Least square plane fit cubic surface to 3d points

Question

I have point cloud and want to get cubic surface fit from it. The cubic polynomial surface equation is ($Ax^2 + By^2 + Cx^3 + Dx^2y + Exy^2 + Fy^3$). I want to get the value of $A, B, C, D, E$ & $F$.

Answer 1

So your model is $z=Ax^2 + By^2 + Cx^3 + Dx^2y + Exy^2 + Fy^3+\epsilon$, where, lets assume at this point that the errors terms are uncorrelated with $\mathbb{E}\epsilon=0$ and finite variance. So, you have to gather at least $n=8$ data points of the form $(z_i, x_i, y_i)$ (preferably much more than $8$). Then construct a design matrix of order $n\times 7$ of the following form $$ [\mathrm{1},x^2, y^2, x^3, x^2y, xy^2, y^3], $$ and then, using your favorite software, you should regress $\mathrm{z}$ on this vectors to obtain the OLS estimators of $A, B, C,...$