Implicitization of parametric curves by matrix annihilation

Abstract

Abstract. Object recognition is a central problem in computer vision. When objects are defined by boundary curves, they can be represented either explicitly or implicitly. Implicit polynomial (IP) equations have long been known to offer certain advantages over more traditional parametric methods. However, the lack of general procedures for obtaining IP models of higher degree has prevented their general use in many practical applications. In most cases today, parametric equations are used to model curves and surfaces. One such parametric representation, elliptic Fourier Descriptors (EFD), has been widely used to represent 2D and 3D curves, as well as 3D surfaces. Although EFDs can represent nearly all curves, it is often convenient to have an implicit algebraic description F(x, y) = 0, for several reasons. Algebraic curves and surfaces have proven very useful in many model-based applications. Various algebraic and geometric invariants obtained from these implicit models have been studied rather extensively, since implicit polynomials are well-suited to computer vision tasks, especially for single computation pose estimation, shape tracking, 3D surface estimation from multiple images and efficient geometric indexing into large pictorial databases. In this paper, we present a new non-symbolic implicitization technique called the matrix annihilation method, for converting parametric Fourier representations to algebraic (implicit polynomial) representations, thereby benefiting from the features of both.

Unification of parametric and implicit methods for shape sensitivity analysis and optimization with fixed mesh

“…An algebraic curve has a parametric representation in the form of rational polynomial if and only if the curve has genus zero [34]. The matrix annihilation method [35] was presented to perform the implicitization for closed curves in the form of elliptical Fourier representation. Now, attempts are made to perform shape sensitivity analysis of a structure with parametric B-rep.…”

A New Affine Invariant Fitting Algorithm for Algebraic Curves

“…Matrix annihilation method [11] established a link between the elliptic Fourier descriptors (EFDs) and implicit polynomials (IP). EFDs are arc-length parameteriza-tions, which are made invariant to changes in location, orientation and scale, that is, they are similarity invariant [12].…”

“…Since this parameterization is affine invariant, we can use the coefficients of this Fourier parameterization as an input to the implicitization method based on matrix annihilation ideas detailed in [11]. We note that the interpolated curve of affineinvariant Fourier descriptors might exhibit Gibbs phenomenon.…”

“…A new matrix annihilation method for converting parametric representations to the implicit algebraic curves has recently been introduced in [11]. This method is numerical and computationally efficient.…”

“…In this paper, we present a new affine invariant fitting algorithm based on the affine invariant Fourier descriptors [13,14] and the implicitization of them by matrix annihilation method [11]. The algorithm is applicable for all free-form shapes.…”