To analyse the 3D surface geometry from the changes in 2D image curves (apparent contours), we must identify the correspondences between the successive apparent contours and 3D surfaces. To do this, the epipolar geometry is useful.
Consider an instantaneous motion of a camera, so that the view point
at time
moves to
at time
. Suppose an apparent contour,
, in an image at time
corresponds to a contour generator,
, on the surface. The two projection centers,
and
, define a family of epipolar planes,
. Then, the contour generators,
and
, at time
and
cross over an epipolar plane,
, at
and
respectively. Since
and
are on the same epipolar plane,
, their projections,
and
, are on the corresponding epipolar lines in images. In the
infinitesimal limit, this provides a natural spatio-temporal
parameterisation of the image and contour generators. This
parameterisation of curved surfaces and image sequence with respect to
and
parameters is called the
epipolar parameterisation
[
5
], and the trajectory of a surface point,
, with a fixed
parameter is called the
epipolar curve
. Since it enables us to identify correspondences between the changes in
apparent contours and the changes in contour generators, the epipolar
parameterisation is very useful for recovering the surface geometry from
apparent contours. The following analyses are based on the epipolar
parameterisation.
J. Sato