2.2.1.9 xyz_shep_nag

Brief Information

Convert XYZ data to matrix using Modified Shepard gridding

Command Line Usage

1. xyz_shep_nag iz:=Col(3);

2. xyz_shep_nag iz:=Col(3) rows:=10 cols:=10;

4. xyz_shep_nag iz:=Col(3) q:=18 w:=9;

5. xyz_shep_nag iz:=Col(3) om:=[MBook]MSheet!Mat(1);

Variables

Display
Name
Variable
Name
I/O
and
Type
Default
Value
Description
Input iz

Input

XYZRange

<active>

Specifies the input XYZ range.

Rows rows

Input

int

20

Rows in the output matrix.

Columns cols

Input

int

20

Columns in the output matrix.

Quadratic q

Input

int

18

The quadratic interplant locality factor, which is used to calculate the influence radius of local approximate quadratic fitted function for each node. By default, q equals to 18. It is better to make q ≈ 2w and it should be satisfy that 0<w≤q. Modifying these factors could increase gridding accuracy, though note that the computation time can be greatly increased for large values (i.e. values that decrease the locality of the method.)

Weight w

Input

int

9

The weight function locality factor, which is used to calculate the weighting radius for each node. By default, w equals to 9. It is better to make q≈2w and it should be satisfy that 0<w<q. Modifying these factors could increase gridding accuracy, though note that the computation time can be greatly increased for large values (i.e. values that decrease the locality of the method.)

Output Matrix om

Output

MatrixObject

<new>

Specify the output matrix object.

See the syntax here.

Description

This function calls NAG library to perform modified Shepard gridding method which described by Franke and Nielson[1]. This is a distance-based method and improves the Shepard's method by some local strategies. During gridding, only the data points that lying within certain ranges, R_q and R_w, to the grid nodes are considered. To make it easier for setting, two integers, N_q and N_w are used to calculate R_q and R_w (parameters q and w of the function, and called Quadratic Interplant Locality Factor and Weight Function Locality Factor, respectively). Increase the value of N_q and N_w will make the calculation more global, vice versa. Generally speaking, setting N_q 2*N_w works fine and by default, Nq=18, Nw=9. However, the following constraints: 0< N_wN_q should be satisfied.

The value of R_q and R_w in this function is fixed and there is another similar X-Function xyz_shep which described by Renka[2] uses a vary R_q and R_w strategy.

Examples

1. Import XYZ Random Gaussian.dat on the \Samples\Matrix Conversion and Gridding folder.

2. Type xyz_shep_nag 3 in the command window. Or type xyz_shep_nag -d to bring up the dialog.

Algorithm

This is a distance-based weighted gridding method which interpolate data by:

Xyz shep nag help English files image002.gif,

where F_i is the underlying function at nodes (x_i, y_i), and W_i(x, y) is the weights. To make the function more local, F_i and W_i are calculated only by the data points lying in the circle with center (x_i, y_i) and some radius R..

Firstly, the weights are defined as:

Xyz shep nag help English files image004.gif.

Given a radius R_w, the relative weight w_k is:

Xyz shep nag help English files image006.gif for

Xyz shep nag help English files image008.gif,

and d_k is the Euclidean distance between (x, y) and (x_k, y_k):

Xyz shep nag help English files image010.gif.

For any R_w>0, we have:

Xyz shep nag help English files image012.gif

Xyz shep nag help English files image014.gif.

Secondly, the nodal function F_i is replaced by a local approximation function Q_k:

Xyz shep nag help English files image016.gif

Q_k is the weighted least-square quadratic fitted function to the data located within R_q of nodal points. So the coefficients minimize:

Xyz shep nag help English files image018.gif

for

Xyz shep nag help English files image020.gif.

It can be seen above that the interpolate function is a local approximation function and depends on the radius of influence about nodal points, R_q and R_w. In this method, two integers N_q and N_w are used to calculate R_q and R_w:

Xyz shep nag help English files image022.gif and Xyz shep nag help English files image024.gif,

where n is the number of data points and D is the maximum distance between any pair of data points. So N_q and N_w can be considered to be the average numbers of data points lying within distance R_q and R_w respectively for each node.

References

[1]. Franke R and Nielson G. smooth Interpolation of Large Sets of Scattered Data. Internat. J.Num. Methods Engrg. 1980, 15, pp:1691-1704.

[2]. Renka, R. J., Multivariate Interpolation of Large Sets of Scattered Data. ACM Transactions on Mathematical Software, Vol. 14, No. 2, June 1988, pp:139-148.

Related X-Functions

xyz_regular, xyz_renka, xyz_renka_nag, xyz_shep, xyz_sparse, xyz_tps


Keywords:worksheet