Why are 8 points needed to compute a fundamental matrix?

It is because in the case of fundamental matrix, each correspondence point relates to only one constraint(i.e it maps a point to a line in other image). Hence 8 correspondence points are required.

1. Why do we need 8 points instead of 9 for computing the fundamental matrix?
2. How many points are needed to estimate a fundamental matrix?
3. What is 8 point algorithm used for?
4. Why are 4 points needed for homography?

Why do we need 8 points instead of 9 for computing the fundamental matrix?

If we like a fast linear solution, then 8-points are required. For formulations using less number of points, the constraints are non-linear and typically involve either determinants or systems of polynomial equations that are solved with some form of Gröbner basis methods.

How many points are needed to estimate a fundamental matrix?

Unlike a homography, where each point correspondence contributes two constraints (rows in the linear system of equations), for estimating the essential/fundamental matrix, each point only contributes one constraint (row). [because the Longuet-Higgins / Epipolar constraint is a scalar eqn.] Thus need at least 8 points.

What is 8 point algorithm used for?

The eight-point algorithm is an algorithm used in computer vision to estimate the essential matrix or the fundamental matrix related to a stereo camera pair from a set of corresponding image points. It was introduced by Christopher Longuet-Higgins in 1981 for the case of the essential matrix.

Why are 4 points needed for homography?

In fact, we need at least 4 points, because the homography matrix has 8 free variables (each point contains and x and a y, for a total of 8 equations). To make the homography more stable, we can utilize more points and derive the least squares equation to solve the overdetermined system.