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Synchronization of Projective Transformations

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Computer Vision – ECCV 2024 (ECCV 2024)

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Abstract

Synchronization involves the task of inferring unknown vertex values (belonging to a group) in a graph, from edges labeled with vertex relations. While many matrix groups (e.g., rotations or permutations) have received extensive attention in Computer Vision, a complete solution for projectivities is lacking. Only the \(3 \times 3\) case has been addressed so far, by mapping the problem onto the Special Linear Group, but the \(4 \times 4\) projective case has remained unexplored and is the focus here. We propose novel strategies to address this task, and demonstrate their effectiveness in synthetic experiments, as well as on an application to projective Structure from Motion.

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Notes

  1. 1.

    With reference to projectivity synchronization, the number of measures used for updating a node \(X_i\) is equal to the degree of node i, that is the number of edges having such a node as endpoint.

  2. 2.

    It is easy to see that A and \(A - \alpha I\) for any scalar \(\alpha \) have the same eigenvectors. Moreover, \(\lambda (A) = - \lambda (-A)\).

  3. 3.

    https://github.com/rakshith95/projective_synchronization.jl.

  4. 4.

    It is worth observing that both Sphere and Euclidean are robust to outliers (being based on the L1 norm), whereas Direction is not (being based on the L2 norm). Therefore, for the first two methods, the IRLS-like scheme has the effect of improving robustness, whereas for the third approach it has the effect of gaining robustness.

  5. 5.

    The datasets can be downloaded from https://www.maths.lth.se/matematiklth/personal/calle/dataset/dataset.html.

References

  1. Aftab, K., Hartley, R., Trumpf, J.: Generalized Weiszfeld algorithms for \(l_q\) optimization. IEEE Trans. Pattern Anal. Mach. Intell. 4(37), 728–745 (2015)

    Article  Google Scholar 

  2. Arie-Nachimson, M., Kovalsky, S.Z., Kemelmacher-Shlizerman, I., Singer, A., Basri, R.: Global motion estimation from point matches. In: Proceedings of the Joint 3DIM/3DPVT Conference: 3D Imaging, Modeling, Processing, Visualization and Transmission (2012)

    Google Scholar 

  3. Arrigoni, F., Rossi, B., Fusiello, A.: Spectral synchronization of multiple views in SE(3). SIAM J. Imag. Sci. 9(4), 1963–1990 (2016)

    Article  MathSciNet  Google Scholar 

  4. Arrigoni, F., Fusiello, A.: Synchronization problems in computer vision with closed-form solutions. Int. J. Comput. Vision 128, 26–52 (2020)

    Article  MathSciNet  Google Scholar 

  5. Arrigoni, F., Pajdla, T.: Motion segmentation via synchronization. In: IEEE International Conference on Computer Vision Workshops (ICCVW) (2019)

    Google Scholar 

  6. Arrigoni, F., Pajdla, T., Fusiello, A.: Viewing graph solvability in practice. In: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp. 8147–8155 (2023)

    Google Scholar 

  7. Arrigoni, F., Rossi, B., Fragneto, P., Fusiello, A.: Robust synchronization in SO(3) and SE(3) via low-rank and sparse matrix decomposition. Comput. Vis. Image Underst. 174, 95–113 (2018)

    Article  Google Scholar 

  8. Bernard, F., Thunberg, J., Gemmar, P., Hertel, F., Husch, A., Goncalves, J.: A solution for multi-alignment by transformation synchronisation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015)

    Google Scholar 

  9. Bernard, F., Thunberg, J., Goncalves, J., Theobalt, C.: Synchronisation of partial multi-matchings via non-negative factorisations. Pattern Recogn. 92, 146–155 (2019)

    Article  Google Scholar 

  10. Bernard, F., Thunberg, J., Swoboda, P., Theobalt, C.: HiPPI: higher-order projected power iterations for scalable multi-matching. In: Proceedings of the International Conference on Computer Vision (2019)

    Google Scholar 

  11. Bhattacharya, U., Govindu, V.M.: Efficient and robust registration on the 3D special euclidean group. In: Proceedings of the International Conference on Computer Vision (2019)

    Google Scholar 

  12. Birdal, T., Arbel, M., Simsekli, U., Guibas, L.J.: Synchronizing probability measures on rotations via optimal transport. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1566–1576 (2020)

    Google Scholar 

  13. Birdal, T., Golyanik, V., Theobalt, C., Guibas, L.: Quantum permutation synchronization. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2021)

    Google Scholar 

  14. Birdal, T., Simsekli, U.: Probabilistic permutation synchronization using the riemannian structure of the birkhoff polytope. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 11105–11116 (2019)

