Thin Plate Spline (TPS)

Python (NumPy & SciPy) implementation of the generalized Polyharmonic Spline interpolation (also known as Thin Plate Spline in 2D). It learns a smooth elastic mapping between two Euclidean spaces with support for:

• Arbitrary input and output dimensions
• Arbitrary spline order k
• Optional regularization

Useful for interpolation, deformation fields, and smooth non-linear regression.

For a faster implementation in PyTorch, see torch-tps.

🚀 Install

Pip

$ pip install thin-plate-spline

From source

git clone git@github.com:raphaelreme/tps.git  # OR https://github.com/raphaelreme/tps.git
cd tps
pip install .

Getting started

import numpy as np
from tps import ThinPlateSpline

# Control points
X_train = np.random.normal(0, 1, (800, 3))  # 800 points in R^3
Y_train = np.random.normal(0, 1, (800, 2))  # Values for each point (800 values in R^2)

# New source points to interpolate
X_test = np.random.normal(0, 1, (300 0, 3))

# Initialize spline model (Regularization is controlled with alpha parameter)
tps = ThinPlateSpline(alpha=0.5)

# Fit spline from control points
tps.fit(X_train, Y_train)

# Interpolate new points
Y_test = tps.transform(X_test)

Examples

See the example/ folder for scripts showing:

• Interpolation in 1D, 2D, 3D
• Arbitrary input and output dimensions
• Image warping with elastic deformation

Image Warping

Example of increasing/decreasing/randomly deforming a dog's face using sparse control points.

Code: example/image_warping.py

🧠 Theory Summary

The model solves the regularized interpolation problem:

min_f \sum_{i=1}^n (y_i - f(x_i))^2 + \int \|\nabla^{\text{order}} f\|_$2^{2}$ dx

With solution:

$$f(x) = P(x) + \sum_{i=1}^n w_i G(\|x - x_i\|_2)$$

Where:

• $G(r)$: radial basis function (RBF) (depends on order and the input dimension d)
• $P(x)$: a polynomial of degree order - 1

Default kernel (TPS):

• $G(r) = r^2 \log(r)$

General kernel:

• $G(r) = r^{(2 \text{order} - d)} \text{ if d is odd}$
• $G(r) = r^{(2\text{order} - d)} \log(r) \text{ otherwise}$

🔧 API

ThinPlateSpline(alpha=0.0, order=2, enforce_tps_kernel=False)

Creates a general polyharmonic spline interpolator (Default to TPS in 2D and natural cubic splines in 1D).

• alpha (float): Regularization strength (default 0.0)
• order (int): Spline order (default is 2 for TPS)
• enforce_tps_kernel (bool): Force TPS kernel r^2 log r, even when mathematically suboptimal

.fit(X, Y)

Fits the model to control point pairs.

• X: (n, d) input coordinates
• Y: (n, v) target coordinates

Returns: self

.transform(X)

Applies the learned mapping to new input points.

• X: (n', d) points

Returns: (n', v) interpolated values

License

MIT License