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Eight-point algorithm (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.