Given **n** corresponding points (**n** is typically hundreds)
--- inevitably the points will be ``** noisy**'' and there will be
** mis-matches** (outliers).

** Linear solution --- ``The 8-point Algorithm''**

This equation will not be satisfied exactly (for **n > 8**). A solution
is obtained by:

This is a standard linear algebra problem, the solution is the eigenvector with minimum eigenvalue of .

This computation is generally poorly conditioned, and it is
very important to ** pre-condition** the matrix: i.e. (affine) transform
the image points:

where and are matrices, such that

and **< >** indicates averages. A fundamental matrix
is then computed from the transformed points
and obtained by

Pre-conditioning can make a difference of an order of magnitude to the average distance of a point from its epipolar lines (e.g. reduce an average of 3.5 to 0.3 pixels) and hundreds of pixels in the position of the epipoles.

** Non-linear solution**

Minimise (average) squared perpendicular distance of points from their epipolar lines:

where

Use Levenberg-Marquardt or Powell for non-linear optimisation.

** Summary Point**

- Epipolar geometry is the fundamental geometric relationship between two images.
- It is represented algebraically by the fundamental matrix .
- can be computed using image correspondences alone (7 or more).

Wed Apr 16 00:58:54 BST 1997