Just have a bit of confusion with understanding the 7-point algorithm for calculating the Fundamental Matrix.
When reading notes, I see that the 7-point algorithm uses 7 point correspondences in the matrix A to calculate the fundamental matrix F. As a result, the notes mentions that the null space is now 2-dimensional. Why is that? Why is the null space two dimensional if 7 points are used?
Also, "note that the singularity constraint enforces $$\det F = 0$$ thus $$\det\left(F = \alpha F_1 +(1-\alpha)F_2\right) = 0$$. This gives a cubic polynomial in α from which we can solve for $α$."
Why does that determinant generate a cubic polynomial in alpha? I'm just confused how the determinant is able to somehow produce that cubic polynomial.
Source: https://staff.fnwi.uva.nl/l.dorst/hz/chap11_13.pdf (slide 17)
I don't know the "fundamental matrix" but in the attached paper, one can see that if 7 points are used, the matrix $A$ has size 7x9. It means that if its rank is maximal, the null space indeed has dimension 9-7=2. I suppose one can ensure a maximal rank by a clever choice of the points.
The paper doesn't give a formula for $F$ but if its size is 3x3, then the determinant $\det(\alpha F_1 + (1 - \alpha) F_2)$ is indeed a cubic polynomial in general.
You forget to mention that the site in question is about a book named Geometric algebra for computer science by Leo Dorst and al