C¹ Positive Surface over Positive Scattered Data Sites

The aim of this paper is to develop a local positivity preserving scheme when the data amassed from different sources is positioned at sparse points. The proposed algorithm first triangulates the irregular data using Delauny triangulation method, therewith interpolates each boundary and radial curve of the triangle by C¹ rational trigonometric cubic function. Half of the parameters in the description of the interpolant are constrained to keep up the positive shape of data while the remaining half are set free for users’ requirement. Orthogonality of trigonometric function assures much smoother surface as compared to polynomial functions. The proposed scheme can be of great use in areas of surface reconstruction and deformation, signal processing, CAD/CAM design, solving differential equations, and image restoration.


Introduction
Data measured or amassed from many engineering and scientific fields, is often positioned at sparse points. For example, meteorological measurements at different weather stations [1], density measurements on different positions within the human body, heart potential measurements at random points in the diagnosis of various ailments of heart [2], 3D photography, aeronautical engineering and industrial design, structural graph networks [3], graph entropy [4], [5], [6]. A visual model is often required to get a clear understanding of underlying phenomena as colossal amount of data is difficult to analyse or communicate a message in raw form. Further, a meticulous visual representation obligates the interpolating function to affirm intrinsic attributes of data like positivity, monotonicity and convexity. Although, tensor product provides a robust medium for fitting surface to rectilinear data sites, it can not be used to fit a surface over sparse data points. This paper addresses the problem of retaining positivity over scattered data points.
Several approaches have been proposed in literature to address the problem of positivity preserving interpolating surfaces. Amidor [7] surveyed method to interpolate scattered data necessitating from electronic imaging system. The author mainly examined radial basis function method, tetrahedral interpolation, cubic triangular interpolation, triangle based blending interpolation, inverse distance method and neutral neighbourhood. The difference between scattered data interpolation and scattered data fitting was also demonstrated in the survey. Cubic and quintic Hermite interpolants were used for preserving monotonicity, positivity and convexity of discrete data by [8]. Piah, Goodman, Unsworth [9] first triangulated the data points by Delaunay triangulation and constructed the interpolating surface consisting of "cubic Bezier triangular patches". Positivity of data was achieved by imposing sufficient conditions on Bezier ordinates in each triangular patch. The proposed scheme was local and C 1 continous. Hussain and Hussain [10] arranged the scattered data over a triangular grid to preserve the positivity and monotonicity. The authors used a cubic interpolant with one parameter to interpolate the boundary of each triangular patch while linear interpolant was used in Nielson side vertex method to obtain radial curves. Final surface patch was obtained by convex combination of interpolants. Positivity and monotonicity was retained by deriving data dependent constraints on free parameters. C 1 Quadratic splines and Powell-Sabin splines were used as interpolating function to tackle the problem of range restricted univariate and bivariate scattered data by Hermann et. al. [11]. The authors obtained a system of inequalities for the gradients and positivity was accomplished by deriving sufficient conditions on this system. A C 1 local rational cubic Bernstein Bezier interpolatory scheme was proposed by Hussain and Hussain [12] to retain positivity of scattered data. In each triangular patch, inner and boundary Bezier ordinates were confined for positivity. If in any triangular grid, Bezier ordinates failed to attain positive shape of data, then these were varied by the weights described in formation of rational cubic Bernstein Bezier interpolant. Sarfraz et. al. [1] established a local C 1 approach to keep up the positivity of scattered data positioned over a triangular domain. They employed C 1 rational cubic function with four parameters in Nielson side vertex technique to formulate the interpolating surface. Two of the four parameters were constrained for positivity.
Although several approaches have been proposed to retain the positivity of data, little attention has been paid towards the use of trigonometric basis function. This paper develops a positivity preserving scheme for scatter data by taking C 1 rational trigonometric function [13] into account. Delaunay triangulation method has been used to place scatter data as vertices of triangle. Nielson side vertex method [14] has been employed in each triangle to construct triangular patches. The C 1 rational trigonometric cubic function [13] with four parameter has been used for the interpolation along boundary and radial curve of the triangle. Positivity is attained by deriving data dependent condition on half of the parameters in the description of C 1 rational trigonometric cubic function [13].
The remainder of the paper is formulated as: Section 2 reviews the ratonal trigonometric cubic function [13]. Nielson side vertex method [14] to formulate triangular patches is detailed in Section 3. Positivity preserving algorithm is developed and explained in Section 4. Section 5 demonstrates the developed algorithm and presents graphical results. Section 5 summarizes this research and draws conclusion.

Rational Trigonometric Cubic Function
Let {(x i , y i ), i = 0,1,2, . . ., n−1} be the given set of data points defined over the interval [a, b] where a = x 0 < x 1 < x 2 < . . . < x n = b. A piecewise rational trigonometric cubic function is defined over each subinterval The rational trigonometric cubic function (Eq 1) satisfy the following properties: d i and d i+1 are derivative at the endpoints of the interval The following result has been proved in [13]. Theorem 2.1 The C 1 piecewise rational trigonometric cubic function preserve the positivity of positive data if in each subinterval I i = [x i ,x i+1 ], the parameters β i and γ i satisfy the following sufficient conditions

Nielson Side Vertex Method
Consider a triangle 4V 1 V 2 V 3 with vertices V 1 , V 2 , V 3 having edges e 1 , e 2 , e 3 and u, v, w be the barycentric coordinates such that any point V on the triangle can be written as: where u þ v þ w ¼ 1 and u; v; w ! 0: The interpolant defined by Nielson [14] to generate surface over each triangular patch is defined as the following convex combination: where Q i 0 s represent line segments joining vertices V 0 i s to points S 0 i s on the opposite boundary. Eq (4) interpolates data at the vertices as well as first order derivatives at the boundary. Since the barycentric coordinates at the vertices of triangle is simultaneously zero, the interpolant Eq (4) takes the following values: where Q i , i = 1,2,3 are the ordinate values at the vertices V i , i = 1,2,3 of triangle.

Positive Scatter Data Interpolation
This section details the derivation of sufficient conditions for C 1 triangular patches to be positive. Let the given positive scattered data set arranged over a triangular domain be {(x i , y i , F i ), i = 1,2, . . ., n}. The resulting surface S(x, y) described as

Domain Triangulation
Triangulation of data is performed by Delaunay triangulation method such that data F i , fall on vertices {V i = (x i , y i ), i = 1,2,3, . . ., n} of the triangles.

Conclusion
In this study, positivity preserving algorithm for scattered data arranged over a triangular domain, is established. The rational trigonometric cubic function [13] with four free parameters is used for the interpolation along each boundary and radial curve. Nielson side vertex has been applied to construct the interpolating surface. Constraints on half of the parameters are obtained to guarantee the positive shape of data while half are set free for users modification. The proposed algorithm, surpasses many prevailing approaches in literature. In [10], authors  Table 1.      Table 2.
doi:10.1371/journal.pone.0120658.g009 utilized a cubic function with one free parameter to retain the positive shape of data. Positive surface was obtained by drawing data dependent constraints on this free parameter, and, hence the scheme did not offer refinement in the shape. The scheme suggested in this paper does not suffer this detriment. The developed algorithm is local and can be applied to data with or without derivatives. Moreover, shape preserving algorithms play an instrumental role in many areas of visualization such as geometric modelling, robot trajectories, evolution game theory, prisoner's dilemma game [16], [17], [18], meshless method and inverse kinemaics etc.