Piecewise Bivariate Hermite Interpolations for Large Sets of Scattered Data

The requirements for interpolation of scattered data are high accuracy and high efficiency. In this paper, a piecewise bivariate Hermite interpolant satisfying these requirements is proposed. We firstly construct a triangulation mesh using the given scattered point set. Based on this mesh, the computational point (í µí±¥, í µí±¦) is divided into two types: interior point and exterior point. The value of Hermite interpolation polynomial on a triangle will be used as the approximate value if point (í µí±¥, í µí±¦) is an interior point, while the value of a Hermite interpolation function with the form of weighted combination will be used if it is an exterior point. Hermite interpolation needs the first-order derivatives of the interpolated function which is not directly given in scatted data, so this paper also gives the approximate derivative at every scatted point using local radial basis function interpolation. And numerical tests indicate that the proposed piecewise bivariate Hermite interpolations are economic and have good approximation capacity.


Introduction
The approximation to higher dimensional scattered data is one of the hot and difficult problems in approximation theory field.Due to the characteristics of scattered data, such as the large amount, irregularity, and high dimensionality, it is difficult to construct the approximation methods for them.However, the approximation method to scattered data has been widely applied in many fields, that is, estimating the parameter of input nonlinear system [1][2][3], solving partial differential equations, surface reconstruction in reverse engineering, data visualization, and so on.The most frequently-used approximation methods include interpolation by spline, interpolation by radial basis function, and the least square approximation.The interpolations by spline have two types, which are global interpolation and local interpolation, respectively.The global spline interpolation [4] is not able to deal with large scale of scattered data, while the local spline interpolation [5][6][7] needs smooth joining between piece and piece.The radial basis function interpolation [8,9] requires solving a large scale of linear system, and the least square approximation [10,11] also requires solving a certain scale of linear system.This is a problem that doesnot easily work out in the computation.It is well known that Hermite interpolation has higher approximation accuracy than interpolation that only interpolates function values.This is because it interpolates not only the function value, but also the derivative value.However, how to construct multivariate Hermite interpolation for large sets of scattered data is an important research consideration which is also a difficult task.At present, some results have been published [12][13][14]; however, these methods all require solving a certain scale of linear system.Therefore, they are not suitable for approximating to large scale of scattered data.
This paper proposes two piecewise bivariate Hermite interpolation methods for large sets of scattered data.One of them uses exact derivative in Hermite interpolation, and the other one does the approximate derivative.Both methods are economic, free us from solving any linear systems, and have better approximation capacity.Therefore, they are especially suitable for the approximation to large sets of scattered data.These interpolation methods firstly use Delaunay triangulation method [15] to make a triangulation mesh based on the given scattered point set and then divide the computational point (, ) into two types of interior point and exterior outer point.If point (, ) is an interior point, then the value of the Hermite interpolation function on the triangle which point (, ) lies on is regarded as the approximate value of the approximated function, while if point (, ) is an exterior point, then the value of a Hermite interpolation with a form of weighted combination is regarded as the approximate value.Hermite interpolation needs the first-order derivatives of the interpolated function; however, the derivative information is not directly given at scatted data, so this paper gives the approximate derivative at every scatted point by using local radial basis function interpolation.
The structure of this paper is organised as follows.The Hermite interpolation on triangle is presented in Section 2. The construction of piecewise bivariate Hermite interpolation is presented in Section 3. The estimation of partial derivative at every scattered point by local radial basis function interpolation is presented in Section 4. Some numerical tests and comparisons between constructed schemes and other interpolation schemes in accuracy and efficiency are carried out in Section 5.This paper ends with a section of brief conclusion.

Hermite Interpolation Based on Triangle
Given the values and partial derivatives of a bivariate function (, ) at three vertices (  ,   ),  = 1, 2, 3 of a triangle  123 (their vertices 1, 2, 3 are arranged counter to clockwise), that is,   , (  )  , (  )  ,  = 1, 2, 3 (see Figure 1).A bivariate Hermite interpolation polynomial  123 (, ) is to be constructed satisfying the following interpolation conditions: In order to infer the expression of Hermite interpolation from interpolation conditions (1), the barycenter coordinate of a triangle needs to be introduced first.It is assumed that (, ) is a point on triangle  123 and its barycentric coordinates is denoted as ( 1 ,  2 ,  3 ).Then the relation between its cartesian coordinates and barycentric coordinates is where  is the area of the triangle  123 and Making use of transformation (2), the interpolation conditions (1) can be changed into the following form under barycentric coordinates: A complete bivariate polynomial of three degrees  3 (, ) = ∑ 3 +=0       has ten undetermined coefficients.Under barycentric coordinate its expression is Journal of Applied Mathematics The exhibition of interior and exterior points.
Since the Hermite interpolation conditions (1) or (4) only provide 9 conditions, we construct a Hermite interpolation polynomial with nine coefficients which meets interpolation condition (4).Then the values of the coefficients in formula ( 6) can be obtained: Combining ( 4), (6), and ( 7), the Hermite interpolation polynomial satisfying (4) can be written as where

