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

The cross ratio of four colinear points is of fundamental importance i n model based vision, because it is the simplest numerical property of an object that is invariant under projection to an image. It provides a basis for algorithms to recognise objects from images without first estimating the position and orientation of the camera. A quantitative analysis of the effectiveness of the cross ratio in model based visio n is made. A given image I of four colinear points is classified by ma king comparisons between the measured cross ratio tau of the four imag e points and the cross ratios stored in the model database. The image I is accepted as a projection of an object O-sigma with cross ratio a if \tau - sigma\ less than or equal to ntu, where n is the standard de viation of the image noise, t is a threshold and u = \del tau\. The pe rformance of the cross ratio is described quantitatively by the probab ility of rejection R, the probability of false alarm F and the probabi lity of misclassification p(sigma)(sigma), defined for two model cross ratios sigma, sigma. The trade off between these different probabilit ies is determined by t. It is assumed that in the absence of an object the image points have identical Gaussian distributions, and that in t he presence of an object the image points have the appropriate conditi onal densities. The measurements of the image points are subject to sm all random Gaussian perturbations. Under these assumptions the trade o ffs between R, F and p(sigma)(sigma) are given to a good approximation by R = 2(1 - Phi(t)), F = r(F) epsilon t, root p(sigma)(sigma) = e(si gma)epsilon t\sigma - sigma\(-1), where epsilon is the relative noise level, Phi is the cumulative distribution function for the normal dist ribution, r(F) is constant, and e(sigma) is a function of a only. The trade off between R and F is obtained in Maybank (1994). In this paper the trade off between R and p(sigma)(S) is obtained. It is conjecture d that the general form of the above trade offs between R, F and p(sig ma)(sigma) is the same for a range of invariants useful in model based vision. The conjecture prompts the following definition: an invariant which has trade offs between R, F, p(sigma)(sigma) of the above form is said to be non-degenerate for model based vision. The consequences of the trade off between R and p(sigma)(sigma) are examined. In partic ular, it is shown that for a fixed overall probability of misclassific ation there is a maximum possible model cross ratio sigma(m), and ther e is a maximum possible number N of models. Approximate expressions fo r sigma(m) and N are obtained. They indicate that in practice a model database containing only cross ratio values can have a size of order a t most ten, for a physically plausible level of image noise, and for a probability of misclassification of the order 0.1.