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How Spline works

Release 9.2
Last modified January 3, 2008
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The basic form of the minimum curvature Spline interpolation imposes the following two conditions on the interpolant:



The basic minimum curvature technique is also referred to as thin plate interpolation. It ensures a smooth (continuous and differentiable) surface, together with continuous first-derivative surfaces. Rapid changes in gradient or slope (the first derivative) can occur in the vicinity of the data points; hence, this model is not suitable for estimating second derivative (curvature).


The basic interpolation technique can be applied by using a value of zero for the {weight} argument to the Spline function.


The REGULARIZED option modifies the minimization criteria so third-derivative terms are incorporated into the minimization criteria. The {weight} argument specifies the weight attached to the third-derivative terms during minimization, referred to as tau in the literature. Higher values of this term lead to smoother surfaces. Values between 0 and 0.5 are suitable. Using the REGULARIZED option ensures a smooth surface together with smooth first-derivative surfaces. This technique is useful if the second derivative of the interpolated surface needs to be computed.


The TENSION option modifies the minimization criteria so first-derivative terms are incorporated into the minimization criteria. The {weight} argument specifies the weight attached to the first-derivative terms during minimization, referred to as phi in the literature. A weight of zero results in the basic thin plate Spline interpolation. Using a larger value of weight reduces the stiffness of the plate, and in the limit as phi approaches infinity, the surface approximates the shape of a membrane, or rubber sheets, passing through the points. The interpolated surface is smooth. First derivatives are continuous but not smooth.


The Spline function uses the following formula for the surface interpolation:


Spline formula

where:

j = 1, 2, ..., N

N is the number of points.

λj are coefficients found by the solution of a system of linear equations.

rj is the distance from the point (x,y) to the jth point.


T(x,y) and R(r) are defined differently, depending on the selected option.


For the REGULARIZED option:

T(x,y) = a1 + a2x + a3y

Spline regularized option


and for the TENSION option:


T(x,y) = a1


Spline tension option

where:

Spline parameter and Spline parameter are the parameters entered at the command line.

r is the distance between the point and the sample.

Bessel function is the modified Bessel function.

c is a constant equal to 0.577215.

Coefficients are coefficients found by the solution of a system of linear equations.


For computational purposes, the entire space of the output raster is divided into blocks or regions equal in size. The number of regions in x and in y directions are the same, and they are rectangular in shape. The number of regions is determined by dividing the total amount of points in input point dataset by the value specified for the number of points. For data less uniformly distributed, the regions may contain a significantly different number of points, with the value for the number of points being only the rough average. If in any region, the number of points is smaller than eight, the region is expanded until it contains a minimum of eight points.


References


Franke, R. 1982. Smooth Interpolation of Scattered Data by Local Thin Plate Splines. Comp. & Maths. with Appls. Vol. 8. No. 4. pp. 237–281. Great Britain.


Mitas, L., and H. Mitasova. 1988. General Variational Approach to the Interpolation Problem. Comput. Math. Applic. Vol. 16. No. 12. pp. 983–992. Great Britain.


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