HAL - INRIA :: [inria-00548505, version 1] On the Absolute Quadratic Complex and its Application to Autocalibration

inria-00548505, version 1

On the Absolute Quadratic Complex and its Application to Autocalibration

Jean Ponce () ¹, Kenton Mc Henry ¹, Théo Papadopoulo ², Monique Teillaud ( Author to contact preferably ) ³, Bill Triggs ⁴

IEEE Conference on Computer Vision & Pattern Recognition (CPRV '05) I (2005) 780--787

Abstract: This article introduces the absolute quadratic complex formed by all lines that intersect the absolute conic. If ω denotes the 3 × 3 symmetric matrix representing the image of that conic under the action of a camera with projection matrix P, it is shown that ω ≈ P~Ω_P~T where V is the 3 × 6 line projection matrix associated with P and Ω_ is a 6 × 6 symmetric matrix of rank 3 representing the absolute quadratic complex. This simple relation between a camera's intrinsic parameters, its projection matrix expressed in a projective coordinate frame, and the metric upgrade separating this frame from a metric one - as respectively captured by the matrices ω, P~ and Ω_ - provides a new framework for autocalibration, particularly well suited to typical digital cameras with rectangular or square pixels since the skew and aspect ratio are decoupled from the other intrinsic parameters in ω.

1: The Beckman Institute for Advanced Science and Technology (Beckman Institute)
University of Illinois
2: ODYSSEE (INRIA Sophia Antipolis)
INRIA – Ecole des Ponts ParisTech – Ecole Normale Supérieure de Paris - ENS Paris
3: GALAAD (INRIA Sophia Antipolis)
INRIA – CNRS : UMR6621 – Université de Nice Sophia Antipolis (UNS)
4: LEAR (IMAG-INRIA Rhône-Alpes / GRAVIR)
CNRS : FR71 – CNRS : UMR5527 – INRIA – Université Joseph Fourier - Grenoble I – Institut National Polytechnique de Grenoble (INPG)

inria-00548505, version 1
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DOI: 10.1109/CVPR.2005.256

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%% inria-00548505, version 1
%% http://hal.inria.fr/inria-00548505
@inproceedings{ponce:inria-00548505,
    hal_id = {inria-00548505},
    url = {http://hal.inria.fr/inria-00548505},
    title = {{On the Absolute Quadratic Complex and its Application to Autocalibration}},
    author = {Ponce, Jean and Mc Henry, Kenton and Papadopoulo, Th{\'e}o and Teillaud, Monique and Triggs, Bill},
    abstract = {{This article introduces the absolute quadratic complex formed by all lines that intersect the absolute conic. If $\omega$ denotes the 3 × 3 symmetric matrix representing the image of that conic under the action of a camera with projection matrix P, it is shown that $\omega$ $\approx$ P\~$\Omega$\_P\~T where V is the 3 × 6 line projection matrix associated with P and $\Omega$\_ is a 6 × 6 symmetric matrix of rank 3 representing the absolute quadratic complex. This simple relation between a camera's intrinsic parameters, its projection matrix expressed in a projective coordinate frame, and the metric upgrade separating this frame from a metric one - as respectively captured by the matrices $\omega$, P\~ and $\Omega$\_ - provides a new framework for autocalibration, particularly well suited to typical digital cameras with rectangular or square pixels since the skew and aspect ratio are decoupled from the other intrinsic parameters in $\omega$.}},
    keywords = {calibration; computational complexity; computational geometry; image processing; matrix algebra},
    language = {English},
    affiliation = {The Beckman Institute for Advanced Science and Technology - Beckman Institute , ODYSSEE - INRIA Sophia Antipolis , GALAAD - INRIA Sophia Antipolis , LEAR - IMAG-INRIA Rh{\^o}ne-Alpes / GRAVIR},
    booktitle = {{IEEE Conference on Computer Vision \& Pattern Recognition (CPRV '05)}},
    publisher = {IEEE Computer Society},
    pages = {780--787},
    address = {San Diego, United States},
    volume = {I},
    audience = {international },
    doi = {10.1109/CVPR.2005.256 },
    year = {2005},
    pdf = {http://hal.inria.fr/inria-00548505/PDF/Ponce-cvpr05.pdf},
}

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