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Pairwise Rigid Registration Based on Riemannian Geometry and Lie Structures of Orientation Tensors

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Abstract

Pairwise rigid registration can be developed by comparing local geometry encoded by intrinsic second-order orientation tensors which allows to model the registration problem using the associated Riemannian geometry or related group structures. Everything starts by representing those tensor fields as multivariate normal models that permit us to manipulate Gaussians in two ways: using the Riemannian manifold elements, that can be embedded into matrix spaces with geometric structures, or through Lie group/algebra techniques. In this paper we discuss some points behind these approaches in the context of rigid registration problems. Firstly, they are not equivalent since, in general, there is no isometry linking them. Secondly, embedding methodologies are not invariant with respect to rigid motion. We discuss these points using two variants of the Iterative Closest Point that use the comparative tensor shape factor (CTSF) to match orientation tensors. We replace the CTSF to different criteria computed through geodesic distance and algebraic embeddings and compare the registration algorithms showing that the latter is more efficient for registration of point clouds.

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References

  1. Almeida, L.R., Giraldi, G.A., Vieira, M.B.: Applying lie groups approaches for rigid registration of point clouds. Technical report, Laboratório Nacional de Computação Científica (2018). arXiv:2006.13341

  2. Almeida, L.R., Giraldi, G.A., Vieira, M.B.: Rigid registration of point clouds based on indirect lie group approach. In: 21st Symposium on Virtual and Augmented Reality (SVR), Rio de Janeiro, RJ, pp. 130–139 (2019)

  3. Amari, S.: Information Geometry and Its Applications, 1st edn. Springer, Berlin (2016)

    Book  Google Scholar 

  4. Baker, A.: Matrix Groups: An Introduction to Lie Group Theory. Springer Undergraduate Mathematics Series. Springer, London (2002)

    Book  Google Scholar 

  5. Berger, M., Tagliasacchi, A., Seversky, L.M., Alliez, P., Guennebaud, G., Levine, J.A., Sharf, A., Silva, C.T.: A survey of surface reconstruction from point clouds. In: Computer Graphics Forum. Wiley Online Library (2016)

  6. Besl, P.J., McKay, N.D.: A method for registration of 3-d shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14(2), 239–256 (1992)

    Article  Google Scholar 

  7. Bridger, M.: Real Analysis: A Constructive Approach Through Interval Arithmetic. American Mathematical Society, Providence (2019)

    MATH  Google Scholar 

  8. Calvo, M., Oller, J.M.: A distance between multivariate normal distributions based on an embedding into the Siegel group. J. Multivar. Anal. 35(2), 223–242 (1990)

    Article  MathSciNet  Google Scholar 

  9. Carmo, M.P.D.: Riemannian Geometry: Theory & Applications, 1st edn. Birkhauser, Boston (1992)

    Book  Google Scholar 

  10. Cejnog, L.W.X., Yamada, F.A.A., Vieira, M.B.: Wide angle rigid registration using a comparative tensor shape factor. Int. J. Image Graph. 17(01), 1–34 (2017)

    Article  Google Scholar 

  11. Chetverikov, D., Svirko, D., Stepanov, D., Krsek, P.: The trimmed iterative closest point algorithm. In: Proceedings of 16th International Conference on Pattern Recognition, vol. 3, pp 545–548 (2002)

  12. Clark, W., Ghaffari, M., Bloch, A.: Nonparametric continuous sensor registration. (2020) arXiv:2001.04286

  13. Deng, Y., Rangarajan, A., Eisenschenk, S., Vemuri, B.C.: A riemannian framework for matching point clouds represented by the schrödinger distance transform. In: 2014 IEEE Conference on Computer Vision and Pattern Recognition, pp. 3756–3761 (2014). https://doi.org/10.1109/CVPR.2014.486

  14. de Araujo Yamada, FA: A shape-based strategy applied to the covariance estimation on the ICP. Master’s thesis, Post-Graduation Program on Computer Science, Federal University of Juiz de Fora, Juiz de Fora, MG, Brazil (2016)

  15. de Araujo Yamada, FA., Giraldi, GA., Vieira, MB., Almeida Jr., L.R.: Frame-to-frame rigid registration of point clouds extracted from depth sequences: Comparing different strategies. Technical report, Laboratório Nacional de Computação Científica, virtual01.lncc.br/\(\sim \)giraldi/extended-svr-paper2017/Relatorio-Registration-27-03-2017.pdf (2017)

  16. de Araujo Yamada, F.A., Giraldi, G., Vieira, M., Almeida, L., Apolinario, A.L.: Comparing seven methodogies for rigid alignment of point clouds with focus on frame-to-frame registration in depth sequences. SBC J. Interact. Syst. 9, 82–104 (2018)

