PROBABILISTIC ANALYSIS OF THE APPLICATION OF THE CROSS RATIO TO MODEL-BASED VISION

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 whic h are closely linked to the six different values of the cross ratio ob tained by permuting the quadruple of points from which the cross ratio is calculated. The density function has logarithmic singularities cor responding to values of the cross ratio for which two of the four poin ts are coincident. The cross ratio forms the basis of a simple system for recognising or classifying quadruples of collinear image points. T he performance of the system depends on the choice of rule for decidin g whether four image points have a given cross ratio sigma. A rule is stated which is computationally straightforward and which takes into a ccount the effects on the cross ratio of small errors in locating the image points. Two key properties of the rule are the probability R of rejection, and the probability F of a false alarm. The probabilities R and F depend on a threshold t in the decision rule, There is a trade off between R and F obtained by varying t. It is shown that the trade off is insensitive to the given cross ratio sigma. Let F-w = max(sigma ){F}. Then R, F-w are related approximately by root ln(R(-1)) = (root 2 epsilon r(F))F--1(w), provided epsilon-F-1(w) greater than or equal to 4. In the equation, epsilon is the accuracy with which image points can be located relative to the width of the image, and r(F) is a cons tant known as the normalised false alarm rate. In the range epsilon(-1 )F(w) less than or equal to 4 the probabilities R and F-w are related approximately by R = 1 - root 2 pi(-1)epsilon(-1)r(F)(-1)F(w). The val ue of r(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 t he trade off between R and F-w holds for a wide class of scalar invari ants 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 mo del based vision.