Eight-Point Algorithm

Epipolar Constraint \[\mathbf{x}_2^TE\mathbf{x}_1 = 0\]

• we know \(\mathbf{x}_i\), and want to solve for \(E\).
• up to nine unknowns
• need eight pairs of points to solve uniquely up to a scalar factor
• the scalar factor cannot be avoided

Kronecker product

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.