---
title: Eight-point algorithm (Lecture)
categories: lecture
---

# 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: $(\cdot)^s$

$$(\mathbf{x}_1\bigotimes\mathbf{x}_2)^TE^s = 0$$

## 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.