I’ve just put my thoughts down elsewhere while trying to explain what I see the maths problem behind this project, and I thought I’d share them here too. It’s quite late at the end of my third 15hr work day so a little slack is appreciated. That said, if what I’m proposing is a complete load of bo**cks feel free to point out the errors – with corrections please! ;-)
My system can't produce exact bearings/'pixels' like the lighthouse system does, but it can measure the relative angles between the various points and, in theory, relate those angles to an absolute distance.
It's similar, or perhaps even the same(?), as the n-point problem I think.
There is only one orientation (and from a perspective point of view, distance) of the target with respect to the laser emitter, which when scanned with the laser line, will give that particular ratio of angles. The problem arises because that orientation could be anywhere on the sweep of the laser, like a toroid in the axis of rotation. However, if you know the attitude of the target with respect to the axis attitude of the laser sweep, it becomes fixed against that axis – there is only one orientation of the target wrt to the laser axis which has that particular target pitch and roll.
The "orientation" reduces down to a relative translation of the target (x,y,z) with a rotation in target yaw/heading - and the whole thing potentially rotated around the vertical axis.
Having a second set results from another characterised laser scanner (characterisation done during the room calibration routine) and it should be possible to calculate the yaw/heading component of the target and fix the target on the XY plane.
The whole reduction thing relies on having known attitudes for the emitters and the targets so they can all relate in the same axis.
Anyway, that's my theory. I've talked to one of the mathematicians at work and he seems interested enough in the system to want to give me a hand getting it working. Perhaps when I give him this talk he'll laugh in my face at my naivety and point out I need 30 scanners to make up for the lack of real bearings.
I hope not.