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Fast Two-View Motion Segmentation Using Christoffel Polynomials

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Computer Vision – ECCV 2022 (ECCV 2022)

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Abstract

We address the problem of segmenting moving rigid objects based on two-view image correspondences under a perspective camera model. While this is a well understood problem, existing methods scale poorly with the number of correspondences. In this paper we propose a fast segmentation algorithm that scales linearly with the number of correspondences and show that on benchmark datasets it offers the best trade-off between error and computational time: it is at least one order of magnitude faster than the best method (with comparable or better accuracy), with the ratio growing up to three orders of magnitude for larger number of correspondences. We approach the problem from an algebraic perspective by exploiting the fact that all points belonging to a given object lie in the same quadratic surface. The proposed method is based on a characterization of each surface in terms of the Christoffel polynomial associated with the probability that a given point belongs to the surface. This allows for efficiently segmenting points “one surface at a time” in \(\mathcal {O}(\text {number of points)}\).

This work was supported in part by NSF grants IIS–1814631 and CNS–2038493, ONR grant N00014-21-1-2431 and U.S. DHS grant 22STESE00001-01-00.

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Notes

  1. 1.

    For a singular \(\textbf{L} \doteq \textbf{U}\begin{bmatrix} \boldsymbol{\Sigma } &{} \textbf{0} \\ \textbf{0} &{} \textbf{0} \end{bmatrix}\textbf{U}^T\), \(\textbf{L}^{-1}\doteq \textbf{U}\begin{bmatrix} \boldsymbol{\Sigma }^{-1} &{} \textbf{0} \\ \textbf{0} &{} \textbf{0} \end{bmatrix}\textbf{U}^T\).

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Authors and Affiliations

  1. ECE Department, Northeastern University, Boston, MA, 02115, USA

    Bengisu Ozbay, Octavia Camps & Mario Sznaier

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  1. Bengisu Ozbay
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  2. Octavia Camps
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  3. Mario Sznaier
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Correspondence to Octavia Camps .

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Editors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Shai Avidan

  2. University College London, London, UK

    Gabriel Brostow

  3. Google AI, Accra, Ghana

    Moustapha Cissé

  4. University of Catania, Catania, Italy

    Giovanni Maria Farinella

  5. Facebook (United States), Menlo Park, CA, USA

    Tal Hassner

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Ozbay, B., Camps, O., Sznaier, M. (2022). Fast Two-View Motion Segmentation Using Christoffel Polynomials. In: Avidan, S., Brostow, G., Cissé, M., Farinella, G.M., Hassner, T. (eds) Computer Vision – ECCV 2022. ECCV 2022. Lecture Notes in Computer Science, vol 13690. Springer, Cham. https://doi.org/10.1007/978-3-031-20056-4_1

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