The application of projective geometry techniques in computer vision
is most notable in the *Stereo Vision* problem which is very
closely related to Structure-from-Motion. Unlike general motion,
stereo vision assumes that there are only two shots of the
scene.
In principle, then, one could apply stereo vision algorithms to a
structure from motion task.

Applying projective geometry to stereo vision is not new and can be traced back from 19th century photogrammetry to work in the late sixties by Thompson [54]. However, interest in the subject was recently rekindled in the computer vision community thanks to important works in projective invariants and reconstruction by Faugeras [16] and Hartley [26].

Figure 4 depicts the imaging situation for
stereo vision. The application of projective geometry to this
situation results in the now popular epipolar geometry approach. The
three points
**[ COP_{1},COP_{2},P]** form what is called an

Here, ** F** is the so-called

Hartley proposes an elegant technique for recovering the parameters of
the fundamental matrix when at least 8 points are observed
[24]. Expanding the expression in
Equation 3 gives one linear constraint on
** F** per observed point as in
Equation 4. Combining

Typically, one solves such a linear system using more than 8 points in
a least squares minimization
subject to the
constraint .
This constraint fixes the scale of the fundamental
which otherwise is arbitrary. In addition, the rank 2 constraint must
also be enforced. The algorithm employed utilizes an SVD computation
but can be quite unstable. One way to alleviate this numerical
ill-conditioning is to normalize pixel coordinates to span
**[-1,1]**. For robust fundamental matrix estimation techniques, refer
to [63].

The fundamental matrix ** F** is recovered independently of the structure
and can be useful on its own, for example in a robotics application
[16]. Hartley also uses it to derive Kruppa
equations for recovering camera internal parameters
[41] [25]. Ultimately, it becomes
possible to recover Euclidean 3D coordinates for the structure which
are often desirable for most typical application purposes.

At this point it is worthwhile to study the stability of such
techniques. The reader should consider the case where the centers of
projection of both images are close to each other (** COP_{1}** and

The linear epipolar geometry formulation also exhibits sensitivity to noise (i.e. in the 2D image measurements) when compared to nonlinear modeling approaches. One reason is that each point can be corresponded to any point along the epipolar line in the other image. Thus, the noise properties in the image are not isotropic with noise along the epipolar line remaining completely unpenalized. Thus, solutions tend to produce high residual errors along the epipolar lines and poor reconstruction. Experimental verification of this can be found in [3].