Simple samples to demonstrate 2D Thin Plate Spline Warping.

The keypoint matches are obtained by lightglue matcher on KeyNetAffNetHardNet keypoints.

2D Thin Plate Spline (TPS) warping is a parametric transformation method to transform a source image to a target image.

TPS mapping function $\textbf{f}(\textbf{p})=\textbf{f}(x,y)$ is estimated using control points with source points ${\left\{ \textbf{p}_i=(x_i,y_i) \right\}}_{i=1}^N$ in the source image that match with target points ${\left\{ \textbf{p'}_i=(x'_i,y'_i) \right\}}_{i=1}^N$.

The loss function in 2D TPS warping is composed of two main components: data term and bending energy

$$ L=\sum_{i=1}^{N}\left| f(\textbf{p}_i)-\textbf{p}'_i \right|^2 \lambda\int\int\left[ \left( \frac{\partial ^2 f}{\partial x^2} \right)^2 2\left( \frac{\partial ^2f}{\partial x\partial y} \right)^2 \left(\frac{\partial ^2f}{\partial y^2} \right)^2 \right]dxdy $$

This optimization problem has a closed form solution as following:

$$ \textbf{f}(\textbf{p})=(f_x(x,y),f_y(x,y)) $$

$$ f_x(x,y)=a_1 a_2x a_3y \sum_{i=1}^{N}w_iU(\left| \textbf{p}-\textbf{p}_i \right|) $$

$$ f_y(x,y)=b_1 b_2x b_3y \sum_{i=1}^{N}v_iU(\left| \textbf{p}-\textbf{p}_i \right|) $$

where $U(r)=r^2log(r^2)$ is radial basis function, with $r=\left\| \textbf{p}-\textbf{p}_i \right\|$

This system of equations has the following constraint:

$$ \sum_{i=1}^{N}w_i=\sum_{i=1}^{N}w_ix_i=\sum_{i=1}^{N}w_iy_i=0 $$

$$ \sum_{i=1}^{N}v_i=\sum_{i=1}^{N}v_ix_i=\sum_{i=1}^{N}v_iy_i=0 $$

Hence, can be transformed to the matrix formulation $\textbf{A}\textbf{c}=\textbf{b}$

$$ A=\begin{bmatrix} \textbf{K} & \textbf{P}\\ \textbf{P}^T & \textbf{0} \end{bmatrix} , c=\begin{bmatrix} \textbf{w} \\ \textbf{a} \end{bmatrix} , b=\begin{bmatrix} \textbf{x}' \\ \textbf{0} \end{bmatrix} $$

$$ K_{ij}=U(\left| \textbf{p}_i-\textbf{p}_j \right|);\textbf{K}\in \textbf{R}^{N \times N} $$

$$ \textbf{P}=\begin{bmatrix} 1 & x_1 & y_1 \\ 1 & x_2 & y_2 \\ ... & ... & ... \\ 1 & x_N & y_N \end{bmatrix}; \textbf{P} \in \textbf{R}^{N \times 3} $$

$$ \textbf{w}=[w_1w_2...w_N]^T;\textbf{a}=[a_1a_2a_3]^T;\textbf{x}'=[x_1'x_2'...x_N']^T $$

python3 scripts/test_tps.py --query_path ./data/car1.jpg --ref_path ./data/car2.jpg

query image

reference image

warped query image

warped query image blended with reference

mamba env create --file environment.yml
mamba activate tps

• Thin Plate Spline