Abstract
This paper discusses the positivity preserving interpolation for positive surfaces data by extending the C1 rational cubic spline interpolant of Karim and Kong to the bivariate cases. The partially blended rational bicubic spline has 12 parameters in the descriptions where 8 of them are free parameters. The sufficient conditions for the positivity are derived on every four boundary curves network on the rectangular patch. Numerical comparison with existing schemes also has been done in detail. Based on Root Mean Square Error (RMSE), our partially blended rational bicubic spline is on a par with the established methods.
1. Introduction
Shape preserving interpolation is a process that involves some mathematical derivation to visualize the two-dimensional (2D) or three-dimensional (3D) data sets in the form of piecewise curves or surfaces that satisfies some degree of smoothness. Notably the shape preserving method will be able to maintain the original shape of the data sets. For example, if the given data is positive, then the final resulting curves or surface will be positive everywhere and it should be visual pleasing for the computer graphics displays. Usually the collected data from some laboratory experiment may contain little unwanted noise. Common cubic spline interpolation is not able to produce the shape preserving interpolation for positive, monotone and convexity of the data sets. Even though the interpolating curves are very smooth, that is, with continuity, there may exist some uncharacteristic behavior such as wiggle that will result in the negative values along the whole intervals. This unwanted flaw must be eliminated before the data are processed by the user. Therefore for the shape preserving interpolation and practical designing purposes, cubic spline and cubic Hermite spline are not recommended. Thus many researchers have proposed the idea of how to preserve the positive data by using some rational spline function with or continuity.
Some literature studies for positivity preserving are as follows. Abbas et al. [1, 2] have discussed the positivity preserving by using partially blended rational bicubic spline function and rational bicubic spline function, respectively. Asim and Brodlie [3], Butt and Brodlie [4], and Brodlie et al. [5] discussed the use of bicubic spline for positivity preserving by inserting one or two extra knots in the region or interval where the negativity of the interpolant is found. Asim et al. [6] have studied the use of modified quadratic Shepard method for positivity preserving of large data sets. Duan et al. [7] have proposed new bivariate rational interpolation based on function values. They utilized the rational cubic spline of the forms of cubic numerator and linear denominator. Meanwhile, Duan et al. [8, 9] investigated the bounded property, point control, and shape control of the bivariate rational cubic spline with linear denominator. Goodman et al. [10] discussed the constrained interpolation by using rational cubic with cubic denominator. Goodman [11] summarized all shape preserving criteria for the univariate cases. Hussain and Sarfraz [12] and M. Z. Hussain and M. Hussain [13] also studied the positivity preserving by utilizing the rational cubic spline of the forms cubic/cubic and cubic/quadratic, respectively. In the study by M. Z. Hussain and M. Hussain [13], there are no free parameters for the refining processes. M. Z. Hussain and M. Hussain [14] have developed the surface construction scheme for positivity preserving of positive scattered data arranged over triangular grids. They utilized the rational cubic spline of Tian et al. [15] without any free parameters. Hussain et al. [16] have developed shape preserving surfaces for positive and convex data using rational biquadratic splines. The degree of smoothness attained is as can clearly be seen from their graphical results. Ibraheem et al. [17] develop a positivity preserving scheme by using trigonometric rational functions which will increase the computation times. Karim and Kong [18] have proposed new rational cubic spline of the form cubic/quadratic with three parameters for local control of interpolating function. Positivity preserving by using this rational cubic spline has been addressed in the study by Karim and Kong [19]. Liu et al. [20] have studied the positivity and monotonicity preserving by using new rational biquartic spline by utilizing the extension cubic Ball basis function with two parameters. Saaban et al. [21] studied the positivity preserving by using quintic triangular Bezier patches. The surfaces are constructed by using convex combination of quintic triangular Bezier patches. They use the real data collected from the rainfall at various stations in West Peninsular of