Estimating Thin-plate Spline Parameters

The unknown UL-TPS parameters $W=[\xi_1, \ldots, \xi_M, a_1, a_2, b]^{T}$ in Eq. 3 are determined by solving the linear system of equations that result by fixing the displacement field values at landmark locations. Let $\phi_{i,j}=\phi(\vert p_i-p_j\vert)$ and build the matrix

$$K = \left[ \begin{array}{cc} \Phi & \Lambda \\ \Lambda^T & O \\ \end{array} \right]$$

$$~~~where ~ ~~$$

$$\Phi = \left[ \begin{array}{cccc} \phi_{1,1} & \phi_{1,2} & \dots... ... \\ \phi_{M,1} & \phi_{M,2} & \dots & \phi_{M,M} \\ \end{array}\right], \quad$$

$$\Lambda = \left[ \begin{array}{ccc} p_{1} & q_{1} & 1 \\ p_{2} &... ...& 1 \\ \vdots & \vdots & \vdots \\ p_{M} & q_{M} & 1 \\ \end{array}\right] ,$$ (7)

where $O$ is a $3\times 3$ matrix of zeros. Also, define the $(M+3) \times 2$ matrix of landmark displacements as $D=[d_1, \ldots, d_M, 0, 0, 0]^{T}$ where $d_i=q_i-p_i$ for $i = 1, \ldots, M$. The equations formed by substituting the landmark constrains into Eq. 3 can be written in matrix form as $D = K W$. The solution $W$ to this matrix equation is determined by least squares estimation since the matrix $K$ is not guaranteed to be full rank.

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