Once the epipolar geometry is recovered, we can apply it to computing the time-to-contact to curved surfaces.
Fig. 4 (a) shows five sequential images of a head observed from a translating camera, and their contour curves extracted by B-spline fitting. The frontier points and the epipolar lines were extracted by using the epipolar bitangency, and are shown in Fig. 4 (a). The time-to-contact to the frontier point (upper frontier point) has been computed, and is shown in Fig. 4 (b) by small circles and solid lines. The exact time-to-contact is shown by dotted lines in (b). As shown in this figure, the computed time-to-contact is quite accurate and reliable.
Fig. 5 shows results from other examples. Fig. 5 (a) and (b) are two sequential images from the Star Wars film. The motion of the spaceship is considered as a pure translation. The epipolar lines and frontier points were computed from the bitangency and are shown by the dashed lines and circles in (c). The intersection point of the dashed lines is the epipole. The time-to-contact to the left and the right frontier points on the spaceship has been computed from the extracted epipole and frontier points, and is shown in (g). (minus means that the point is going away from the observer). Fig. 5 (d), (e) and (f) show results from the 2001 film.
Since the spaceships in Fig. 5 are curved surfaces, we cannot compute the exact time-to-contact just from the changes in area of apparent contours. However, as we have seen, the exact time-to-contact is computed efficiently by using the epipolar bitangency.
Figure 4: Results of computing the time-to-contact.
(a) shows five sequential images of a head taken from a
translational camera. Extracted contour curves are shown by
black lines. (b) shows the computed time-to-contact to the
head (i.e. curved surface).
Figure 5: Results of the Computation of Epipolar Geometry and
the time-to-contact. (a) and (b) show two sequential images of
a spaceship extracted from the Star Wars movie. (c) show
the epipolar geometry recovered from the bitangency.
The time-to-contact to the spaceship has been computed
from the frontier points and the epipole, and is shown in
(g). (d), (e) and (f) show another example extracted from
the 2001 movie.