### BMVC 2004, Kingston, 7th-9th Sept, 2004

On the Probabilistic Epipolar Geometry

S. S. Brandt (Helsinki University of Technology, Finland)

In this paper, we are going to answer the following question: assuming
that

we have estimates for the epipolar geometry and its uncertainty between
two

views, how probable it is that a new, independent point pair will
satisfy the

true epipolar geometry and be, in this sense, a feasible candidate
correspondence

pair? If we knew the true fundamental matrix, the answer would be

trivial but in reality it is not because of estimation errors. So, as a
point in the

first view is given, we will show that we may compute a probability
density

for the feasible correspondence locations in the second view that
describes

the current level of knowledge of the epipolar geometry between the
views.

We will thus have a point-probability-density relation which can be
understood

as a probabilistic form of the epipolar constraint; it also approaches
the

true point-line relation as the number of training correspondences
tends to in-

finity. We will also show that the eigenvectors of the epipolar line
covariance

matrix have certain interpretations on the image plane, of which one is
the

previously observed, narrowest point of the epipolar envelope. The
results

of this paper are novel and important since the uncertainty of the
epipolar

constraint can be now taken into account in a sound way in applications.

(pdf article)