﻿ 顾及异向性的局部径向基函数三维空间插值
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 武汉大学学报·信息科学版  2015, Vol. 40 Issue (5): 632-637

#### 文章信息

DUAN Ping, SHENG Yehua, ZHANG Siyang, LV Haiyang, WANG Ting

A 3D Local RBF Spatial Interpolation Considering Anisotropy

Geomatics and Information Science of Wuhan University, 2015, 40(5): 632-637
http://dx.doi.org/10.13203/j.whugis20130422

### 文章历史

A 3D Local RBF Spatial Interpolation Considering Anisotropy
DUAN Ping, SHENG Yehua , ZHANG Siyang, LV Haiyang, WANG Ting
Key Laboratory of Virtual Geographic Envirionment, MOE, Nanjing Normal University, Nanjing 210023, China
Abstract:Aiming at the searching range of interpolation points in the process of three-dimensional(3D), local radial basis function (RBF) interpolation based on exploring spatial anisotropy with variogram analysis was proposed. Firstly, three axes of the data was solved by constructing covariance matrix of the sampling point data and then the data was transformed into the new coordinate system by rotating transformation; the range of each direction was calculated using geostatistical variograms; the three values of range was set as three axes of the ellipsoid; at last, node RBF at each sample point was built. The attribute values of interpolation were solved by linear combination of node RBF. Experimental results show that the proposed method is a feasible method for 3D spatial interpolation considering anisotropy with high accuracy and reliable interpolation result.
Key words: anisotropy     variogram     RBF     spatial interpolation

1 三维空间局部RBF插值方法 1.1 方向性探索

 图 1 三维椭球体搜索区域Fig. 1 Searching Region of 3-Dimensional Ellipsoid

1) 三维空间数据协方差矩阵为:

2) 对矩阵ΣX进行特征值分解:

3) 将原始数据旋转变换:

1.2 空间异向性结构探索

1) 点对预处理。计算所有点对{(xi，xj)，i ≠ j}的距离dij、变异函数值γ(dij)、点对中的最大距离dmax，剔除大于0.5dmax的点对。

2) 点对分类。设置每个轴向的角度容差为45°，比较数据点对[(xi，yi，zi)，(xj，yj，zj)]的分量在所对应轴x、y、z的投影距离，记：xpd =|xi-xj|，ypd = |yi -yj|，zpd = |zi-zj|，求取三个分量中的投影距离最大值max{xpd，ypd，zpd}。将数据点对归为最大值所对应的轴，对所有数据点对进行操作，数据点对将分为三个集合且分别对应不同的三个轴。

3) 点对分组。如果集合中点对较少时，不需要分组；点对较多时，则进行分组。设分组数为n，求得分组距离容差值lagsize = 0.5dmax/ n，进行分组计算m= dij/lagsize 为第m组，对步骤2中三个集合进行相同的操作。

4) 实验变异函数的拟合。本文选取球状模型，使用RANSAC算法[5]进行球状模型的拟合，对步骤3中每个集合的分组平均值进行拟合，得到三个方向的变程分别对应于椭球体的三个轴长。

1.3 局部RBF插值方法

 图 2 邻近范围的确定Fig. 2 Determination of Neighboring Region

，文献[9]证明了Wk插值权函数满足以下3个性质：

2 实验验证

 图 3 局部RBF插值流程图Fig. 3 Flow Chart of Local RBF Interpolation

 图 4 采样数据的空间分布Fig. 4 Spatial Distribution of the Sampling Data

 插值方法 εmin εmax εme εrmse LRBF -0.116 3 0.369 2 -0.003 7 0.060 7 OK -0.077 1 0.166 2 0.004 3 0.069 2 AOK -0.051 2 0.236 1 0.002 1 0.038 7 本文方法 -0.060 3 0.130 6 0.001 8 0.037 5

 图 5 4种插值结果Fig. 5 Four Interpolation Results
3 结 语

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