I need to solve a time-parametrised parabola fitting problem with $N$ observations $(x_i, y_i, z_i, t_i)$ with $i\in\{1\cdot\cdot\ N\}$, allowing to recover $(x^*_i, y^*_i, z^*_i, t_i)$ with the following relation:
$$\left[\begin{matrix}x^*_i\\y^*_i\\z^*_i\end{matrix}\right] = \left[\begin{matrix}x_0\\y_0\\z_0\end{matrix}\right] + \left[\begin{matrix}v^x_0\\v^y_0\\v^z_0\end{matrix}\right]t_i - \left[\begin{matrix}0\\0\\g\end{matrix}\right]\dfrac{{t_i}^2}{2}$$
By formulating this problem into the following linear system,
$$A\left[\begin{matrix}x_0\\y_0\\z_0\\v^x_0\\v^y_0\\v^z_0\end{matrix}\right] \approx b\qquad\text{with}\quad A=\left[\begin{matrix}1&0&0&t_1&0&0\\0&1&0&0&t_1&0\\0&0&1&0&0&t_1\\ &\vdots&&&\vdots&\\ 1&0&0&t_N&0&0\\0&1&0&0&t_N&0\\0&0&1&0&0&t_N\end{matrix}\right]\qquad\text{and}\quad b=\left[\begin{matrix}x_1\\y_1\\z_1+g\ {t_1}^2/2\\\vdots\\x_N\\y_N\\z_N+g\ {t_N}^2/2\end{matrix}\right]$$
the MSE solution is given by:
$$\left[\begin{matrix}x_0\\y_0\\z_0\\v^x_0\\v^y_0\\v^z_0\end{matrix}\right] = (A^TA)^{-1}A^Tb$$
However, this solution minimises the MSE in the 3D space. Instead, I would like to formulate the problem by minimising the MSE in the 2D space, by projecting the points with the associated projection matrix $P_{3\times 4}$!
Question: How should the multiplication by the projection matrix be used, knowing that for a point $(x,y,z)$ given in carthesian coordinates, the projected point associated to $P$ is $(i,j)$ given by the following relationship (parametrised by $\lambda$ in homogenous coordinates!):
$$\lambda\left[\begin{matrix}i\\j\\1\end{matrix}\right]=P\left[\begin{matrix}x\\y\\z\\1\end{matrix}\right]$$
Path towards a solution: I think it boils down to describing the whole system in homogenous coordinates and multiplying by $P$ on both sides of the $\approx$ sign. However, I have trouble doing even the first step without creating inconsistencies in the homogenous relationships (adding or subtracting the homogenous component…)
Does anybody know how to proceed? I capitulate after multiple un-successful attempts.
By projecting, the problem becomes non-linear (because of the division by the homogenous component computed with the input data). Therefore, the solution for linear systems cannot be used and a solver must be used instead.