Probabilistic analysis of the application of the cross ratio to model based vision

Abstract

The probability density function for the cross ratio is obtained under the hypothesis that the four image points have independent, identical, Gaussian distributions. The density function has six symmetries which are closely linked to the six different values of the cross ratio obtained by permuting the quadruple of points from which the cross ratio is calculated. The density function has logarithmic singularities corresponding to values of the cross ratio for which two of the four points are coincident. The cross ratio forms the basis of a simple system for recognising or classifying quadruples of collinear image points. The performance of the system depends on the choice of rule for deciding whether four image points have a given cross ratio σ. A rule is stated which is computationally straightforward and which takes into account the effects on the cross ratio of small errors in locating the image points. Two key properties of the rule are the probabilityR of rejection, and the probabilityF of a false alarm. The probabilitiesR andF depend on a thresholdt in the decision rule. There is a trade off betweenR andF obtained by varyingt. It is shown that the trade off is insensitive to the given cross ratio σ. LetF _w =max _o {F}. ThenR, F _w are related approximately by$\sqrt {\ln (R^{ - 1} )} = (\sqrt 2 \varepsilon r_F )^{ - 1} F_w$, provided ε⁻¹ F _w ≥4. In the equation, ε is the accuracy with which image points can be located relative to the width of the image, andr _F is a constant known as the normalised false alarm rate. In the range ε⁻¹ F _w ≤4 the probabilitiesR andF _w are related approximately by$R = 1 - \sqrt {2\pi ^{ - 1} } \varepsilon ^{ - 1} r_F^{ - 1} F_w$. The value ofr _F is 14.37. The consequences of these relations between R and F_w are discussed. It is conjectured that the above general form of the trade off betweenR andF _w holds for a wide class of scalar invariants that could be used for model based object recognition. Invariants with the same type of trade off between the probability of rejection and the probability of false alarm are said to be nondegenerate for model based vision.