Positive data modeling using spline function

Abstract

A rational cubic function with two parameters has been constructed to visualize the positive data. The main focus of the work is the representation of the data in such a way that its view looks smooth and attractive. In the first step simple data dependent constraints are derived on the parameters in the description of the rational cubic function to visualize the shape of positive data then, it is extended to a rational bi-cubic partially blended functions (Coons-patches) and derived constraints on parameters to visualize the shape of positive surface data. The developed scheme is locally positive and economical. The approximation order of rational cubic spline function is $O\left(h_i^3\right)$.

Introduction

One of the significant features of interpolatory methods that make sense to study is its positivity. The goal of the paper is to preserve the inherited feature called, positivity of data. More specifically, it has useful discussion on the problem of visualizing data where underlying constrained must be established. To clarify the idea e.g. for a data which is naturally positive, it shows how the spline which interpolates the data can be constrained to preserve the positivity. Paper presents a new model of spline curve and surface. A rational cubic spline with quadratic denominator has been constructed. Here it is important to mention that spline interpolation is a powerful tool in curve and surface designing. In recent years, the rational spline, particularly the rational cubic spline and its application to shape control, design and preservation has received attention in literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [16], [17].

Shape control [22], shape design [21] and shape preservation [16], [17], [18], [19] are important areas for graphical presentation of data. The problem of shape preservation has been discussed by a number of authors. Brodlie and Butt [3] preserved the shape of convex data by piecewise cubic interpolation. In any interval where convexity was lost, the authors divided the interval into two subintervals by inserting extra knot in that interval. The presented method was C¹. The authors used the same technique in [4] to preserve the shape of positive data. Fuhr and Kallay [7] used a C¹ monotone rational B-spline of degree one to preserve the shape of monotone data. Goodman, Ong and Unsworth [8] presented two interpolating schemes to preserve the shape of data lying on one side of the straight line using rational cubic function. The first scheme preserved the shape of data lying above the straight line by scaling the weights by some scale factor. The second scheme preserves the shape of data by the insertion of a new interpolation point. Goodman [10] surveyed the shape preserving interpolating algorithms for 2D data. Gregory and Sarfraz [11] introduced a rational cubic spline with one tension parameter in each subinterval, both interpolatory and rational B-spline forms. The authors also analyzed the effect of variation of tension parameter on the shape of the curve. Hussain and Sarfraz [13] used rational cubic function in its most generalized form to preserve the shape of positive planar data. The authors used same rational cubic function in [14] to preserve the shape of monotone data. In [13], [14], the authors developed data dependent sufficient conditions on free parameters to preserve the shape of planar data. Lamberti and Manni [16] used cubic Hermite in parametric form to preserve the shape of data. The step length was used as tension parameters to preserve the shape of planar functional data. The first order derivatives at the knots were estimated by a tridiagonal system of equations which assured C² continuity at the knots. Schmidt and He$\beta$ [20] developed sufficient conditions on derivatives at the end points of interval to assure positivity of cubic polynomial over the whole interval.

The theory, in this paper, has a number of advantageous features. It produces C¹ interpolant. No additional points are inserted. Generic interpolatory schemes, although smoother but violates natural characteristics of data. For example, for data in Table 1, the corresponding Fig. 1 may not be wanted by user for the positive data, the user would be ambitious to view it as in Fig. 2. Thus undesirable wiggles which completely deviate the data from its natural features are required to be eliminated. This paper scrutinizes the problem of positive data. Various authors have worked in area of shape preservation. In past few years; several researchers have started using curves and surface of high degree as geometric models or shape descriptors in different model-based-computer vision tasks. Typically speaking, in everyday life many physical situations exist where entities only have sense if their values are positive e.g. in probability distribution, area in the observation of gas discharge etc. It has special usage in image processing, high performance computing, meteorological monitoring, maps, data plots, drawing and many other areas.

The paper has been arranged in a manner so that Section 2 describes the construction of the spline. Section 3 discusses the error of approximation. Constraints for positive curve model have been described in Section 4. The bi-cubic partially blended surfaces have been suggested in Section 5 which is followed by Section 6 devoted to the constraints on the parameters for positive surfaces. The last Section 7 concludes the paper.