Malaysia. Sarfraz et al. [22] also have developed rational cubic spline scheme for positivity preserving of 3D data. Their scheme has 8 parameters without any free parameter. Furthermore Abbas et al. [1] indicated that Sarfraz et al. [22] works do not produce positive surfaces. The surfaces may only have continuity. Bastian-Walther and Schmidt [23] discussed the range restricted data interpolation for scattered data by using rational cubic spline of Gregory [24]. Wu et al. [25] studied the positivity preserving data approximation by using compactly supported radial basis functions (CSRBFs). Their scheme requires the solving of optimization problems in order to calculate the parameters that will guarantee that the positivity of the data is preserved. Peng et al. [26] have used the rational cubic spline of Hussain and Sarfraz [12] to construct the positive rational cubic Bezier interpolant for scattered positive data. Their methods require the modification of the first derivatives if the positivity of the rational interpolant is not met. The bicubic Hermite spline [27] interpolation cannot preserve the positivity of the surface data sets, since there maybe exist some negative values on the constructed surfaces. This paper is a continuation of our previous works. The univariate rational cubic spline interpolation in the studies by Karim and Kong [18, 19] is extended to the bivariate cases. Positivity preserving interpolation is our main object in this paper. We compare the performance between our rational bicubic spline and that of the work of Abbas et al. [1] by using Root Mean Square Error (RMSE).
We identify several advantages of our proposed partially blended rational bicubic spline interpolation for positive data as follows.(i)Our schemes have 12 parameters and 8 of them are free parameters; meanwhile, in the study by M. Z. Hussain and M. Hussain [13] the bivariate spline has 8 parameters and none of them are free parameters.(ii)Our schemes are applicable for both equally or unequally spaced data; meanwhile, the rational cubic spline of Duan et al. [7–9] requires that the data be equally spaced.(iii)The scheme can also be used whether the first derivative is given or not. Meanwhile, the works of Duan et al. [7, 9] require the true function values without the first derivative values.(iv)Our rational bicubic spline is on a par with the works of Abbas et al. [1, 2] and Hussain and Sarfraz [12]. Furthermore the presented scheme is easy to use and not involving any trigonometric functions compared with the work of Ibraheem et al. [17].(v)When , our partially blended rational bicubic spline is reduced to the work of M. Z. Hussain and M. Hussain [13].(vi)The partially blended rational bicubic spline is unique and the free parameters in its description provide the flexibility in controlling the final shape of the positive surface data sets.(vii)Furthermore based on Root Mean Square Error (RMSE) our partially blended rational bicubic spline gives numerical results as good as the numerical results from the study by Abbas et al. [1].The remainder of the paper is organized as follows. Section 2 is devoted to the derivative estimation for 2D and 3D data by using Arithmetic Mean Method (AMM). Meanwhile, Section 3 reviews the univariate of rational cubic spline initiated by Karim and Kong [18, 19]. Positivity preserving for 2D data also is discussed in this section. The partially bicubic rational cubic spline interpolation is discussed in Section 4. Section 5 discusses the positivity preserving for 3D positive data sets by using the constructed partially bicubic rational cubic spline interpolation with numerical demonstrations with comparison to the existing schemes. Finally a summary and conclusions are given in Section 6.
2. Derivative Estimation
The first derivative that depends on the original data can be chosen by using Arithmetic Mean Method (AMM). The details are as follows:
For 2D Data
At the interior points, , , the values of are given as
For 3D data where
3. Review of Rational Cubic Spline Interpolant
In this section the rational cubic spline of Karim and Kong [18, 19] is discussed briefly before it extends to the bivariate cases.
Given the set of data points such that , let , , and , where . For , ,whereThe rational function in (5) satisfies continuity as follows:It can be shown that the unknowns , , are given as follows: where denotes derivative with respect to and denotes the derivative value which is given at the knot , . The parameters are chosen such that , . When , , the rational cubic interpolant in (5) is reduced to cubic Hermite spline [19].
Positivity Preserving. For the strictly positive set of data , , ,Karim and Kong [19] have developed the following results for strictly positive data sets given in (9).