    Google Scholar 

  17. Department of Statistics, Stanford University (SUD), Lene Theil Skovgaard, National Science Foundation (US) (1981) A Riemannian Geometry of the Multivariate Normal Model. Statistical Research Unit, https://books.google.com.br/books?id=VQpfGwAACAAJ

  18. Díez, Y., Roure, F., Lladó, X., Salvi, J.: A qualitative review on 3d coarse registration methods. ACM Comput. Surv. 47(3), 45 (2015)

    Article  Google Scholar 

  19. do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, Hoboken (1976)

    MATH  Google Scholar 

  20. Dong, J., Peng, Y., Ying, S., Hu, Z.: Lietricp: an improvement of trimmed iterative closest point algorithm. Neurocomputing 140, 67–76 (2014)

    Article  Google Scholar 

  21. Dryden, I.L., Koloydenko, A., Zhou, D.: Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann Appl Stat 3(3), 1102–1123 (2009). https://doi.org/10.1214/09-AOAS249

    Article  MathSciNet  MATH  Google Scholar 

  22. Gao, Q.H., Wan, T.R., Tang, W., Chen, L.: Object registration in semi-cluttered and partial-occluded scenes for augmented reality. Multimed. Tools Appl. 78(11), 15079–15099 (2019)

    Article  Google Scholar 

  23. Gojcic, Z., Zhou, C., Wegner, J., Wieser, A.: The perfect match: 3d point cloud matching with smoothed densities. In: 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp 5540–5549 (2019)

  24. Gong, L., Wang, T., Liu, F.: Shape of Gaussians as feature descriptors. In: 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp 2366–2371 (2009)

  25. Group, C.V.: RGB-D SLAM dataset and benchmark. Technical report, https://vision.in.tum.de/data/datasets/rgbd-dataset/download (2012)

  26. Handa, A., Whelan, T., McDonald, J., Davison, A.J.: A benchmark for RGB-D visual odometry, 3D reconstruction and SLAM. In: 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 1524–1531 (2014). https://doi.org/10.1109/ICRA.2014.6907054

  27. Horn, B.K.: Closed-form solution of absolute orientation using unit quaternions. JOSA A 4(4), 629–642 (1987)

    Article  Google Scholar 

  28. Huynh, D.Q.: Metrics for 3d rotations: comparison and analysis. J. Math. Imaging Vis. 35(2), 155–164 (2009)

    Article  MathSciNet  Google Scholar 

  29. Levoy, M., Gerth, J., Curless, B., Pull, K.: Bunny model (1994). https://graphics.stanford.edu/data/3Dscanrep/

  30. Li, P., Wang, Q., Zeng, H., Zhang, L.: Local log-Euclidean multivariate Gaussian descriptor and its application to image classification. IEEE Trans. Pattern Anal. Mach. Intell. 39, 803–817 (2017)

    Article  Google Scholar 

  31. Lima, E.: Variedades diferenciáveis. Monografias de matemática. Instituto Matemática Puro e Aplicada, Conselho Nacional de Pesquisas (1973)

  32. Lovrić, M., Min-Oo, M.: Multivariate normal distributions parametrized as a Riemannian symmetric space. J. Multivar. Anal. 74(1), 36–48 (2000)

    Article  MathSciNet  Google Scholar 

  33. Luong, H.Q., Vlaminck, M., Goeman, W., Philips, W.: Consistent ICP for the registration of sparse and inhomogeneous point clouds. In: 2016 IEEE Sixth International Conference on Communications and Electronics (ICCE), pp 262–267 (2016). https://doi.org/10.1109/CCE.2016.7562647

  34. Olver, P.J.: Applications of Lie Groups to Differential Equations, p. 1986. Springer-Verlag, Berlin (1986)

    Book  Google Scholar 

  35. Rusu, R.B., Blodow, N., Marton, Z.C., Beetz, M.: Aligning point cloud views using persistent feature histograms. In: 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3384–3391 (2008) https://doi.org/10.1109/IROS.2008.4650967

  36. Rusu, R.B., Blodow, N., Beetz, M.: Fast point feature histograms (FPFH) for 3D registration. In: 2009 IEEE International Conference on Robotics and Automation, pp. 3212–3217 (2009)

  37. Salvi, J., Matabosch, C., Fofi, D., Forest, J.: A review of recent range image registration methods with accuracy evaluation. Image Vis. Comput. 25(5), 578–596 (2007)

