Bayesian methods for the use of implicit polynomials and algebraic invariants in practical computer vision

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Abstract
Implicit higher degree polynomials in x, y, z (or in x, y for curves in images) have considerable global and semiglobal representation power for objects in 3D space. (Spheres, cylinders, cones and planes are special cases of such polynomials restricted to second degree). Hence, they have great potential for object recognition and position estimation and for object geometric-property measurement. In this paper we deal with four problems pertinent to using these polynomials in real world robust systems: (1) Characterization and fitting algorithms for the subset of these algebraic curves and surfaces that is bounded and exists largely in the vicinity of the data; (2) The aposteriori distribution for the possible polynomial coefficients given a data set. This measures the extent to which a data set constrains the coefficients of the best fitting polynomial; (3) Geometric Invariants for determining whether one implicit polynomial curve or surface is a rotation and translation of another, or whether one implicit polynomial curve is an affine transformation of another; (4) A Mahalanobis distance for comparing the coefficients or the invariants of two polynomials to determine whether the curves or surfaces that they represent are close over a specified region. In addition to handling objects with easily detectable features such as vertices, high curvature points, and straight lines, the polynomials and tools discussed in this paper are ideally suited to smooth curves and smooth curved surfaces which do no have detectable features.

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