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Section 7.2 Binocular Reconstruction 201

T

is also clear that l = E p is the coordinate vector representing the epipolar line l
associated with p in the second image. Essential matrices are singular because t is
T
T
parallel to the coordinate vector e of the first epipole, so that E e = −R [t × ]e =0.
Likewise, it is easy to show that e is in the nullspace of E.As shown by Huang

and Faugeras (1989), essential matrices are in fact characterized by the fact that
they are singular with two equal nonzero singular values (see the problems).
7.1.3 The Fundamental Matrix
The Longuet–Higgins relation holds in normalized image coordinates. In native

image coordinates, we can write p = Kˆ p and p = K ˆ p ,where K and K are the

3 × 3 calibration matrices associated with the two cameras. The Longuet–Higgins
relation holds for these vectors, and we obtain
T
p Fp =0, (7.3)

−T −1
where the matrix F = K EK ,called the fundamental matrix,isnot,ingeneral,
T
an essential matrix. It has again rank two, and the eigenvector of F (resp. F ) cor-
responding to its zero eigenvalue is as before the position e (resp. e) of the epipole.

T

Likewise, l = Fp (resp. l = F p) represents the epipolar line corresponding to
the point p (resp. p) in the first (resp. second) image.

The matrices E and F can readily be computed from the intrinsic and extrinsic
parameters. Let us close this section by noting that Equations (7.2) and (7.3) also
provide constraints on the entries of these matrices, irrespective of the 3D position
of the observed points. In particular, this suggests that E and F can be computed
from a sufficient number of image correspondences without the use of a calibration
chart. We will come back to this issue in Chapter 8. For the time being, we will
assume that the cameras are calibrated and that the epipolar geometry is known.
7.2 BINOCULAR RECONSTRUCTION
Given a calibrated stereo rig and two matching image points p and p ,itisinprin-

ciple straightforward to reconstruct the corresponding scene point by intersecting
the two rays R = Op and R = O p (Figure 7.2). However, the rays R and R never

actually intersect in practice, due to calibration and feature localization errors. In
this context, various reasonable approaches to the reconstruction problem can be
adopted. For example, consider the line segment perpendicular to R and R that

intersects both rays (Figure 7.5): its mid-point P is the closest point to the two

rays and can be taken as the preimage of p and p .
Alternatively, one can reconstruct a scene point using a purely algebraic ap-

proach: given the projection matrices M and M and the matching points p and

p , we can rewrite the constraints Zp = MP and Z p = MP as

p ×MP =0 [p ]M
⇐⇒ × P =0.
p ×M P =0 [p ]M

×
This is an overconstrained system of four independent linear equations in the ho-
mogeneous coordinates of P that is easily solved using the linear least-squares tech-
niques introduced in Chapter 22. Unlike the previous approach, this reconstruction