Bivariate Piecewise Hermite Interpolation for Large Sets of Scattered Data
Given a set of scattered data {(  ,   ,   )}  =1 ⊂  3 , which are assumed to be sampled from a function(, ),(, ) ∈ Ω.The partial derivatives of (, ) at the set of points {(  ,   )}  =1 ⊂ Ω ⊂  2 are given as (  )  , (  )  ,  = 1, . . ., .Taking the  scattered points as nodes, a triangulation mesh, , is constructed in domain Ω using Delaunay triangulation method.If point (, ) ∈ Ω ⊂  2 lies on a triangle of , it is called as an interior point, otherwise, an exterior point.For example, points  1 ,  2 in Figure 2 are interior points, while point  3 is an exterior point.Next we will give the approximate value   (, ) of (, ) according to the category of point (, ).

Exterior Point.
When point (, ) doesnot belong to any triangle within , we construct the following form of Hermite interpolation: satisfying interpolation conditions In this paper, we take  , (, ) as a Hermite interpolation polynomial with the same form of (8) on triangle   with the least area among all triangles with vertex (  ,   ), satisfying interpolation conditions (10). , (, ) is called as node basis function and function   (, ) is called as weighted function.In order to satisfy interpolation conditions (12), these weighted functions are required for satisfying conditions In the paper we take the weighted functions in (11) as those in the literature [16]: where The value of   in (17) can be selected by user properly.It can be seen that when (, ) → (  ,   ), (1/ 2  ) → + ∝, thereby   (, ) → 1; when (, ) → (  ,   ),  ̸ =  and (1/ 2  ) → + ∝, thereby   (, ) → 0. Thus   satisfies condition (13).  in formula (16) satisfies condition (14) obviously, and the proof of which can be found in literature [16].If 1/  is determined by (17), then when   (, ) >   ,   (, ) = 0.So in case of   being appropriately selected (  is also called as influence radius of node basis function), there are only several nonzero terms in sum ∑  =1   (, ).Then it can be found that interpolation function (11) which takes functions ( 16) and ( 17) as its weight functions is a local Hermite interpolation.When applying (11), we can use different influence radius at every node, so we can use the same radius .The following computational formulation of uniform influence radius  in (18) is from [16]: is the number of points used inside a circle of radius .where   is a triangle within  with node (  ,   ), (  ,   ), (  ,   ) as its vertices and   is a Hermite interpolation polynomial on triangle   .We now summarize the description of the constructed interpolation function, written as Algorithm 1.

Bivariate
Algorithm 1.Consider the following.
Step 1. Generate a triangle mesh T in domain Ω using Delaunay triangulation method based on given scattered point set {(  ,   )}  =1 ⊂ Ω.
Step 2. Judge the category of point (, ): interior point or exterior point.
Step 3. If point (, ) is an interior point, then find out the triangle   which includes point (, ), compute the value of the interpolation function   at point (, ) using formulas (2), (8), and (10), and take the value as the value of   (, ). Step

The Estimation of Partial Derivative
In the process of using bivariate piecewise Hermite interpolation   , it demands the first-order derivatives of the interpolated function (, ) at every point, but the scattered data {(  ,   ,   )}  =1 doesnot provide the derivatives.Thus we use the first-order derivative of a local radial basis interpolation to approximate the derivative of the interpolated function (, ) and replace this exact derivative in (19) with the approximate derivative.The Delaunay triangulation  is a convex subdomain of Ω.Every scattered point   = (  ,   ) is either a boundary point or an interior node point of .We put the vertices of all triangles with point (  ,   ) as into a set, written as   = (  ,   ) ∪ {(   ,    )}  =1 , see Figure 3.We take the point set   as local radial basis function interpolation point set and multiquadric radial basis functions . . .
The bivariate piecewise interpolation function   using the previous approximate derivatives is written as Algorithm 2 summarizes the computational process of interpolation Q (, ).
Algorithm 2. Consider the following.
Step 3. Judge the category of point (, ): interior point or exterior point.value of the interpolation function   at point (, ) using formulas ( 2), (8), and (10), and assign the value to the function Q (, ).

