When are simple LS estimators enough? An empirical study of LS, TLS and GTLS

Arvind Nayak, Emanuele Trucco, Neil A. Thacker

    Research output: Contribution to journalArticle

    6 Citations (Scopus)

    Abstract

    A variety of least-squares estimators of significantly different complexity and generality are available to solve over-constrained linear systems. The most theoretically general may not necessarily be the best choice in practice; problem conditions may be such that simpler and faster algorithms, if theoretically inferior, would yield acceptable errors. We investigate when this may happen using homography estimation as the reference problem. We study the errors of LS, TLS, equilibrated TLS and GTLS algorithms with different noise types and varying intensity and correlation levels. To allow direct comparisons with algorithms from the applied mathematics and computer vision communities, we consider both inhomogeneous and homogeneous systems. We add noise to image co-ordinates and system matrix entries in separate experiments, to take into account the effect on noise properties (heteroscedasticity) of pre-processing data transformations. We find that the theoretically most general algorithms may not always be worth their higher complexity; comparable results are obtained with moderate levels of noise intensity and correlation. We identify such levels quantitatively for the reference problem, thus suggesting when simpler algorithms can be applied with limited errors in spite of their restrictive assumptions.
    Original languageEnglish
    Pages (from-to)203-216
    Number of pages14
    JournalInternational Journal of Computer Vision
    Volume68
    Issue number2
    DOIs
    Publication statusPublished - Jun 2006

    Fingerprint

    Computer vision
    Linear systems
    Experiments

    Keywords

    • Least squares
    • Total least squares
    • Generalized total least squares
    • 2-D homography
    • Correlated noise

    Cite this

    @article{c5f77f715e514ee2b9d71bf9dbd94031,
    title = "When are simple LS estimators enough? An empirical study of LS, TLS and GTLS",
    abstract = "A variety of least-squares estimators of significantly different complexity and generality are available to solve over-constrained linear systems. The most theoretically general may not necessarily be the best choice in practice; problem conditions may be such that simpler and faster algorithms, if theoretically inferior, would yield acceptable errors. We investigate when this may happen using homography estimation as the reference problem. We study the errors of LS, TLS, equilibrated TLS and GTLS algorithms with different noise types and varying intensity and correlation levels. To allow direct comparisons with algorithms from the applied mathematics and computer vision communities, we consider both inhomogeneous and homogeneous systems. We add noise to image co-ordinates and system matrix entries in separate experiments, to take into account the effect on noise properties (heteroscedasticity) of pre-processing data transformations. We find that the theoretically most general algorithms may not always be worth their higher complexity; comparable results are obtained with moderate levels of noise intensity and correlation. We identify such levels quantitatively for the reference problem, thus suggesting when simpler algorithms can be applied with limited errors in spite of their restrictive assumptions.",
    keywords = "Least squares, Total least squares, Generalized total least squares, 2-D homography, Correlated noise",
    author = "Arvind Nayak and Emanuele Trucco and Thacker, {Neil A.}",
    note = "dc.publisher: Springer Verlag The original publication is available at www.springerlink.com",
    year = "2006",
    month = "6",
    doi = "10.1007/s11263-006-6486-z",
    language = "English",
    volume = "68",
    pages = "203--216",
    journal = "International Journal of Computer Vision",
    issn = "0920-5691",
    publisher = "Springer Verlag",
    number = "2",

    }

    When are simple LS estimators enough? An empirical study of LS, TLS and GTLS. / Nayak, Arvind; Trucco, Emanuele; Thacker, Neil A.

    In: International Journal of Computer Vision, Vol. 68, No. 2, 06.2006, p. 203-216.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - When are simple LS estimators enough? An empirical study of LS, TLS and GTLS

    AU - Nayak, Arvind

    AU - Trucco, Emanuele

    AU - Thacker, Neil A.

    N1 - dc.publisher: Springer Verlag The original publication is available at www.springerlink.com

    PY - 2006/6

    Y1 - 2006/6

    N2 - A variety of least-squares estimators of significantly different complexity and generality are available to solve over-constrained linear systems. The most theoretically general may not necessarily be the best choice in practice; problem conditions may be such that simpler and faster algorithms, if theoretically inferior, would yield acceptable errors. We investigate when this may happen using homography estimation as the reference problem. We study the errors of LS, TLS, equilibrated TLS and GTLS algorithms with different noise types and varying intensity and correlation levels. To allow direct comparisons with algorithms from the applied mathematics and computer vision communities, we consider both inhomogeneous and homogeneous systems. We add noise to image co-ordinates and system matrix entries in separate experiments, to take into account the effect on noise properties (heteroscedasticity) of pre-processing data transformations. We find that the theoretically most general algorithms may not always be worth their higher complexity; comparable results are obtained with moderate levels of noise intensity and correlation. We identify such levels quantitatively for the reference problem, thus suggesting when simpler algorithms can be applied with limited errors in spite of their restrictive assumptions.

    AB - A variety of least-squares estimators of significantly different complexity and generality are available to solve over-constrained linear systems. The most theoretically general may not necessarily be the best choice in practice; problem conditions may be such that simpler and faster algorithms, if theoretically inferior, would yield acceptable errors. We investigate when this may happen using homography estimation as the reference problem. We study the errors of LS, TLS, equilibrated TLS and GTLS algorithms with different noise types and varying intensity and correlation levels. To allow direct comparisons with algorithms from the applied mathematics and computer vision communities, we consider both inhomogeneous and homogeneous systems. We add noise to image co-ordinates and system matrix entries in separate experiments, to take into account the effect on noise properties (heteroscedasticity) of pre-processing data transformations. We find that the theoretically most general algorithms may not always be worth their higher complexity; comparable results are obtained with moderate levels of noise intensity and correlation. We identify such levels quantitatively for the reference problem, thus suggesting when simpler algorithms can be applied with limited errors in spite of their restrictive assumptions.

    KW - Least squares

    KW - Total least squares

    KW - Generalized total least squares

    KW - 2-D homography

    KW - Correlated noise

    U2 - 10.1007/s11263-006-6486-z

    DO - 10.1007/s11263-006-6486-z

    M3 - Article

    VL - 68

    SP - 203

    EP - 216

    JO - International Journal of Computer Vision

    JF - International Journal of Computer Vision

    SN - 0920-5691

    IS - 2

    ER -