Essential Matrix
An essential matrix, denoted \(\displaystyle \mathbf{E}\), is a \(\displaystyle 3 \times 3\) matrix relating camera parameters.
You can compute the essential matrix based on features matches between two images.
Using the essential matrix, you can extract the relative rotation and translation between two cameras.
Given feature points \(\displaystyle \mathbf{x}\) and \(\displaystyle \mathbf{x'}\) from two images, the essential matrix satisfies the equation \(\displaystyle \mathbf{x}'^T \mathbf{E} \mathbf{x} = 0\)
Properties
- A \(\displaystyle 3 \times 3\) matrix is an essential matrix iff two of its singular values are equal and the third value is \(\displaystyle 0\)
Proof
See Bartoli and Olsen[1].
Calculating the Essential Matrix from two images
Determining rotation \(\displaystyle R\) and translation \(\displaystyle t\)
3D points
See Wikipedia: Essential_matrix