Practical reliable bayesian recognition of 2d and 3d objects using implicit polynomials and algebraic invariants

Patches of quadric curves and surfaces such as spheres, planes, and cylinders have found widespread use in modeling and recognition of objects of interest in computer vision. In this paper, we treat use of more complex higher degree polynomial curves and surfaces of degree higher than 2, which have many desirable properties for object recognition and position estimation, and attack the instability problem arising in their use with partial and noisy data. The scenario discussed in this paper is one where we have a set of objects that are modeled as implicit polynomial functions, or a set of representations of classes of objects with each object in a class modeled as an implicit polynomial function, stored in the database. Then, given partial data from one of the objects, we want to recognize the object (or the object class) or collect more data in order to get better parameter estimates for more reliable recognition. Two problems arising in this scenario are discussed in this paper: 1) the problem of recognizing these polynomials by comparing them in terms of their coefficients, which are global descriptors, or in terms of algebraic invariants, i.e., functions of the polynomial coefficients that are independent of translations, rotations, and general linear transformation of the data; and 2) the problem of where to collect data so as to improve the parameter estimates as quickly as possible. We solve these problems by formulating them within a probabilistic framework. We use an asymptotic Bayesian approximation which results in computationally attractive solutions to the two problems. Among the key ideas discussed in this paper are the intrinsic dimensionality of polynomials and the use of the Mahalanobis distance as an effective tool for comparing polynomials in terms of their coefficients or algebraic invariants.