Numerical Test
In this section,   (, ) and Q (, ) are used to approximate Franke function ) , 0 ≤ ,  ≤ 1. ( Firstly, a scattered data set {(  ,   ,   )}  =1 is sampled from (, ); then, the interpolation functions   (, ) and Q (, ) are generated using the sampled data.Aiming at different sampled data sets, mean square error (MSE) and maximum error (MME) of the two approximate functions are computed and their approximate capacities are also  compared.Meanwhile, the approximate accuracy and computational efficiency of   (, ) and Q (, ) are compared with Shepard interpolation [17] and Weighted Local RBF interpolant (abbr.WLRBF) [9].Finally, the CPU time of these four methods is compared which are devised by MAT-LAB (7.12.0 R2011a) installed in a computer with Processor: Intel(R) Q8400 2.66 GHz, RAM: 4 GB.The computational results show that the methods introduced in this paper need less CPU time and have higher accuracy.Figure 4 shows the 300 scattered points which are randomly selected from [0, 1] × [0, 1], and Figure 5  Besides, with the number of scattered data increasing, it is found that approximation capability of Q (, ) is closer to that of   (, ).In case of large set of scattered points, Q (, ) can be hence absolutely superior to WLRBF.It can be seen from Table 1 that the approximation accuracy of   (, ) is higher than that of Q (, ) at the same scattered points.Meanwhile, MSE and MME of the two methods decrease with the increase of the point number.Besides, the approximating accuracies to (, ) of   (, ) and Q (, ) get closer at the same time.These results indicate that the local radial function interpolation method can be used to estimate the partial derivative when the number of data points is large enough.Therefore, Q (, ) can be used to approximate (, ) when the partial derivative information of the scattered data is unknown.
It can be seen from Table 2 that the errors of approximation functions   (, ) and Q (, ) increase with the radius of influence decreasing.
Tables 1 and 3 indicate that the approximating accuracy to (, ) using   (, ) and Q (, ) is much more accurate than that using WLRBF method and Shepard method with the same number of scattered points.The data in Table 4 indicates that the computation efficiencies of   (, ) and Q (, ) are much higher than those of WLRBF and Shepard methods.

Conclusion
In this paper, Delaunay triangulation based on planar point set is used to obtain the piecewise bivariate Hermite interpolation function in order to approximate three-dimensional scattered data sets.When point (, ) falls on a triangle of the triangulation, the Hermite interpolation on the triangle is used for approximate calculation.If point (, ) falls outside the triangulation area, the node basis function value weighted sum of the points that are closer to (, ) is used as the approximate value of (, ).Since the constructed interpolation function needs the first derivative value of the approximated function, however, the given scattered data set does not provide such information, our interpolation scheme uses local radial basis interpolation function to estimate the first derivative of each scattered point.Numerical experiments show that our methods have strong approximation ability to scattered data and the estimation of derivatives by local radial basis interpolation has high accuracy.Owing to no demand for solving linear system and the weight functions with local support, our methods are easily implemented and have high computational efficiency, so it is better than B-Spline least square fitting which needs solving a big enough linear system.The use of local radial basis function interpolation to estimate the derivatives is still consuming time.Therefore, how to construct a simple and high-accurate numerical differential formula is one of our future works.Meanwhile, for nonuniform distributed scattered data, how to more reasonably approximate them is another task in further work.
In addition, in the process of numerical experiments, we found that the triangulation based on scattered point set has a greater impact on the test results.Hence, how to construct a triangulation more suitable for approximate schemes also needs to be considered in the future work.

Figure 3 :Figure 4 :
Figure 3: The local radial basis interpolation point set   around point   , (a)   is a boundary point of  and (b)   is an interior node point of .

Figure 9 :
Figure 9: The graph of WLRBF based on Figure 4.

Figure 10 :Figure 11 :
Figure 10: The graph of Shepard interpolation based on Figure 4.

Figure 14 :
Figure 14: The error graph of Shepard interpolation.

Table 3 :
The comparison of MSE and MME of WLRBF method and Shepard method for the same scattered points.

Table 4 :
The computing time of the four methods on the 50 × 50 grid points in the unit square.