TPSINT

Description

Generates a raster image by interpolating image values between specified pixel locations using a Radial Basis Interpolation Algorithm. This module implements the Thin Plate Spline scheme.

Parameters

Name	Type	Length	Value range
Input: Input vector segment *	Vector port	1 - 1
Output: Output raster channel	Raster port	0 - 1
Field Name	String	0 - 1	Default: ATTRIBUTE
Pixel X Size	Float	0 - 2	0.0 -
Pixel Y Size	Float	0 - 2	0.0 -

* Required parameter

Parameter descriptions

Input: Input vector segment

Specifies the vector segment in which the attribute value or the z-coordinate will be used as the basis for interpolation.

Output: Output raster channel

Specifies the output channel to receive the interpolated image.

Field Name

Specifies the field name that contains the elevation values.

If this parameter is specified as ZCOORD, the actual z-coordinates of the vectors are used. Field names are not case-sensitive, and they do not need to be specified in complete form. If more than one match exists, the first name is used.

The Field Name (FLDNME) parameter is required only for vector layers that contain 3-D points or contours. For 3-D lines and 3-D polygons the z-coordinates are automatically used. For 2-D layers, the elevation values are not specified.

The same name applies to all contour and 3-D point segments specified in the input vectors (DBVS) parameter. The segments that do not satisfy this requirement can be converted to the required format using ZVALTRNS.

If a field name other than ZCOORD is specified but is not found, this function attempts to use the z-coordinates of each vertex or point; this may, however, lead to unexpected results.

Pixel X Size

Specifies, in meters, the X (horizontal) pixel size of the output image. The specified X and Y pixel sizes determine the resolution of the output file. A smaller pixel size results in a larger output file and increases computation time.

Pixel Y Size

Specifies, in meters, the Y (vertical) pixel size of the output image. The specified X and Y pixel sizes determine the resolution of the output file. A smaller pixel size results in a larger output file and increases computation time.

Details

TPSINT reads the gray-level values for an arbitrary number of pixel locations to generate a raster image based on the Thin Plate Spline method of interpolation between the specified gray levels.

The gray-level values are read from an input vector segment (DBVS).

This method is computationally expensive. The coefficients must be calculated by solving a system of equations. The number of equations is proportional to the size of the data set. After the coefficients are calculated, the interpolation is calculated. The complexity of this calculation is based on the number of data points related to each point to be interpolated. For example, for N data points on an MxM grid, approximately NxMxM calculations are required to calculate the interpolation.

This method is also memory demanding and a maximum limit of 4000 pixel locations has been set. This limit is further lowered depending on the amount of free memory available.

Algorithm

TPSINT implements the Thin Plate Spline radial basis function interpolation scheme. This interpolation scheme is defined by mathematical series (sum of terms).

In the Thin Plate Spline scheme, the interpolator is:

                                         N
                                         
      f(x, y) =  a + a x + a y  + 0.5 * SUM b  * sqr(R ) * log(R )
                  0   1     2                j        j         j
                                       j = 1

where:

     a , a , a , and b 
     0   1   2       j

are the coefficients defining the particular problem and

     R  
     j

is the distance between the point (X,Y) and the jth control point.

TPSINT determines the set of coefficients for the interpolator by solving a system of linear equations defined at a set of control points. The result is then substituted back into the interpolator and interpolation is computed at any point by summing the series. Further mathematical justifications on how these interpolators are derived and why they work can be found in the papers cited in References.

References

"Theory And Applications Of The Multiquadric-Biharmonic Method", Vol. 19, No. 8/9 pp. 163-208, 1990

"Warping Digital Images Using Thin Plate Splines", in Pattern Recognition, Vol. 26, No. 2 pp. 375-376, 1993