quadric reco example
Basic Problem
Reconstruction of ground truth ellipsoid Q from two projections (along z and y-axis) following this approach:
https://www.robots.ox.ac.uk/~vgg/publications/1998/Cross98/
More details:
https://math.stackexchange.com/questions/4474878/reconstruct-quadric-from-projections-on-plane-conics
Positive Definite Constraint
The solution space Qsol evidently has two free parameters (see below). However, the parameter b can be chosen arbitrarily due to the homogenous nature of the quadric.
To limit the search space for c, we observe, that for a quadric to be ellipsoidal, all upper-left sub-determinants Ak need to be positive as detailed here:
https://www.math.utah.edu/~zwick/Classes/Fall2012_2270/Lectures/Lecture33_with_Examples.pdf
By writing out the equations for the determinats, we obtain the two constraints:
1.
2.
Details on the constraints here: https://www.geogebra.org/m/mctkq6ex
This interactive plot allows to explore the solution space within the bounds set by the contraint that the resulting quadric shall be positive definite.