Applications of invariants to model-based vision

This research presents a new approach to model-based object recognition from images that combines information about the position and orientation (pose) of the object relative to the camera with knowledge about the model itself.

Abstract

Previous approaches to model-based object recognition from images have required information about the position and orientation (pose) of the object relative to the camera. Object recognition has been carried out in parallel with pose estimation, leading to algorithm which are error prone and computationally expensive. Object recognition can be decoupled from pose estimation by using geometrical properties of the object that are unchanged or invariant under projection to the image. Thus, the computational cost of object recognition is drastically reduced. A number of invariants of current interest in computer Interest in computer vision are described. The simplest and most fundamental projective invariant, the cross ratio of four collinear points, is then investigated in detail. A simple system is defined for recognizing objects on the basis of the cross ratio alone. The system has a database of models. Each model is a single cross ratio value. The performance of the system is characterized by the probability R of rejection, the probability po(s) of misclassification and the probability F of a false alarm. Formulae for R, po(g) and Fare stated. The probability density function p(t) for the cross ratio t of four collinear points with independent, identical Gaussian distribution is stated. Experiments have been carried out to see how well the formulae for R, F and p(t) apply in practice. The results are extremely encouraging. The cumulative distribution function for p(r) is closely matched by the cumulative distribution function estimated from natural images. The experimental estimates of R agree well with the theoretical predictions. However, the experimental estimates of F are below the theoretical predictions. Two possible reasons for the discrepancy are suggested: (1) it is due to the finite resolution of the corner detector; and (2) it is due to deviations from the Gaussian distributions assumed in the theoretical calculations. The experimental investigation of R has led to a new, simple and theoretically well-founded way of estimating the accuracy of corner detectors.