Suppose I have two ellipsoids $A$ and $B$ respectively represented as $(x-C_A)^T M_A(x-C_A)$ and $(x-C_B)^T M_B(x-C_B)$ in the matrix representation.
What's the best way to find a a function that attaches a real number to any pair of ellipsoids as a way of measuring their similarity?
Here's an example:
I've come up with the following but was hoping for a simpler method:
- Minimize the volume of intersection (how?)
- Convert the ellipse to the canonical form $\frac{(x-C_x)^2}{a}+\frac{(y-C_y)^2}{b}+\frac{(z-C_z)^2}{c} = 1$ by obtaining the SVD of $M$ such that $M = U*S*V^T$ and extracting $a,b,c$ from $S$ and the axis-angle representation of the rotation matrix $V$. Then, I (somehow) use this to compare two ellipsoids.
Any ideas?
EDIT:
Here's an The example:
This example's I've provided is a little extreme since the smaller ellipsoid is "very" different from the others. But as you can see, the thin skinny ones overlap over a similar space and are similar in dimensions and orientation - I'm trying to quantify this in some way.
A fairly detailed explanation of a good distance metric would be appreciated!