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Semidefinite Relaxations of Truncated Least-Squares in Robust Rotation Search: Tight or Not

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Computer Vision – ECCV 2022 (ECCV 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13683))

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Abstract

The rotation search problem aims to find a 3D rotation that best aligns a given number of point pairs. To induce robustness against outliers for rotation search, prior work considers truncated least-squares (TLS), which is a non-convex optimization problem, and its semidefinite relaxation (SDR) as a tractable alternative. Whether or not this SDR is theoretically tight in the presence of noise, outliers, or both has remained largely unexplored. We derive conditions that characterize the tightness of this SDR, showing that the tightness depends on the noise level, the truncation parameters of TLS, and the outlier distribution (random or clustered). In particular, we give a short proof for the tightness in the noiseless and outlier-free case, as opposed to the lengthy analysis of prior work.

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Notes

  1. 1.

    The catch is that the fast methods might not always be correct (e.g., at extreme outlier rates).

  2. 2.

     [5, 12, 39, 62] analyzed SDRs under noise but they are not for geometric vision problems.

  3. 3.

    Alternatively, if \((\boldsymbol{w}_0^*)^\top \boldsymbol{Q}_{j}\boldsymbol{w}_0^*\) is small, then \(\boldsymbol{Q}_{j}\) might be treated as noisy data rather than an outlier. We consider such noisy case in Sects. 3.3 (without outliers) and 3.4 (with outliers).

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Acknowledgments

This work was supported by grants NSF 1704458, NSF 1934979 and ONR MURI 503405-78051.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, USA

    Liangzu Peng & Mahyar Fazlyab

  2. Mathematical Institute for Data Science, Johns Hopkins University, Baltimore, USA

    Liangzu Peng, Mahyar Fazlyab & René Vidal

  3. Department of Biomedical Engineering, Johns Hopkins University, Baltimore, USA

    René Vidal

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  1. Liangzu Peng
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  2. Mahyar Fazlyab
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  3. René Vidal
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Correspondence to Liangzu Peng .

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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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Peng, L., Fazlyab, M., Vidal, R. (2022). Semidefinite Relaxations of Truncated Least-Squares in Robust Rotation Search: Tight or Not. 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 13683. Springer, Cham. https://doi.org/10.1007/978-3-031-20050-2_39

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