise cubic interpolation to convex data. Also, Passow and Roulier [2] considered the piecewise polynomial interpolation to monotonic and convex data. In particular, an algorithm for a quadratic spline interpolation is given by McAllister and Roulier [1]. An alternative to the use of polynomials, for the interpolation of monotonic and convex data, is the application of piecewise rational quadratic and cubic functions by Gregory [4]. Rational functions have been discussed by Sarfraz [9] in a parametric context.

The theory of methods, in this paper, has number of advantageous features. It produces C1 interpolant. No additional points (knots) are needed. In contrast, the quadratic spline methods of Schumaker [6] and the cubic interpolation method of Brodlie and Butt [7] require the introduction of additional knots when used as shape preserving methods. The interpolant is not concerned with an arbitrary degree. It is a rational cubic with a cubic numerator and a cubic denominator. The rational spline curve representation is bounded and unique in its solution.

The paper begins with a definition of the rational function in Section 2. The description of the rational cubic spline, which does not preserve the given shape of a data, is made in this section. Although this rational spline was discussed in [15], it was in the parametric context which was useful for the designing applications in Computer Aided Design (CAD). A scalar version of the same spline method was discussed in [16] to preserve the shape of a monotone data. The positivity problem is discussed in Section 3 for the generation of a C1 spline which can preserve the shape of a positive data. The sufficient constraints, on the shape parameters, have been derived to preserve and control the positive interpolant. The convexity problem is discussed in Section 4 for the generation of a C1 spline which can preserve the shape of a convex data. The sufficient constraints, in this section, lead to a convex spline solution. Section 5 concludes the paper.

Section snippets

Rational cubic spline with shape control

Let (xi,fi),i=1,2,…,n, be a given set of data points, where x1<x2<⋯<xn. Lethi=xi+1−xi,Δi=fi+1−fihi,i=1,2,…,n−1.Consider the following piecewise rational cubic function:s(x)≡si(x)=Ui(1−θ)3+viViθ(1−θ)2+wiWiθ2(1−θ)+Ziθ3(1−θ)3+viθ(1−θ)2+wiθ2(1−θ)+θ3,whereθ=x−xihi.To make the rational function (2) be C1, one needs to impose the following interpolatory properties:s(xi)=fi,s(xi+1)=fi+1,s(1)(xi)=di,s(1)(xi+1)=di+1,which provide the following manipulations:Ui=fi,Zi=fi+1,Vi=fi+hidivi,Wi=fi+1hidi+1wi,

Positive spline interpolation

The rational spline method, described in the previous section, has deficiencies as far as positivity preserving issue is concerned. For example, the rational cubic in Section 2 does not always preserve the shape of the positive data (see Fig. 1). It is required to assign appropriate values to the shape parameters so that it generates a data preserved shape. Thus it looks as if ordinary spline schemes do not provide the desired shape features and hence some further treatment is required to

Convex spline interpolation

The rational cubic, in Section 2, does not preserve the shape of the convex data (see Fig. 6). Thus, it looks as if ordinary spline schemes do not provide the desired shape features and hence some further treatment is required to achieve a shape preserving spline for convex data. This requires an automated computation of suitable shape parameters and derivative values. To proceed with this strategy, some mathematical treatment is required which will be explained in the following paragraphs.

For

Concluding remarks

A rational cubic interpolant, with two families of shape parameters, has been utilized to obtain C1 positivity and/or convexity preserving interpolatory spline curves. The shape constraints are restricted on shape parameters to assure the shape preservation of the data. For the C1 interpolant, the choices on the derivative parameters have been defined. The solution to the shape preserving spline exists and provides a unique solution.

The rational spline scheme has been implemented successfully

Acknowledgements

The author acknowledges the support of King Fahd University of Petroleum and Minerals in the development of this work. The author also likes to thank for the anonymous referee’s valuable comments in the improvement of this manuscript.

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