3 Computing the epipole

We now show how Equation ( 5 ) leads to a simple method for estimating the epipole's location.

If are the coordinates of the epipole, it is well known [ 6 , 10 ] that, at any image point ( x , y ), the dot product between the motion field and the vector , or

does not depend on the 3-D structure. This important property is at the basis of several methods for the reconstruction of 3-D motion and structure from calibrated optical flow (see [ 5 , 6 , 10 ] for example).

In our notation, the dot product above takes the form

with

Equation ( 7 ) suggests immediately a scheme for estimating the location of the epipole, based on the estimates of the motion field at six or more image points. For each image point, call it , we write six or more instances of Equation ( 7 ), solve the resulting system by least squares for the five unknowns , and compute the associated residual, . The epipole, being the point at which R =0, can be located by finding the minimum of the surface , where and span ideally the whole image plane, using a standard optimization procedure.

In the next section we present some preliminary experimental results obtained with a simple gradient descent minimization to locate the minimum of the surface of residuals; before that, however, it is useful to address two important questions.

1. Is the epipole the only absolute minimum of the surface of residuals?
2. Does this surface have local minima?

Both questions are open at the time of writing. As to the first one, it seems that six flow estimates are enough to generate residuals different from zero away from the epipole, and therefore to locate the epipole (where the residual is zero) by minimization. Our experiments with synthetic flows seem to support this conjecture, unless of course the 3-D points form degenerate configurations: for example, if all the points lie on a plane, Equation ( 5 ) is equally satisfied by any . In other words, the observed optical flow is compatible with an epipole located arbitrarily on the image plane. Notice however that the minimum number of points has very little practical relevance, as many more than six flow estimates are available from any optical flow.

As to the second question, the experimental evidence to date is that the method never got stuck in local minima, in any sequence tried. However, we do not yet have an analytic proof of the presence or absence of local minima, neither globally nor in a region surrounding the epipole.