Implicit polynomial representation through a fast fitting error estimation

Mohammad Rouhani, Angel Domingo Sappa

    Research output: Contribution to journalArticleResearchpeer-review

    14 Citations (Scopus)


    This paper presents a simple distance estimation for implicit polynomial fitting. It is computed as the height of a simplex built between the point and the surface (i.e., a triangle in 2-D or a tetrahedron in 3-D), which is used as a coarse but reliable estimation of the orthogonal distance. The proposed distance can be described as a function of the coefficients of the implicit polynomial. Moreover, it is differentiable and has a smooth behavior. Hence, it can be used in any gradient-based optimization. In this paper, its use in a Levenberg-Marquardt framework is shown, which is particularly devoted for nonlinear least squares problems. The proposed estimation is a generalization of the gradient-based distance estimation, which is widely used in the literature. Experimental results, both in 2-D and 3-D data sets, are provided. Comparisons with state-of-the-art techniques are presented, showing the advantages of the proposed approach. © 2011 IEEE.
    Original languageEnglish
    Article number6031920
    Pages (from-to)2089-2098
    JournalIEEE Transactions on Image Processing
    Issue number4
    Publication statusPublished - 1 Apr 2012


    • Curve/surface fitting
    • geometric distance estimation
    • implicit polynomial (IP)
    • residual error minimization


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