Theorem 1. For strictly positive data defined in (9), the rational cubic interpolant defined over the interval is positive if in each subinterval , , the involving parameters , , and satisfy the following sufficient conditions:
3.1. Numerical Demonstrations
In this section the positivity shape preserving using rational cubic spline of Karim and Kong [19] is discussed. The positive data is taken from the study by Hussain et al. [28].
Example 2. See Tables 1 and 2 and Figure 1.
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4. Rational Bicubic Spline Interpolant
The piecewise rational cubic spline in (5) is extended to bicubic partially blended rational function over rectangular domain . The partition of the intervals and is defined as and . The partially blended rational bicubic function over each rectangular patch , and , is defined as follows: where where, , , and are rational cubic functions (5) defined on the boundary of rectangular patch , , , and are given as follows: With , , , , and ;with , , , , and ;with , , , , and ;with , , , , and .
Theorem 3. The partially blended rational bicubic spline defined by (11) is -continuous degree-seven piecewise rational surface.
Proof. The results trivially follow from Proposition 3 in the study by Casciola and Romani [29].
5. Positivity Preserving of 3D Positive Data
Let be positive data defined over rectangular grid , , , such that Casciola and Romani [29] have developed the following results: bicubic partially blended surface patch inherits all the properties of network of boundary curves.Thus the partially bicubic surface patch in (21) is positive if boundary curves defined in (22)–(25) are positive; that is, , , , and , respectively. By simple algebraic manipulation we have the following theorem.
Theorem 4. The piecewise rational partially bicubic function in (21) defined over the rectangular mesh , , , preserves the positive surface data if the parameters satisfy the following sufficient conditions:
Proof. From (14)–(17), , if , . Now for , , and , , if Combining the conditions in (20) results in the following sufficient conditions for the positivity of :Likewise the remaining three boundary curves , , and are positive if the following sufficient conditions are satisfied:Combining the conditions in (22) and (23), (24) and (25), and (26) and (27) leads us to the following:Then from (21) and (28), sufficient conditions for the positivity of network of all four curves , , , and are obtained. Thus the proof is complete.
The sufficient conditions in (19) can be rewritten aswhere , , , and .
Theorem 5. The partially blended rational bicubic spline constructed by using shape parameters calculated from sufficient condition for positivity in (29) is -continuous degree-seven piecewise rational surface with positivity preserving properties to the positive data sets.
Proof. The proof also trivially follows from the work of Casciola and Romani [29].
5.1. Algorithm for Computer Implementations
The algorithm is for the purpose of computer implementation of positivity preserving by using our partially rational bicubic spline presented in Theorem 4.
Input. The input is the positive data points , and .
Output. The output is the positive surfaces.
Step 1. Calculate the first derivative values , , by using AMM.
Step 2. Choose suitable values for free parameters , , , , , , , and , and compute the values of parameters , , , by using Theorem 4.
Step 3. Construct the positive surfaces by substituting all the required parameters into the partially blended rational bicubic spline function (11).
Repeat Steps 2 and 3 to generate different positive surfaces through different positive data sets.
5.2. Numerical Demonstrations
In this section, numerical examples for positivity preserving by using partially blended rational bicubic spline interpolation of positive surface data will be discussed in detail. Four positive data sets are taken from the studies by Abbas et al. [1, 2] and Hussain and Sarfraz [12], respectively.
Example 6. Positive data from the following function is truncated to five decimal places [1]: Figure 2 shows the default bicubic Hermite spline for the positive data given in Table 3. Figures 3(a) and 3(b) show the -view and -view for Figure 2, respectively. Figure 4 shows the positivity preserving by using the proposed rational bicubic spline with , . Meanwhile, Figures 5(a) and 5(b) show the -view and -view for Figure 4, respectively.
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Example 7. Positive data from the following function is truncated to five decimal places [1]: Figure 6 shows the default bicubic Hermite spline for the positive data given in Table 4. Figures 7(a) and 7(b) show the -view and -view for Figure 6, respectively. Figure 8 shows the positivity preserving by using the proposed rational bicubic spline with , . Meanwhile, Figures 9(a) and 9(b) show the -view and -view for Figure 8, respectively.