    Article  Google Scholar 

  38. Set, D.: File formats. Technical report (2012) https://vision.in.tum.de/data/datasets/rgbd-dataset/file_formats

  39. Sharp, G.C., Lee, S.W., Wehe, D.K.: ICP registration using invariant features. IEEE Trans. Pattern Anal. Mach. Intell. 24(1), 90–102 (2002)

    Article  Google Scholar 

  40. Shih, S., Chuang, Y., Yu, T.: An efficient and accurate method for the relaxation of multiview registration error. IEEE Trans. Image Process. 17(6), 968–981 (2008). https://doi.org/10.1109/TIP.2008.921987

    Article  MathSciNet  Google Scholar 

  41. Skovgaard, L.T.: A Riemannian geometry of the multivariate normal model. Scand. J. Stat. 11(4), 211–223 (1984)

    MathSciNet  MATH  Google Scholar 

  42. Sturm, J., Engelhard, N., Endres, F., Burgard, W., Cremers, D.: A benchmark for the evaluation of RGB-D SLAM systems. In: International Conference on Intelligent Robot Systems (IROS) (2012)

  43. Tam, G., Cheng, Z.Q., Lai, Y.K., Langbein, F., Liu, Y., Marshall, D., Martin, R., Sun, X.F., Rosin, P.: Registration of 3D point clouds and meshes: a survey from rigid to nonrigid. IEEE Trans. Vis. Comput. Graph. 19(7), 1199–1217 (2013)

    Article  Google Scholar 

  44. Wang, Y., Solomon, M.J..: Deep closest point: learning representations for point cloud registration. In: International Conference on Computer Vision, pp. 3523–3532 (2019)

  45. Xiong, H., Szedmak, S., Piater, J.: A study of point cloud registration with probability product kernel functions. In: International Conference on 3D Vision, pp. 207–214. IEEE (2013). https://doi.org/10.1109/3DV.2013.35, https://iis.uibk.ac.at/public/papers/Xiong-2013-3DV.pdf

  46. Yamada, F.A., Giraldi, G.A., Vieira, M.B., Almeida Jr., L.R.: Comparing seven methodologies for rigid alignment of point clouds with focus on frame-to-frame registration in depth sequences. SBC J. Interact. Syst. 9(2), 82–104 (2018)

    Google Scholar 

  47. Yang, H., Shi, J., Carlone, L.: Teaser: fast and certifiable point cloud registration (2020). arXiv:2001.07715

  48. Yang, J., Li, H., Jia, Y.: Go-icp: Solving 3d registration efficiently and globally optimally. In: 2013 IEEE International Conference on Computer Vision (ICCV), pp. 1457–1464. IEEE (2013)

  49. Ying, S., Peng, J., Du, S., Qiao, H.: Lie group framework of iterative closest point algorithm for n-d data registration. Int. J. Pattern Recognit. Artif. Intell. 23, 1201–1220 (2009). https://doi.org/10.1142/S0218001409007533

    Article  Google Scholar 

  50. Zeng, A., Song, S., Nießner, M., Fisher, M., Xiao, J., Funkhouser, T.: 3dmatch: learning local geometric descriptors from rgb-d reconstructions. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 199–208 (2017)

  51. Zhang, B., Li, Y.F.: Automatic Calibration and Reconstruction for Active Vision Systems. Springer, Berlin (2014)

    Google Scholar 

  52. Zhou, Y., Li, H., Kneip, L.: Canny-vo: visual odometry with rgb-d cameras based on geometric 3-d 2-d edge alignment. IEEE Trans. Robot. 35(1), 184–199 (2019). https://doi.org/10.1109/TRO.2018.2875382

    Article  Google Scholar 

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Acknowledgements

We thanks PCI-LNCC for the financial support.

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Authors and Affiliations

  1. National Laboratory for Scientific Computing, Petrópolis, RJ, Brazil

    Liliane Rodrigues de Almeida & Gilson Antonio Giraldi

  2. Federal University of Juiz de Fora, Juiz de Fora, MG, Brazil

    Marcelo Bernardes Vieira

  3. Federal University of Sergipe, São Cristóvão, SE, Brazil

    Gastão Florêncio Miranda Jr

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  1. Liliane Rodrigues de Almeida
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  4. Gastão Florêncio Miranda Jr
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Correspondence to Liliane Rodrigues de Almeida.

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de Almeida, L.R., Giraldi, G.A., Vieira, M.B. et al. Pairwise Rigid Registration Based on Riemannian Geometry and Lie Structures of Orientation Tensors. J Math Imaging Vis 63, 894–916 (2021). https://doi.org/10.1007/s10851-021-01037-z

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