Question : will a second order polynomial be rich enough to be able to express motion vector fields stemming from affine transformations in 3 dimensions as seen through a pin-hole camera?
$$\hat v_x = c_1x^2+c_2y^2+c_3xy +c_4x +c_5y + c_6$$
$$\hat v_y = d_1x^2+d_2y^2+d_3xy +d_4x +d_5y + d_6$$
Background :
Consider the simplified 1:st order polynomial parametric model / approximation
$$\hat v = \begin{bmatrix}\hat v_x&\hat v_y\end{bmatrix} = \begin{bmatrix}x&y&1\end{bmatrix} \begin{bmatrix}a_1&a_2\\b_1&b_2\\c_1&c_2\end{bmatrix}$$
for a vector field $$v = [v_x,v_y]$$
My intuition tells me and I would like to think that the above parametric model is rich enough to be able to express affine motion. In particular : rotation, translation, scaling.
Own work (limited to the 2D special case with 1st order polynomial) :
- Pure translation is obvious, let : $c_1,c_2$ to the scalars of translation and set $a_i,b_i = 0$.
- Rotation (around origo) almost equally so, just set : $a_1 = b_2 = \cos(\phi), b_1=-b_2 = \sin(\phi)$
- Scaling as well, just set $a_j,b_j = S, c_j = 0$