STOCHASTIC PROPERTIES OF THE CROSS RATIO

In computer vision the cross ratio is the basis of many recognition al gorithms which do not require estimates of the position and orientatio n of the camera. The cross ratio is also employed in vehicle navigatio n, in which it is necessary to identify artificial landmarks or beacon s placed in the environment. A stochastic model is constructed in orde r to assess the false alarm rate in a recognition system based on the cross ratio. The evolution of a quadruple of measurements is modelled by a Brownian motion in R(4). Let tau(sigma), be the first time at whi ch the Brownian motion hits the set of measurement vectors with cross ratio sigma. It is shown that P(tau(sigma) < infinity) = 1, and bounds are obtained for the exponent a/2 for which 1 - P(tau(sigma) less tha n or equal to t) = O(t(-a/2)).