As shown in some previous work [ 4 ], time-to-contact can provide a useful visual cue for navigating robots. It has been shown that if the object is planar, the time-to-contact can be computed from the first order derivatives of image flow, i.e. image divergence and deformation [ 4 ]. Unfortunately, these analyses are limited to fixed features on planar surfaces. In this section, we show that it is also possible to compute time-to-contact to curved surfaces from apparent contours .
From (
5
) and since
, we find that the time-to-contact,
, to the frontier point on a surface is computed simply from
as follows:
Thus,
to frontier points is computed from the
first
derivatives of image curves with respect to time,
. Furthermore, from (
8
) and (
9
), we find that
to non-frontier points on a surface can be computed from the
second
derivatives of image curves with respect to time,
, as follows:
Note, since
cancels out, the time-to-contact to curve surfaces can be computed
uniquely just from their apparent contours in images. The
time-to-contact does not depend on camera calibration either.
J. Sato
