Abstract
A biquartic rational interpolation spline surface over rectangular domain is constructed in this paper, which includes the classical bicubic Coons surface as a special case. Sufficient conditions for generating shape preserving interpolation splines for positive or monotonic surface data are deduced. The given numeric experiments show our method can deal with surface construction from positive or monotonic data effectively.
1. Introduction
In most of computer graphics applications and scientific visualization, the construction of shape preserving interpolation spline for positive or monotonic surface data is the essential problem. This problem has been considered by many authors. In [1], the authors gave an algorithm for determining a monotone quadratic spline surface interpolating monotone data over a rectangular grid. In [2], Costantini and Fontanella proposed a method for constructing shape preserving surfaces interpolating arbitrary sets of data on rectangular grids. The surfaces are tensor product splines of arbitrary continuity class. In [3], based on the Boolean sum of cubic interpolating operators, Costantini proposed a local method for the construction of differentiable functions which interpolate a set of gridded data and are monotonicity preserving. In [4], Han and Schumaker derived sufficient conditions on the Bézier net of a Bernstein-Bézier polynomial defined on a triangle in the plane to insure that the corresponding surface is monotone. Then, they applied these conditions to construct a new algorithm for fitting a monotone surface to gridded data. In [5], Hussain and Sarfraz gave simple constraints on the free parameters in the description of rational bicubic spline to preserve the shape of positive surface data and to preserve the shape of the data that lie above a plane. A rational bicubic partially blended patch (Coons patch) to visualize the monotone data in the view of monotone surfaces was developed in [6]. The scheme is economical to compute and visually pleasant. In [7], the authors developed a scheme for visualizing positive data set by using a kind of rational cubic trigonometric function. Recently, in [8], Peng et al. developed a nonnegativity preserving interpolation spline for nonnegative surface data by using a kind of bivariate rational functions.
The purpose of this paper is to present a kind of quartic rational shape preserving interpolation spline for positive and monotone surface data. It has low computation cost and can generate satisfying shape preserving interpolation spline surface. The rest of this paper is organized as follows. We will construct a class of quartic rational interpolation spline and its error bounds in Section 2. Based on it, Section 3 constructs a biquartic rational interpolation spline over a rectangular domain. After that, the sufficient conditions for constructing shape preserving interpolation splines for positive and monotone surface data are deduced in Sections 4 and 5. The numeric experiments given in Section 6 demonstrate that the biquartic rational interpolation splines are effective for visualizing the positive data and monotone data. Conclusions are given in Section 7.
2. Quartic Rational Interpolation Spline and Its Error Bounds
Let , , be data given at the distinct knots , , with interval spacing , and let denote the first derivative values at the knots. The proposed quartic rational interpolation spline is as follows: where , , , , , and the quartic rational basis functions are given by
The spline given in (1) is a Hermite interpolation spline as it satisfies the following interpolatory properties: And it can be easily checked that, for , the quartic rational interpolation spline is exactly the classical cubic Hermite interpolation spline.
Now let us assume that the data , , being interpolated are generated from a function . Since the developed interpolation (1) is local, without loss of generality, we only consider the error of approximation in the subinterval . The absolute interpolation error in the subinterval can be expressed in terms of Peano-Kernel [9] as follows: where is known as Peano-Kernel and is the truncated power function. We have for and for . Therefore, the integral involved in (4) can be expressed as
For the proposed quartic rational interpolation spline (1), we have where and denote the quartic rational basis functions and given in (2), respectively.
In order to compute the integral of absolute values in (5), the roots of and are calculated. It is observed that, for all , . Substituting in (6) and after some simplification, we have Therefore, the roots of in are , , and
Similarly, we have . And it is obvious that the roots of in are also , , and . To compute the roots of , we rewrite it as the following form: Thus, we can get the roots of as follows: where
Moreover, we can easily obtain the following roots of : The above discussion provides the different values of the absolute error as follows.
Case 1. For , the absolute error (4) in is where
Case 2. For , the absolute error (4) in is where
Summarizing the above discussion, we conclude the following theorem.
Theorem 1. For , let be the quartic rational interpolating function of in defined by (1). For the nonnegative parameter , the error of the interpolating function satisfies with
3. Biquartic Rational Interpolation Spline
Let be a given set of data points defined over the rectangular domain , where and . The biquartic rational interpolation spline is defined over each rectangular patch as where with Here, are known as the given data, , are the first derivatives, and are the mixed derivatives. Obviously, for all , the new proposed patch is exactly the classical bicubic Coons patch.
