Who's afraid of the cross-ratio ?

THE CROSS-RATIO The cross-ratio remains invariant under perspective projection. It is defined for 4 colinear points as follows: with AB being the distance between point A and B, etc.

Hence, irrespective of a camera's viewpoint with respect to these 4 points, the cross-ratio will be the same. This is not a new result of course. VIVA is about assessing the practicality of old and mainly novel invariants for a wide variety of shapes.

The effectiveness of the cross ratio in model based vision is being assessed within VIVA. The cross ratio is chosen because it is the simplest and most fundamental projective invariant. Each object Os is a set of four colinear points with a given cross ratio s. The database of models is a list of cross ratio values. An object Os is detected by finding in an image four colinear points with a cross ratio s. The measured cross ratio can be used as an index into the database because the cross ratio is unchanged under projection to an image.

The performance of the detection algorithm is characterised by the probability of rejection (R), the probability of misclassification (M), and the probability of a false alarm (F). There are trade offs between R, M and F obtained by varying a threshold t in the detection algorithm. A theoretical description of the trade offs has been obtained. The calculations give the best possible values of F and M compatible with a given value of R. In addition, it has been shown that for a fixed overall probability P of misclassification the maximum number of models is finite. The result is quantitative. For example, if P = 0.1 then the maximum number of models, at a typical level of image noise, is twelve. Steve Maybank (GEC) and Paul Beardsley (Oxford)