    Google Scholar 

  15. Birdal, T., Simsekli, U., Eken, M.O., Ilic, S.: Bayesian pose graph optimization via bingham distributions and tempered geodesic MCMC. In: Advances in Neural Information Processing Systems, vol. 31. Curran Associates, Inc. (2018)

    Google Scholar 

  16. Boumal, N., Singer, A., Absil, P.A., Blondel, V.D.: Cramer-Rao bounds for synchronization of rotations. Inf. Inference J. IMA 3(1), 1–39 (2014)

    MathSciNet  Google Scholar 

  17. Buss, S.R., Fillmore, J.P.: Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20(2), 95–126 (2001)

    Article  Google Scholar 

  18. Carlone, L., Tron, R., Daniilidis, K., Dellaert, F.: Initialization techniques for 3D SLAM: a survey on rotation estimation and its use in pose graph optimization. In: Proceedings of the IEEE International Conference on Robotics and Automation (2015)

    Google Scholar 

  19. Chatterjee, A., Govindu, V.M.: Efficient and robust large-scale rotation averaging. In: Proceedings of the International Conference on Computer Vision (2013)

    Google Scholar 

  20. Chen, Y., Guibas, L., Huang, Q.: Near-optimal joint object matching via convex relaxation. In: Proceedings of the International Conference on Machine Learning, pp. 100–108 (2014)

    Google Scholar 

  21. Crandall, D., Owens, A., Snavely, N., Huttenlocher, D.P.: Discrete-continuous optimization for large-scale structure from motion. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3001–3008 (2011)

    Google Scholar 

  22. Dellaert, F., Rosen, D.M., Wu, J., Mahony, R., Carlone, L.: Shonan rotation averaging: global optimality by surfing \(SO(p)^n\). In: Vedaldi, A., Bischof, H., Brox, T., Frahm, J.-M. (eds.) ECCV 2020. LNCS, vol. 12351, pp. 292–308. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58539-6_18

    Chapter  Google Scholar 

  23. Eriksson, A., Olsson, C., Kahl, F., Chin, T.J.: Rotation averaging and strong duality. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 127–135 (2018)

    Google Scholar 

  24. Gojcic, Z., Zhou, C., Wegner, J.D., Guibas, L.J., Birdal, T.: Learning multiview 3D point cloud registration. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2020)

    Google Scholar 

  25. Govindu, V.M.: Combining two-view constraints for motion estimation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2001)

    Google Scholar 

  26. Govindu, V.M.: Lie-algebraic averaging for globally consistent motion estimation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 684–691 (2004)

    Google Scholar 

  27. Govindu, V.M., Pooja, A.: On averaging multiview relations for 3D scan registration. IEEE Trans. Image Process. 23(3), 1289–1302 (2014)

    Article  MathSciNet  Google Scholar 

  28. Hartley, R., Aftab, K., Trumpf, J.: L1 rotation averaging using the Weiszfeld algorithm. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3041–3048 (2011)

    Google Scholar 

  29. Hartley, R.I., Trumpf, J., Dai, Y., Li, H.: Rotation averaging. Int. J. Comput. Vision (2013)

    Google Scholar 

  30. Huang, J., et al.: Multibodysync: multi-body segmentation and motion estimation via 3D scan synchronization. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7108–7118 (2021)

    Google Scholar 

  31. Huang, X., Liang, Z., Zhou, X., Xie, Y., Guibas, L.J., Huang, Q.: Learning transformation synchronization. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2019)

    Google Scholar 

  32. Iglesias, J.P., Olsson, C., Kahl, F.: Global optimality for point set registration using semidefinite programming. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2020)

    Google Scholar 

  33. Kasten, Y., Geifman, A., Galun, M., Basri, R.: GPSfM: global projective SFM using algebraic constraints on multi-view fundamental matrices. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3259–3267 (2019)

    Google Scholar 

  34. Lee, S.H., Civera, J.: Hara: a hierarchical approach for robust rotation averaging. In: 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2022)

    Google Scholar 

  35. Leonardos, S., Zhou, X., Daniilidis, K.: Distributed consistent data association via permutation synchronization. In: 2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 2645–2652 (2017)

    Google Scholar 

  36. Leonardos, S., Zhou, X., Daniilidis, K.: A low-rank matrix approximation approach to multiway matching with applications in multi-sensory data association. In: 2020 IEEE International Conference on Robotics and Automation (ICRA), pp. 8665–8671 (2020)

