Should we use all $N$ correspondence points to compute Homography or just $4$ points?

Question

Given correspondences of $(x,x')$, we want to find a Homography matrix $H$ that maps $x' = Hx$. If we use 4 corresponding non - collinear points and the matrix $H$ has rank $n-1$, then the homogenous equation $Ah =0$ has a unique solution up to a constant. Suppose we find more than $4$ corresponding pairs of points, we get an overdetermined system of equation and we have to use least squares minimisation to find the solution.

I am curious as to whether it is better to just use $4$ points or $>4$ points to find the Homography ?

Also, it makes sense that from a linear algebra perspective, I am unable to solve the homogenous equation exactly when we have an overdetermined system. However, from a planar perspective, given my correspondence points are good, shouldn't there be an exact solution ?

Answer 1

It's generally better to use all the points you have, since often, errors are independent per point.

given my correspondence points are good, shouldn't there be an exact solution

Sure, but in realistic situations, it's not usually the case and/or you don't know that the points are good / which points are good.

A typical approach is to use all the points you have, but reject "bad" points using outlier-rejectiong (e.g. RANSAC ) the MIT 6.869 - Advances in Computer Vision - Lecture 13 - Homographies and RANSAC slides have an example of such a procedure.