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

Abstract

The cross ratio of four colinear points is of fundamental importance in 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 vision is made. A given imageI of four colinear points is classified by making comparisons between the measured cross ratio τ of the four image points and the cross ratios stored in the model database. The imageI is accepted as a projection of an objectO _σ with cross ratio σ if |τ−σ|≤ntu, wheren is the standard deviation of the image noise,t is a threshold andu=∥∇τ∥. The performance of the cross ratio is described quantitatively by the probability of rejectionR, the probability of false alarmF and the probability of misclassificationp _σ(ς), defined for two model cross ratios σ, ς. The trade off between these different probabilities is determined byt.

It is assumed that in the absence of an object the image points have identical Gaussian distributions, and that in the presence of an object the image points have the appropriate conditional densities. The measurements of the image points are subject to small random Gaussian perturbations. Under these assumptions the trade offs betweenR,F andp _σ(ς) are given to a good approximation byR=2(1−Ф(t)),F=r _F ∈t,$\sqrt {p_\sigma (\varsigma )} = e_\sigma \in t\left| {\sigma - \varsigma } \right|^{ - 1}$∈t|σ−ς|⁻¹, where ε is the relative noise level, Ф is cumulative distribution function for the normal distribution,r _F is constant, ande _σ is a function of σ only. The trade off betweenR andF is obtained in Maybank (1994). In this paper the trade off betweenR andp _σ(ς) is obtained.

It is conjectured that the general form of the above trade offs betweenR,F andp _σ(ς) 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 betweenR,F,p _σ(ς) of the above form is said to benon-degenerate for model based vision.

The consequences of the trade off betweenR andp _σ(ς) are examined. In particular, it is shown that for a fixed overall probability of misclassification there is a maximum possible model cross ratio σ _m , and there is a maximum possible numberN of models. Approximate expressions for σ _m andN are obtained. They indicate that in practice a model database containing only cross ratio values can have a size of order at most ten, for a physically plausible level of image noise, and for a probability of misclassification of the order 0.1.