I wonder if there is a technique for fitting a surface to a given mesh grid of data points? I've seen interpolating a polynomial to $2$D data, but not $3$D.
E.g. say I was given the matrix $$ H=\left(\begin{array}{cccc} 1 & 32 & 245 & 45 \\ 5 & 145 & 134 & 54 \\ 9 & 2 & 67 & 24\\ 12 & 99 & 121 & 201 \end{array}\right), $$
could I construct a function $f(x,y)=z$ where $x,y$ are the discrete points and $z$ is the corresponding matrix value? Many thanks.
Try bicubic Bézier surface patch.
$z=f(u,v)=U\cdot C\cdot P\cdot C\cdot V^T$, where $U=[u^3\ u^2\ u\ 1],\ V=[v^3\ v^2\ v\ 1]$, $C=\left[\begin{array}{cccc} -1 & 3 & -3 & 1 \\ 3 & -6 & 3 & 0 \\ -3 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$, $P=\left[\begin{array}{cccc} P_{00} & P_{01} & P_{02} & P_{03} \\ P_{10} & P_{11} & P_{12} & P_{13} \\ P_{20} & P_{21} & P_{22} & P_{23} \\ P_{30} & P_{31} & P_{32} & P_{33} \end{array}\right]$.
To find 16 control points $P_{00}\dots P_{33}$, consider given matrix
$H=\left[\begin{array}{cccc} 1 & 32 & 245 & 45 \\ 5 & 145 & 134 & 54 \\ 9 & 2 & 67 & 24\\ 12 & 99 & 121 & 201 \end{array}\right]= \left[\begin{array}{cccc} f(0,0) & f(0,\frac13) & f(0,\frac23) & f(0,1) \\ f(\frac13,0) & f(\frac13,\frac13) & f(\frac13,\frac23) & f(\frac13,1) \\ f(\frac23,0) & f(\frac23,\frac13) & f(\frac23,\frac23) & f(\frac23,1) \\ f(1,0) & f(1,\frac13) & f(1,\frac23) & f(1,1) \end{array}\right]$.
Hint: it is easier to find the outer points first and then solve a system for $P_{11},P_{12},P_{21},P_{22}$.
The picture shows how the function $f(u,v)$ looks like for the given $H$: