In this section, we discuss the recovery of epipolar geometry from apparent contours viewed from uncalibrated cameras with unknown translational motions.
Figure 2:
Epipolar geometry under pure translations. If the motion of the camera
is a pure translation, The epipole,
, and the epipolar lines,
, in images from two viewpoints coincide as sown in (b). In this case,
the bi-tangent lines coincide with epipolar lines,
, and the bi-tangent points coincide with the projection of frontier
points,
, in images.
Consider an uncalibrated camera, i.e. a camera whose calibration matrix
is unknown. It is known that if the camera motion is pure translation,
then the corresponding points observed from two different viewpoints are
auto-epipolar
[
12
], i.e. the epipoles and epipolar lines before and after the camera
motion coincide. Since the frontier points on contour generators can be
regarded as fixed features, the above fact can be extended for the
apparent contours viewed from the same but uncalibrated cameras. Suppose
and
are the projections of a frontier point,
, in two views (i.e. before and after a translation) as shown in Fig.
2
:
Since the projections of a frontier point are observed as a tangent point of an epipolar line to the apparent contour in the image, we have the following properties:
Thus, if the camera motions are limited to translation, epipolar lines
and epipoles can be computed uniquely from the bi-tangency of apparent
contours. We, in the next section, use the extracted epipole,
, and frontier points for reconstruction.
J. Sato