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Hyper least squares fitting of circles and ellipses

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  • Kanatani, Kenichi
  • Rangarajan, Prasanna

Abstract

This work extends the circle fitting method of Rangarajan and Kanatani (2009) to accommodate ellipse fitting. Our method, which we call HyperLS, relies on algebraic distance minimization with a carefully chosen scale normalization. The normalization is derived using a rigorous error analysis of least squares (LS) estimators so that statistical bias is eliminated up to second order noise terms. Numerical evidence suggests that the proposed HyperLS estimator is far superior to the standard LS and is slightly better than the Taubin estimator. Although suboptimal in comparison to maximum likelihood (ML), our HyperLS does not require iterations. Hence, it does not suffer from convergence issues due to poor initialization, which is inherent in ML estimators. In this sense, the proposed HyperLS is a perfect candidate for initializing the ML iterations.

Suggested Citation

  • Kanatani, Kenichi & Rangarajan, Prasanna, 2011. "Hyper least squares fitting of circles and ellipses," Computational Statistics & Data Analysis, Elsevier, vol. 55(6), pages 2197-2208, June.
    • Handle: RePEc:eee:csdana:v:55:y:2011:i:6:p:2197-2208
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    References listed on IDEAS

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    1. Chi-Lun Cheng & Alexander Kukush, 2006. "Non-Existence of the First Moment of the Adjusted Least Squares Estimator in Multivariate Errors-in-Variables Model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 64(1), pages 41-46, August.
    2. Chernov, N. & Lesort, C., 2004. "Statistical efficiency of curve fitting algorithms," Computational Statistics & Data Analysis, Elsevier, vol. 47(4), pages 713-728, November.
    3. Kukush, Alexander & Markovsky, Ivan & Van Huffel, Sabine, 2004. "Consistent estimation in an implicit quadratic measurement error model," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 123-147, August.
    4. Kanatani, Kenichi & Sugaya, Yasuyuki, 2007. "Performance evaluation of iterative geometric fitting algorithms," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 1208-1222, October.
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    Citations

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    Cited by:

    1. Manolopoulou, Ioanna & Kepler, Thomas B. & Merl, Daniel M., 2012. "Mixtures of Gaussian wells: Theory, computation, and application," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3809-3820.
    2. Al-Sharadqah, A. & Chernov, N., 2012. "A doubly optimal ellipse fit," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2771-2781.
    3. Greco, Luca & Pacillo, Simona & Maresca, Piera, 2023. "An impartial trimming algorithm for robust circle fitting," Computational Statistics & Data Analysis, Elsevier, vol. 181(C).

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