MATLAB Implementation of Thin Plate Spline Interpolation

Thin plate spline is an interpolation method based on radial basis functions, commonly used for data fitting in high-dimensional spaces and surface reconstruction. This approach finds the optimal smooth interpolation function by minimizing bending energy, making it particularly suitable for fitting scattered data. In MATLAB, thin plate spline can be implemented by constructing a radial basis function matrix. The implementation requires preparing known sample point data, which includes positional coordinates and corresponding function values. The core of thin plate spline lies in solving a linear system of equations that incorporates distance information between sample points. Key MATLAB functions involved include `pdist2` for calculating pairwise distances and the backslash operator (`\`) for efficient linear system solution. The computational process primarily involves two components: calculation of radial basis functions and determination of weight coefficients. Radial basis functions typically adopt logarithmic or polynomial forms, while weight coefficients are solved through least squares methods or other optimization approaches. In MATLAB, this can be efficiently handled using matrix operations like `reshape` for data organization and `svd` or `pinv` for stable matrix inversion, ultimately generating smooth and accurate fitting results. For practical implementation, the algorithm follows these steps: 1. Compute pairwise distances between control points using Euclidean distance 2. Apply the radial basis function φ(r) = r²log(r) to distance matrix 3. Construct the linear system with regularization constraints 4. Solve for weights using MATLAB's linear algebra capabilities In practical applications, thin plate spline is suitable for image warping, terrain modeling, and other domains where it effectively handles non-uniformly distributed sample points while providing excellent smoothness properties. The method's MATLAB implementation typically involves creating custom functions for basis computation and leveraging built-in matrix solvers for optimal performance.