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title: Eight-point algorithm (Lecture)
categories: lecture
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# 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
Kronecker product: $\bigotimes$
Serialisation of a matrix: $(\cdot)^s$
$$(\mathbf{x}_1\bigotimes\mathbf{x}_2)^TE^s = 0$$
## Preparing for the eight-point algorithm
$$\mathbf{a} = \mathbf{x}_1\bigotimes\mathbf{x}_2$$
$$\chi = [\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n]$$
We can solve $\chi E^s = 0$ for $E^s$.
With eight points, we have unique solutions up to a scalar factor.
## Projection onto the essential space
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.