Generalized bounds for time to collision from first-order image motion

This paper addresses the problem of estimating time to collision from focal motion field measurements in the case of unconstrained relative rigid motion and surface orientation. It is first observed that, as long as time to collision is regarded as a scaled depth, the above problem does not admit a solution unless a narrow camera field of view is assumed. By a careful generalization of the time to collision concept, it is then expounded how to compute novel solutions which hold however wide the field of view. The formulation, which reduces to known literature approaches in the narrow field of view case, extends the applicability range of time to collision based techniques in areas such as mobile robotics and visual surveillance. The experimental validation of the main theoretical results includes a comparison of narrow- and wide-field of view time to collision approaches using both dense and sparse motion estimates