I observed particle motion within an experiment with two cameras. One was on the left, the other was on the right. I need to link identical trajectories viewed in the left camera to their counterpart in the right camera.
The left view obtains a set of $N_L$ trajectories $\{l_i\}_{i=1}^{N_L}$. The right view obtains a set of $N_R$ trajectories $\{r_i\}_{i=1}^{N_R}$. Probably $N_L\neq N_R$ due to measurement error.
Each pairing of trajectories $l_i$ and $r_j$ defines a cost $C_{ij}$. Since $N_L\neq N_R$, some trajectories will not pair up. Therefore a dummy row and dummy column must be appended to the cost matrix $[C_{ij}]$, and it has dimension $(N_L+1)\times (N_R+1)$.
I need to find an assignment $x_{ij}$. I believe its constraints should be
$x_{ij} \in \{0,1\}$
$\sum_{i=1}^{N_L}x_{ij} = 1 $ if $j\neq N_R+1$
$\sum_{j=1}^{N_R}x_{ij} = 1 $ if $i\neq N_L+1$
$\sum_{i=1}^{N_L}x_{ij} \leq N_L $ if $j = N_R+1$
$\sum_{j=1}^{N_R}x_{ij} \leq N_R $ if $i = N_L+1$
The difference between this problem and a standard assignment problem is that more than one particle can be unlinked.
I'm certain this problem is well-studied, but I don't know much about this type of mathematics. Maybe someone could point me in the right direction for a solution? Any help or background would be appreciated. I will probably be implementing any solution in python, if that's relevant. Thanks!