I'm having trouble solving the following problem. Hopefully someone can help!
The following is in homogeneous coordinates. I am projecting a point into a pinhole camera. $\lambda$ is an unknown scaling factor, all other variables are known.
$\mathbf{x} = \mathtt{P}(\lambda \mathbf{X} + \mathbf{Q})$
P is a 3x4 projection matrix. I'd like to solve for $\lambda$. Here's some example data:
x = [-0.4052 -0.0502 1.0000]' P = [1 0 0 0 0 1 0 0 0 0 1 0] X = [0.3925 0.0465 -0.9313 1.0000]' Q = [-18.4299, 0, -7.7678, 1.0000]'
And for this data, I know $\lambda$ is about 1430 if I just brute force search. Plugging the numbers and this result into the equation gives a correct solution. However, surely there must be a direct solution for $\lambda$? I can't seem to work it out!
Thanks,
John