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Example 8. Positive data from the following function is truncated to four decimal places [2]: Figure 10 shows the default bicubic Hermite spline for the positive data given in Table 5. Figures 11(a) and 11(b) show the -view and -view for Figure 10, respectively. Figure 12 shows the positivity preserving by using the proposed rational bicubic spline with , . Meanwhile, Figures 13(a) and 13(b) show the -view and -view for Figure 12, respectively.
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Example 9. Positive data from the following function is truncated to four decimal places [12]: Figure 14 shows the default bicubic Hermite spline for the positive data given in Table 6. Figures 15(a) and 15(b) show the -view and -view for Figure 14, respectively. Figure 16 shows the positivity preserving by using the proposed rational bicubic spline with , . Meanwhile, Figures 17(a) and 17(b) show the -view and -view for Figure 16, respectively.
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Meanwhile, Figures 18(a) and 18(b) show the positivity preserving by using the study by Abbas et al. [1] for positive data in Tables 3 and 4, respectively. Figure 19 shows the positivity preserving by using the study by Abbas et al. [2] for positive data in Table 5. Finally Figures 20 and 21 show the positive surfaces by using Hussain and Sarfraz [12] rational bicubic spline and Liu et al. [20] rational biquartic spline surfaces.
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We compare the performance between our proposed partially blended rational bicubic spline interpolation and that of the work of Abbas et al. [1] by calculating Root Mean Square Error (RMSE). Table 7 summarized the results of RMSE estimation. It can be seen clearly that our rational bicubic spline interpolation is on a par with the work of Abbas et al. [1]. The value for RMSE for all four data sets is almost the same. Thus we conclude that the proposed partially blended rational bicubic spline interpolation produces the positive surfaces through positive data sets with smaller RMSE. Furthermore we believe that the computation is less by using our rational bicubic spline interpolation compared with the other rational bicubic spline interpolation of Hussain and Sarfraz [12] and Sarfraz et al. [22].
6. Discussions
The proposed partially blended rational bicubic spline with 12 parameters has been used for positivity preserving of 3D positive data. The scheme has been tested on 4 different data sets. The bicubic Hermite spline does not have the capability to preserve the positivity of the data without changing the original data points. Figures 2, 6, 10, and 14 show that there exist some negative values when the data sets are interpolated by using bicubic Hermite spline. From Figures 4, 8, 12, and 16 it can be seen clearly that the proposed scheme gives comparable results with existing schemes such as those of Abbas et al. [1, 2], Hussain and Sarfraz [12], and Liu et al. [20]. Similar to the work of Abbas et al. [1], our scheme also is partially blended bicubic rational spline. Meanwhile, Abbas et al. [2] and Hussain and Sarfraz [12] have used rational bicubic spline interpolation. Table 7 summarized the RMSE values comparison between the proposed partially blended rational bicubic spline interpolation and that of the work of Abbas et al. [1].
7. Summary and Conclusions
In this paper the rational cubic spline of Karim and Kong [18, 19] has been extended to the bivariate cases. The partially blended bicubic rational spline has 12 parameters in the descriptions and 8 of them are free parameters. These free parameters can be used to change the shape of the final surfaces of the positive data. Graphical comparison between the proposed schemes and existing schemes also has been done. From all numerical results, we concluded that the proposed partially blended bicubic rational spline works well for all tested data sets. Future works will be developing the rational bicubic spline for scattered data interpolation without the need to modify the first derivative. This will improve the work of Peng et al. [26] that requires the modification of the first derivative in which positivity preserving is not met. Work on monotonicity and convexity preservation by using the proposed partially blended rational bicubic spline interpolation is underway.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The first author would like to acknowledge Universiti Teknologi PETRONAS (UTP) for the financial support received in the form of a research grant: Short Term Internal Research Funding (STIRF) no. 35/2012 including the Mathematica Software.