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title: Eight-point algorithm (Lecture)
categories: lecture
---
# Eight-Point Algorithm
$$\mathbf{x}_2^TE\mathbf{x}_1 = 0$$
Kronecker product: $\bigotimes$
Serialisation of a matrix: $(\cdot)^s$
$$(\mathbf{x}_1\bigotimes\mathbf{x}_2)^TE^s = 0$$
$$\mathbf{a} = \mathbf{x}_1\bigotimes\mathbf{x}_2$$
$$\chi = [\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n]$$
We can solve $\chi E^s = 0$ for $E^s$.
With eight points, we have unique solutions up to a scalar factor.
The solution is not necessarily a valid essential matrix, but we can
project onto the space of such matrices and correct the sign to
get positive determinant.