In most applications, the derivative parameters , , and are not given and hence must be determined either from given data or by some other means. These methods are the approximation based on various mathematical theories. We use a common choice as follows: where and . These arithmetic mean methods are computationally economical and suitable for visualization of shaped data.
For convenience, by using the quartic rational basis functions (2), we rewrite the quartic rational interpolation spline given in (20) as follows: where with with with with
4. Positivity Preserving Interpolation
Let be positive data defined over the rectangular domain such that The quartic rational interpolation spline (24) preserves the shape of positive data if
For any , since the quartic rational basis functions given in (2) have the property of nonnegativity, from (24) we can see that a sufficient condition for is
For , , a sufficient condition is
For , , a sufficient condition is
For , a sufficient condition is
For , , a sufficient condition is
Summarizing the above discussion, we can conclude the following theorem.
Theorem 2. The biquartic rational interpolation spline defined in (24) visualizes positive data in the view of positive surface if in each rectangular patch the shape parameters , , , and satisfy the following conditions: where , , , are left to users for interactively controlling the shape of the obtained positivity preserving interpolation spline surface.
5. Monotonicity Preserving Interpolation
Let be a monotone data defined over the rectangular domain . The necessary conditions for monotonicity of the data are The biquartic rational interpolation spline preserves the shape of monotone surface data if
Straightforward computation gives that where with with with Similarly, we have where with with with
From (43) and (50), we can see that a sufficient condition for and is as follows:
For , , a sufficient condition is
For , , a sufficient condition is
In order to give a simple sufficient condition for , , we rewrite the expressions of , , as the following forms:
Thus, we can easily obtain the following sufficient condition for , :
By comparing the expressions of with those of , we can easily obtain sufficient conditions for analogous to those for . We summarize the above discussion as the following theorem.
Theorem 3. The biquartic rational interpolation spline defined in (24) is monotone if the shape parameters , , , and satisfy the following conditions: where , , , are free parameters for interactively adjusting the shape of the monotonicity preserving interpolation spline surface.
6. Numeric Experiments
In this section, we will construct a positivity preserving surface and two monotonicity preserving surfaces from the corresponding data with the biquartic rational interpolation spline and show the surfaces generated by the methods in [5, 6] for comparisons. Our method provides a set of simpler basis functions to represent the positive or monotonic surface data than [5, 6]. In particular, it is worth noting that only four parameters are used in our method, that is, half of those used in [5, 6].
The positive surface data set in Table 1 is obtained from [5]. In Figure 1, the first three rows show the biquartic rational interpolation spline with different shape parameters for the positive surface data set given in Table 1, and the last row shows the surface constructed by the method in [5]. The images on the right column present the -view of the left. From the results, it can be seen that the biquartic interpolation spline describes the positive surface data set as well as [5].
The monotonic surface data sets in Table 2 are derived from [6]. In Figure 2, for visualizing the monotonic surface data set given in Table 2, the first three rows are the biquartic interpolation spline surfaces with different shape parameters, and the last row is the spline surface in [6]. The right column images are the -views of the left images. From the figure, it can be seen that the biquartic interpolation spline visualizes the monotonic surface data set as well as [6].
7. Conclusion
As stated above, the biquartic rational spline surface over rectangular domain includes the classical bicubic Coons surface as a special case. By using the quartic rational basis function, it can be rewritten neatly. Moreover, by selecting suitable parameters on the spline, it can be used to nicely visualize positive or monotonic surface data. Moreover, in our future works, we will concentrate on trying to make use of the modeling technique in practical applications, such as shape preserving surface reconstruction from 3D scattered data.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research is supported by the National Natural Science Foundation of China (nos. 61173119 and 11271376), the Open Project Program of the State Key Lab of CAD&CG (no. A1414) at Zhejiang University, Program for New Century Excellent Talents in University (no. NCET-13-0590), Graduate Students Scientific Research Innovation Project of Hunan Province (no. CX2012B111), and Mathematics and Interdisciplinary Sciences Project, Central South University. Science and technology project of Hunan Province (no. 2014FJ2008).