Polyharmonic |
Interpolation method for scattered data in any dimension based on radial basis functions.
In 2D this is the so called Thin Plate Spline, which is an interpolation method that finds a "minimally bended"
smooth surface that passes through all given points.
The polyharmonic spline has an arbitrary number of dimensions and arbitrary derivative order.
Note: The allocation space requirement is in the order (N+3)*(N+3), where N is the number of control points. Thus it is not applicable for too many points.
Inheritance HierarchyNamespace: Altaxo.Calc.Interpolation
Assembly: AltaxoCore (in AltaxoCore.dll) Version: 4.8.3261.0 (4.8.3261.0)
SyntaxC#
public class PolyharmonicSpline : IInterpolationFunction, IInterpolationCurve
The PolyharmonicSpline type exposes the following members.
Constructors| Name | Description | |
|---|---|---|
 | PolyharmonicSpline | Initializes the spline with a regularization parameter of zero and an derivative order of 2. |
Properties| Name | Description | |
|---|---|---|
 | DerivativeOrder | Derivative order of the spline. Effectively, the L2-norm of this derivative is minimized. |
| NumberOfControlPoints | Number of control points. |
| NumberOfDimensions | Number of dimensions of the control points. |
| RegularizationParameter | Regularization parameter (>=0). If the regularization parameter is zero, interpolation is exact. As it approaches infinity, the resulting spline is reduced to a least squares linear fit (in 2D this is a plane, the bending energy is 0). |
Methods| Name | Description | |
|---|---|---|
| Construct(IReadOnlyListDouble, IReadOnlyListDouble) | Constructs the interpolation (1 dimensional). The values and the corresponding coordinates of the values are given in separate vectors. |
| Construct(IReadOnlyListDouble, IReadOnlyListDouble) | Constructs the interpolation (any dimension). The values and the corresponding coordinates of the values are given in separate vectors. |
| Construct(IReadOnlyListDouble, IReadOnlyListDouble, IReadOnlyListDouble) | Constructs the interpolation (2 dimensional). The values and the corresponding coordinates of the values are given in separate vectors. |
| Construct(IReadOnlyListDouble, IReadOnlyListDouble, IReadOnlyListDouble, IReadOnlyListDouble) | Constructs the interpolation (3 dimensional). The values and the corresponding coordinates of the values are given in separate vectors. |
| Equals | Determines whether the specified object is equal to the current object. (Inherited from Object) |
 | Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object) |
| GetBendingEnergy | Gets the bending energy of the interpolation. |
| GetHashCode | Serves as the default hash function. (Inherited from Object) |
| GetInterpolatedValue | Gets the interpolation value of given coordinates x and y. |
| GetType | Gets the Type of the current instance. (Inherited from Object) |
| MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object) |
| ToString | Returns a string that represents the current object. (Inherited from Object) |
Fields| Name | Description | |
|---|---|---|
  | DefaultDerivativeOrder | Default value of the DerivativeOrder. |
| DefaultRegularizationParameter | Default value of the RegularizationParameter. |
Remarks
Literature:
http://en.wikipedia.org/wiki/Polyharmonic_spline
Extension to any number of dimensions:
TIM GUTZMER AND JENS MARKUS MELENK, MATHEMATICS OF COMPUTATION, Volume 70, Number 234, Pages 699{703, S 0025-5718(00)01299-0, Article electronically published on October 18, 2000
See Also