    Google Scholar 

  37. Levi, N., Werman, M.: The viewing graph. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 518 – 522 (2003)

    Google Scholar 

  38. Li, H., Cui, Z., Liu, S., Tan, P.: Rago: recurrent graph optimizer for multiple rotation averaging. In: 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2022)

    Google Scholar 

  39. Li, S., Shi, Y., Lerman, G.: Fast, accurate and memory-efficient partial permutation synchronization. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2022)

    Google Scholar 

  40. Li, X., Ling, H.: Pogo-net: pose graph optimization with graph neural networks. In: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp. 5895–5905 (2021)

    Google Scholar 

  41. Mankovich, N., Birdal, T.: Chordal averaging on flag manifolds and its applications. In: Proceedings of the International Conference on Computer Vision (2023)

    Google Scholar 

  42. Martinec, D., Pajdla, T.: Robust rotation and translation estimation in multiview reconstruction. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2007)

    Google Scholar 

  43. Maset, E., Arrigoni, F., Fusiello, A.: Practical and efficient multi-view matching. In: Proceedings of IEEE International Conference on Computer Vision, pp. 4568–4576 (2017)

    Google Scholar 

  44. Moreira, G., Marques, M., Costeira, J.A.P.: Rotation averaging in a split second: a primal-dual method and a closed-form for cycle graphs. In: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp. 5452–5460 (2021)

    Google Scholar 

  45. Olsson, C., Enqvist, O.: Stable structure from motion for unordered image collections. In: Heyden, A., Kahl, F. (eds.) SCIA 2011. LNCS, vol. 6688, pp. 524–535. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21227-7_49

    Chapter  Google Scholar 

  46. Ozyesil, O., Sharon, N., Singer, A.: Synchronization over cartan motion groups via contraction. SIAM J. Appl. Algebra Geom. 2(2), 207–241 (2018)

    Article  MathSciNet  Google Scholar 

  47. Ozyesil, O., Voroninski, V., Basri, R., Singer, A.: A survey of structure from motion. Acta Numer. 26, 305–364 (2017)

    Article  MathSciNet  Google Scholar 

  48. Pachauri, D., Kondor, R., Singh, V.: Solving the multi-way matching problem by permutation synchronization. In: Advances in Neural Information Processing Systems, vol. 26, pp. 1860–1868 (2013)

    Google Scholar 

  49. Parra, A., Chng, S.F., Chin, T.J., Eriksson, A., Reid, I.: Rotation coordinate descent for fast globally optimal rotation averaging. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4298–4307 (2021)

    Google Scholar 

  50. Porfiri Dal Cin, A., Magri, L., Arrigoni, F., Fusiello, A., Boracchi, G.: Synchronization of group-labelled multi-graphs. In: IEEE International Conference on Computer Vision (ICCV) (2021)

    Google Scholar 

  51. Purkait, P., Chin, T.-J., Reid, I.: NeuRoRA: neural robust rotation averaging. In: Vedaldi, A., Bischof, H., Brox, T., Frahm, J.-M. (eds.) ECCV 2020. LNCS, vol. 12369, pp. 137–154. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58586-0_9

    Chapter  Google Scholar 

  52. Rosen, D.M., DuHadway, C., Leonard, J.J.: A convex relaxation for approximate global optimization in simultaneous localization and mapping. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 5822 – 5829 (2015)

    Google Scholar 

  53. Rosen, D.M., Carlone, L., Bandeira, A.S., Leonard, J.J.: Se-sync: a certifiably correct algorithm for synchronization over the special euclidean group. Int. J. Robot. Res. 38, 95–125 (2019)

    Article  Google Scholar 

  54. Santellani, E., Maset, E., Fusiello, A.: Seamless image mosaicking via synchronization. ISPRS Ann. Photogram. Remote Sens. Spatial Inf. Sci. IV-2, 247–254 (2018)

    Google Scholar 

  55. Schroeder, P., Bartoli, A., Georgel, P., Navab, N.: Closed-form solutions to multiple-view homography estimation. In: IEEE Workshop on Applications of Computer Vision (WACV), pp. 650–657 (2011)

    Google Scholar 

  56. Sengupta, S., et al.: A new rank constraint on multi-view fundamental matrices, and its application to camera location recovery. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2413–2421 (2017)

    Google Scholar 

  57. Shen, Y., Huang, Q., Srebro, N., Sanghavi, S.: Normalized spectral map synchronization. In: Advances in Neural Information Processing Systems, vol. 29, pp. 4925–4933. Curran Associates, Inc. (2016)

    Google Scholar 

  58. Shi, Y., Lerman, G.: Message passing least squares framework and its application to rotation synchronization. In: III, H.D., Singh, A. (eds.) Proceedings of the 37th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 119, pp. 8796–8806. PMLR (2020)

