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

# Eight-Point Algorithm

Input
: At least eight pairs of points $(\mathbf{x}_1,\mathbf{x}_2)$ (from stereo view).

Output
: The essential matrix $E$ (or rather an estimate)

Main Principle
: **Epipolar Constraint** $\mathbf{x}_2^TE\mathbf{x}_1 = 0$

## Outline

+ The epipolar constraint is a linear equation, but it is written
  in an unusual form, with the unknown $E$ in the middle.
    + We know $\mathbf{x}_i$, and want to solve for $E$.
    + **Note** if $E$ is a solution, then $c\cdot E$ is a solution
      for any $c\in\mathbb{R}$
+ Up to nine unknowns in the $3\times3$ matrix $E$, but one is the
  indeterminable $c$.
    + Scaling up all distances by the same factor gives the same solution
+ Step 1. Rewrite the constraint in a more regular format.
+ Step 2. Formulate a system of linear equations using eight pairs of points.
    + Eight pairs of points give eight equations

## Kronecker product


+ **Serialisation** of a matrix:
  $(\cdot)^s$ maps a $3\times 3$ matrix into a 9-dimensional vector.  
    + You can scan row by row or column by column; and seriously, 
      Matlab does it one way and Python (numpy) the other.
    + Numpy method: `.flatten()`
    + You need to be consistent, and it is worth running a test to
      see what your system does.
+ **Kronecker product**: $\bigotimes$
    + $\vec{a}\bigotimes \vec{b} = (\vec{a}\vec{b}^\mathrm{T})^s$
+ Rewrite the **epipolar constraint**:
  $$(\mathbf{x}_1\bigotimes\mathbf{x}_2)^TE^s = 0$$
    + Note: one equation in nine unknowns,

## Preparing for the eight-point algorithm

+ Each point of pairs gives rise to a nine-dimensional vector:
  $$\mathbf{a} = \mathbf{x}_1\bigotimes\mathbf{x}_2$$
    + view it as a column
+ The epipolar constraint is $\mathbf{a}^\mathrm{T}\cdot E^s = 0$.
+ This gives a matrix equation $\chi\cdot E^s=0$ where
  $$\chi = [\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n]^T$$
+ We can solve $\chi E^s = 0$ for $E^s$.
    + With eight points, we have unique solutions up to a scalar factor.
+ With additional points, we can minimise the squared errors
    + $\min_E ||\chi E^s||^2$
    + i.e. eigenvector of $\chi^T\chi$ of smallest eigenvalue
    + (this is a general approach to solve linear systems of equations *with noise*)
+ The scale ambiguity is intrinsic.
    + The distance $||T||$ cannot be determined from the images

## 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.

+ Write $F$ for the solution of $\chi E^s=0$.
  This is called the **fundamental matrix**
+ Write $F = U\Sigma_FV^T$ for the 
<details>
  <summary>singular value decomposition </summary>
    + $\Sigma_F$ is a diagonal matrix
    + $U$ and $V$ are orthogonal matrices (assuming they are real)
    + canonical decomposition with the entries on $\Sigma_F$ in descending order
    + use an API to compute
</details>
+ Use $E=U\Sigma V^T$ for the essential matrix, where
    + $\Sigma=\mathsf{diag}(1,1,0)$

## Recover the transform from the essential matrix

$$
\begin{cases}
  (\hat T_1,R_1) &=
  (UR_Z(+\frac\pi2)\Sigma U^T, UR_Z(+\frac\pi2)V^T)
\\
  (\hat T_2,R_2) &=
  (UR_Z(-\frac\pi2)\Sigma U^T, UR_Z(-\frac\pi2)V^T)
\end{cases}
$$
where
$$R_Z(+\frac\pi2) = 
\begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}
$$
is a rotation by $\pi/2$ radians around the $z$-axis.

+ **Four solutions**
    + Two solutions $\pm F$ to the equation $\chi E^s=0$
    + Two poses from each candidate matrix
    + Only one of the four solutions is in front of both cammeras

## Notes

### Number of points.

+ $(R,T)$ only has five degrees of freedom when $T$
  is determined up to a scalar factor.
+ thus five points suffices in theory
+ $\mathsf{det}(E)=0$ removes one degree of freedom
+ linear algorithm exists for six points

### Requirements

**General position**
We have to assume linear independence in the equation
$\chi E^s=0$

**Sufficient parallax**
If $T=0$, then $E=0$, hence the algorithm requires
$$T\neq0$$
The algorithm may work even if $T=0$, due to noise, but the
resulting estimate for $T$ (direction) would be meaningless.

### Variants

+ Infinitesimal viewpoint change, see Section 5.4
+ Multiple motion, see Chapter 7

### Scale ambiguity (Ma (2004) Section 5.2.2)

+ Depth scales $\lambda_1,\lambda_2$ (for each object point)
+ Motion scale $\gamma=||T||$

$$\lambda_2^j\mathbf{x}_2^j = \lambda_1^jR\mathbf{x}_2^j + \gamma T$$ 

+ Recall that the scale ambiguity is intrinsic
+ $\gamma$ cannot be determined from the images

# Optimal pose in the presence of noise

**Reading** Ma (2004) Section 5.2.3

+ Noise is everywhere - Two problems:
    + error in $(R,T)$
    + no consistent 3D reconstruction from noisy image points
+ The epipolar constraint assumes ideal co-ordinates
    - but applied to noisy measurements
+ **Two approaches**
    1.  Maximum likelihood 
    2.  Minimal sum of squared errors
+ We only discuss option 2.

## Optimal Pose

Consider an observed point
$\tilde{\textbf{x}}^j=(\tilde x^j_1, \tilde x^j_2, \tilde x^j_3)$ and
an unknown true point $\textbf{x}^j=( x^j_1,  x^j_2,  x^j_3)$.
The corresponding 3D point is $\textbf{X}^j$.

+ **Observations** $\tilde x_i^j$
+ **Independent variables** $x_i^j,R,T$
+ **Sum of squared errors**
  $$\phi(\mathrm{X}^j,R,T) = 
    \sum_j ||w_1^j||^2 + ||w_2^j||^2 =
    \sum_j|| \tilde x_1^j-x_1^j||^2 + ||\tilde x_2^j-x_2^j||^2 =
    \sum_j|| \tilde x_1^j-\pi_1(\textbf{X}^j)||^2 + ||\tilde x_2^j-\pi_2(\textbf{X}^j)||^2$$

**Task** 
$$\min_{\textbf{X}^j}\phi(\mathrm{X}^j,R,T) = \sum_j\sum_{i\in\{1,2\}} ||w_i^j||^2$$ 
under the conditions that
$$ x_2^j\hat TR x_1^j = 0$$
and
$$x_1^{jT}e_3 = 1\quad\text{for } i=1,2, j=1,2,\ldots,n$$

+ Lagrangian optimisation 

**Goal** find image point pairs which minimise the reprojection error
for that particular object point. (Structure Triangulation)