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The Classical Theory of Invariants and Object Recognition Using Algebraic Curve and Surfaces

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Abstract

Combining implicit polynomials and algebraic invariants for representing and recognizing complicated objects proves to be a powerful technique. In this paper, we explore the findings of the classical theory of invariants for the calculation of algebraic invariants of implicit curves and surfaces, a theory largely disregarded in the computer vision community by a shadow of skepticism. Here, the symbolic method of the classical theory is described, and its results are extended and implemented as an algorithm for computing algebraic invariants of projective, affine, and Euclidean transformations. A list of some affine invariants of 4th degree implicit polynomials generated by the proposed algorithm is presented along with the corresponding symbolic representations, and their use in recognizing objects represented by implicit polynomials is illustrated through experiments. An affine invariant fitting algorithm is also proposed and the performance is studied.

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Authors and Affiliations

  1. AT Kearny Consulting, Istanbul, Turkey

    Hakan Civi

  2. Department of Mathematics, Bogazici University Bebek, Istanbul, Turkey

    Colin Christopher

  3. Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey

    Aytul Ercil

Authors
  1. Hakan Civi
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  2. Colin Christopher
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  3. Aytul Ercil
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Civi, H., Christopher, C. & Ercil, A. The Classical Theory of Invariants and Object Recognition Using Algebraic Curve and Surfaces. Journal of Mathematical Imaging and Vision 19, 237–253 (2003). https://doi.org/10.1023/A:1026233121583

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  • DOI: https://doi.org/10.1023/A:1026233121583

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