Implicitization of Parametric Curves by Matrix Annihilation

  • Authors:

  • Hulya Yalcin;Mustafa Unel;William Wolovich

  • Affiliations:

  • Division of Engineering, Brown University, Providence, RI, USA. hulya_yalcin@brown.edu;Center for Computational Vision and Control, Yale University, CT, USA. mu@sysc.eng.yale.edu;Division of Engineering, Brown University, Providence, RI, USA. waw@lems.brown.edu

  • Venue:
  • International Journal of Computer Vision - Special Issue on Computational Vision at Brown University
  • Year:
  • 2003

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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.