Thin plate spline

Remarks

The conditions form a square linear system, and it has exactly one solution when the points are not collinear. This is the polyharmonic spline with $d = 2$ and $m = 2$, built from the radial basis function $\varphi$. Among all functions $f$ with square integrable second derivatives and $f(x_i) = y_i$, the spline $s$ minimizes the bending energy $$\iint_{\R^2} \big( f_{x_1 x_1}^2 + 2 f_{x_1 x_2}^2 + f_{x_2 x_2}^2 \big)\, dx.$$ Away from the points, $s$ satisfies the biharmonic equation $\Delta^2 s = 0$. Example: if $y_i = c_0 + c^T x_i$ for an affine function, then $w = 0$, $a_0 = c_0$, $a = c$ satisfy all three clauses, so $s$ is that affine function. Non-example: $f(x) = \varphi(\norm{x})$ with the single centre $0$ fails clause 2, since its only weight is $1$. Thin plate splines are also used with one map per coordinate to define smooth deformations of the plane from matched landmark points.