Time to collision from first-order spherical image motion

In the absence of constraints on either object motion or surface slant, a narrow field of view constraint has to be assumed to compute time to collision as a scaled depth from first-order image motion measurements. In this work, time to collision and scaled depth are regarded as different visual entities, and it is shown that a bound for time to collision can always be computed regardless of the field of view, thus extending the range of applicability of time to collision based techniques in areas such as mobile robotics and visual surveillance. The method relies on computing in closed form the spherical motion field and the associated parallax from image plane measurements obtained with either conventional cameras or space-variant sensors. An experimental validation of the main theoretical results highlights the difference between time to collision and scaled depth, and addresses a comparison of time to collision approaches using both dense and sparse motion estimates.