RbfSurface Class

public sealed class RbfSurface : Surface

Public NotInheritable Class RbfSurface
	Inherits Surface

public ref class RbfSurface sealed : public Surface

[<SealedAttribute>]
type RbfSurface = 
    class
        inherit Surface
    end

Inheritance

Object  →  Surface  →  RbfSurface

The RbfSurface class implements radial basis function interpolation for scattered data in arbitrary dimensions. RBF methods construct a smooth global interpolant as a weighted sum of radial basis functions centered at each data point.

The interpolant has the form:

For strictly positive definite (PD) kernels:

s(x) = Σᵢ wᵢ φ(||x - pᵢ||)

For conditionally positive definite (CPD) kernels:

s(x) = Σᵢ wᵢ φ(||x - pᵢ||) + p(x)

where φ is the radial basis function kernel, pᵢ are the data point locations, wᵢ are the interpolation weights, and p(x) is an augmenting polynomial.

Kernel types:

• PD kernels (Gaussian, Multiquadric, InverseMultiquadric, InverseQuadratic): Do not require polynomial augmentation and produce unconditionally solvable systems.
• CPD kernels (ThinPlate, Linear, Cubic, Quintic): Require an affine polynomial tail p(x) = c₀ + c₁x₁ + ... + cₐxₐ to ensure well-posedness and exact reproduction of linear functions.

Construction and evaluation:

• Given N sample points pᵢ ∈ ℝᴰ with values fᵢ, weights and polynomial coefficients are computed by solving a dense linear system.
• Construction cost is O(N³) due to the dense linear solve.
• Evaluation cost is O(N) per query point.
• The interpolant is defined everywhere in ℝⁿ and smoothly extrapolates beyond the data points.

RBF interpolation produces smooth, infinitely differentiable surfaces (for smooth kernels) and is particularly effective for scattered data in multiple dimensions. However, it becomes computationally expensive for large datasets due to the cubic construction cost.

The RbfSurface class inherits from Surface and provides both evaluation and gradient computation. Create instances using the ScatteredRbfInterpolator factory methods, which accept various input formats including vectors, arrays, and matrices.

RBF options such as kernel type, shape parameter, and regularization can be specified using the RbfOptions class.

Reference