Background
I have two projection images of an ellipsoid in 3d-space. My goal is to reconstruct the quadric given the parametrized projected conic outlines (ellipses). I am trying to recreate the approach described in this paper. I visualized an example geometry in this plot (not the one from the numerical example below, sorry).
It is stated, that a Quadric $Q$ can be reconstructed from two view-conics $C_{1}, C_{2}$ up to one free parameter. The resulting family of Quadrics can be written as:
$Q^{\ast}(\lambda)=Q_{1}^{\ast}+\lambda Q_{2}^{\ast}$
An additional view or point correspondences would define the reconstruction unambiguously.
Example
As an example, I define a quadric as: $$Q=\begin{bmatrix} 1 & -0.5 & 0 & 0\\ -0.5 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1 \end{bmatrix}$$.
Using the orthogonal projection along the y- and z-axis, $P_z, P_y$, I obtain the conics $C_z, C_y$ following this approach. Constructing and solving the equation system $Mw=v$ (see paper for details), yields this solution space:
$$Q_{sol}=\begin{bmatrix} -b & b/2 & 0 & 0\\ b/2 & -b & c & 0\\ 0 & c & -b & 0\\ 0 & 0 & 0 & b \end{bmatrix}$$.
Here I visualized this example and made an interactive plot of the solution space. I think this is a 1d solution space, as parameter $b$ is homogenous.
Question
In the example, we can clearly see, that $c=0$. Is there a way to find that out generally? According to the visualization, some choices of $c$ seem to violate the projected conics and thus dont solve the equation system. Why is that?
Can we define further constraints on the problem to make it solvable from two views? For example:
- Constraint shape of quadric (positive-definite for ellipsoids?)
- Does the angle between projections matter?
- Quadric with maximal volume (as constraint not optimization afterwards)
Any thoughts and comments would be appreciated!
