Zisserman - 3 
X 
Figure 1: Epipolar Geometry. A 3D point X is imaged at x and x 7 in the left 
and right images respectively. The two optical centres, O and O', and X define an 
epipolar plane , II, for X. The epipolar plane contains the rays joining X to each 
optical centre, and the baseline joining the optical centres. The baseline intersects 
the left and right image planes at the epipoles e and e 7 respectively. Given the point 
x, the corresponding point in the right image must lie on the epipolar line 1' which 
is the image of the ray from the optical centre O through x. 
2.1 Epipolar Geometry 
The epipolar constraint is the fundamental geometric constraint existing between 
two images with non-coincident optical centres. It arises from the epipolar geometry 
between two cameras illustrated in figure 1. The primary use of epipolar geometry 
is in providing a disambiguating constraint for correspondences between images 
planes—given a point in one image the corresponding point in the other image is 
constrained to lie on a straight line. The essence of epipolar geometry is that it 
depends only on the relative location of the optical centres and image planes; there 
is no dependence on extrinsic 3D structure. 
Conventionally, epipolar geometry is computed from the interior orientation and 
relative exterior orientation of the cameras. But this is not necessary - it can be 
computed solely from point matches between the two images. 
2.1.1 The fundamental matrix 
The fundamental matrix F is the algebraic representation of epipolar geometry, its 
computation requires no knowledge of calibration (neither interior or exterior). The 
matrix has the following properties: 
• F is a 3 x 3 matrix of maximum rank 2.