Essential Matrix - Derivation and Definition

Derivation and Definition

This derivation follows the paper by Longuet-Higgins.

Two normalized cameras project the 3D world onto their respective image planes. Let the 3D coordinates of a point P be and relative to each camera's coordinate system. Since the cameras are normalized, the corresponding image coordinates are

\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \frac{1}{x_3} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} and

A homogeneous representation of the two image coordinates is then given by

and

which also can be written more compactly as

\mathbf{y} = \frac{1}{x_{3}} \, \tilde{\mathbf{x}} and \mathbf{y}' = \frac{1}{x'_{3}} \, \tilde{\mathbf{x}}'

where and are homogeneous representations of the 2D image coordinates and and are proper 3D coordinates but in two different coordinate systems.

Another consequence of the normalized cameras is that their respective coordinate systems are related by means of a translation and rotation. This implies that the two sets of 3D coordinates are related as

where is a rotation matrix and is a 3-dimensional translation vector.

Define the essential matrix as

where is the matrix representation of the cross product with .

To see that this definition of the essential matrix describes a constraint on corresponding image coordinates multiply from left and right with the 3D coordinates of point P in the two different coordinate systems:

(\tilde{\mathbf{x}}')^{T} \, \mathbf{E} \, \tilde{\mathbf{x}} \, \stackrel{(1)}{=} \,(\tilde{\mathbf{x}} - \mathbf{t})^{T} \, \mathbf{R}^{T} \, \mathbf{R} \, _{\times} \, \tilde{\mathbf{x}} \, \stackrel{(2)}{=} \, (\tilde{\mathbf{x}} - \mathbf{t})^{T} \, _{\times} \, \tilde{\mathbf{x}} \, \stackrel{(3)}{=} \, 0
  1. Insert the above relations between and and the definition of in terms of and .
  2. since is a rotation matrix.
  3. Properties of the matrix representation of the cross product.

Finally, it can be assumed that both and are > 0, otherwise they are not visible in both cameras. This gives

0 = (\tilde{\mathbf{x}}')^{T} \, \mathbf{E} \, \tilde{\mathbf{x}} = \frac{1}{x'_{3}} (\tilde{\mathbf{x}}')^{T} \, \mathbf{E} \, \frac{1}{x_{3}} \tilde{\mathbf{x}} = (\mathbf{y}')^{T} \, \mathbf{E} \, \mathbf{y}

which is the constraint that the essential matrix defines between corresponding image points.

Read more about this topic:  Essential Matrix

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