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Jacob
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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: alt text

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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: alt text

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!

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?

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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 example: alt text

This example's 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!

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: alt text

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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.

The example 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!

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Jacob
  • 35
  • 6

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?

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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 example: alt text

This example's 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!

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?

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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?

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?

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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 example: alt text

This example's 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!

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Jacob
  • 35
  • 6

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 degreea function that attaches a real number to any pair of geometric similarity between these ellipsesellipsoids as a way of measuring their similarity? I've

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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?

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 degree of geometric similarity between these ellipses? I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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?

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?

I've come up with the following but was hoping for a simpler method:

  1. Minimize the volume of intersection (how?)
  2. 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?

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Jacob
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