    Google Scholar 

  59. Singer, A.: Angular synchronization by eigenvectors and semidefinite programming. Appl. Comput. Harmon. Anal. 30(1), 20–36 (2011)

    Article  MathSciNet  Google Scholar 

  60. Sun, Y., Huang, Q.: Pose synchronization under multiple pair-wise relative poses. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2023)

    Google Scholar 

  61. Tejus, G., Zara, G., Rota, P., Fusiello, A., Ricci, E., Arrigoni, F.: Rotation synchronization via deep matrix factorization. In: 2023 IEEE International Conference on Robotics and Automation (ICRA) (2023)

    Google Scholar 

  62. Torsello, A., Rodolà, E., Albarelli, A.: Multiview registration via graph diffusion of dual quaternions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2441 – 2448 (2011)

    Google Scholar 

  63. Tron, R., Danilidis, K.: Statistical pose averaging with varying and non-isotropic covariances. In: Proceedings of the European Conference on Computer Vision (2014)

    Google Scholar 

  64. Tron, R., Zhou, X., Daniilidis, K.: A survey on rotation optimization in structure from motion. In: Computer Vision and Pattern Recognition Workshops (CVPRW) (2016)

    Google Scholar 

  65. Visual Geometry Group - University of Oxford: Multiview datasets. https://www.robots.ox.ac.uk/~vgg/data/

  66. Wang, H., Liu, Y., Dong, Z., Guo, Y., Liu, Y.S., Wang, W., Yang, B.: Robust multiview point cloud registration with reliable pose graph initialization and history reweighting. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2023)

    Google Scholar 

  67. Wilson, K., Bindel, D.: On the distribution of minima in intrinsic-metric rotation averaging. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6030–6038 (2020)

    Google Scholar 

  68. Wilson, K., Bindel, D., Snavely, N.: When is rotations averaging hard? In: Proceedings of the European Conference on Computer Vision, pp. 255 – 270 (2016)

    Google Scholar 

  69. Yang, H., Carlone, L.: Certifiably optimal outlier-robust geometric perception: semidefinite relaxations and scalable global optimization. IEEE Trans. Pattern Anal. Mach. Intell. 45(3), 2816–2834 (2023)

    Google Scholar 

  70. Yang, L., Li, H., Rahim, J.A., Cui, Z., Tan, P.: End-to-end rotation averaging with multi-source propagation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 11774–11783 (2021)

    Google Scholar 

  71. Yu, J.G., Xia, G.S., Samal, A., Tian, J.: Globally consistent correspondence of multiple feature sets using proximal Gauss-Seidel relaxation. Pattern Recogn. 51, 255–267 (2016)

    Article  Google Scholar 

  72. Zhang, G., Larsson, V., Barath, D.: Revisiting rotation averaging: uncertainties and robust losses. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2023)

    Google Scholar 

  73. Zhou, X., Zhu, M., Daniilidis, K.: Multi-image matching via fast alternating minimization. In: Proceedings of the International Conference on Computer Vision, pp. 4032–4040 (2015)

    Google Scholar 

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Acknowledgements

This paper is supported by PNRR-PE-AI FAIR project funded by the NextGeneration EU program. The authors would like to thank Gaia Trebucchi for her help on a preliminary project related to this research.

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Authors and Affiliations

  1. DEIB, Politecnico di Milano, Milan, Italy

    Rakshith Madhavan & Federica Arrigoni

  2. DPIA, University of Udine, Udine, Italy

    Andrea Fusiello

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  1. Rakshith Madhavan
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  2. Andrea Fusiello
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  3. Federica Arrigoni
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Correspondence to Rakshith Madhavan .

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Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Aleš Leonardis

  2. University of Trento, Trento, Italy

    Elisa Ricci

  3. Technical University of Darmstadt, Darmstadt, Germany

    Stefan Roth

  4. Princeton University, Princeton, NJ, USA

    Olga Russakovsky

  5. Czech Technical University in Prague, Prague, Czech Republic

    Torsten Sattler

  6. École des Ponts ParisTech, Marne-la-Vallée, France

    Gül Varol

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Madhavan, R., Fusiello, A., Arrigoni, F. (2025). Synchronization of Projective Transformations. In: Leonardis, A., Ricci, E., Roth, S., Russakovsky, O., Sattler, T., Varol, G. (eds) Computer Vision – ECCV 2024. ECCV 2024. Lecture Notes in Computer Science, vol 15095. Springer, Cham. https://doi.org/10.1007/978-3-031-72913-3_2

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