I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires.
The model I am working with has multiple equations each with a number of parameters that are unknown and have to be found. The equations share a number of unknown parameters between them. I.e.
$$ y_1 = u^2 + v^2 $$
$$ y_2 = u^2 + v^2 + k^2 $$
where u, v and k are unknowns.
Am I correct in assuming that the first thing I should do is work out the Jacobian of the set of equations? Once I work out the Jacobian can I run that through the LM algorithm with the observed results to find out my parameters?
The equations work out a value that is a 2D coordinate. Do I have to split up the equations into $2$ separate $x$ and $y$ components and add them both as separate equations into the jacobian? or can I use the $(x,y)$ coordinate in a single equation?
Finally I realise that will be a lot of parameters that are not shared a lot of the equations, meaning that a lot of the first order derivatives will be $0$ values. Does that matter for LM?
Thanks in advance, hopefully this all makes sense.