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Implicitization of Algebraic Curves by Matrix Annihilation
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Hulya Yalcin,
Mustafa Unel,
William Wolovich
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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 , especially for determining whether given points lie on the curve. Various invariants associated with EFDs can be computed either from a purely mathematical point of view or from some geometrical considerations. A! lgebraic curves and surfaces also 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 to provide a means of benefiting from the features of both. [ More about this... ] Download the paper [pdf] |
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Carnegie Mellon University, Robotics Institute
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