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On Geometric Invariants, Learning, and Recognition of Shapes and Forms

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Handbook of Variational Methods for Nonlinear Geometric Data

Abstract

Extracting meaningful representations from geometric data has prime importance in the areas of computer vision, computer graphics, and image processing. Classical approaches use tools from differential geometry for modeling the problem and employed efficient and robust numerical techniques to engineer them for a particular application. Recent advances in learning methods, particularly in the areas of deep-learning and neural networks provide an alternative mechanism of extracting meaningful features and doing data-engineering. These techniques have proven very successful for various kinds of visual and semantic cognition tasks achieving state-of-the art results. In this chapter we explore the synergy between these two seemingly disparate computational methodologies. First, we provide a short treatise on geometric invariants of planar curves and a scheme to discover them from data in a learning framework, where the invariants are modelled using neural networks. Secondly, we also demonstrate the reverse, that is, imputing principled geometric invariants like geometric moments into standard learning architectures enables a significant boost in performance. Our goal would not only be to achieve better performance, but also to provide a geometric insight into the learning process thereby establishing strong links between the two fields.

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References

  1. Ackerman, M.: Sophus Lie’s 1884 Differential Invariant Paper. Math Sci Press, Berkeley (1976)

    Google Scholar 

  2. Atzmon, M., Maron, H., Lipman, Y.: Point convolutional neural networks by extension operators. ACM Trans. Graph. 37(4), 71:1–71:12 (2018)

    Google Scholar 

  3. Ben-Shabat, Y., Lindenbaum, M., Fischer, A.: 3DmFV: three-dimensional point cloud classification in real-time using convolutional neural networks. IEEE Robotics Autom. Lett. 3(4), 3145–3152 (2018)

    Article  Google Scholar 

  4. Bromley, J., Guyon, I., LeCun, Y., Säckinger, E., Shah, R.: Signature verification using a “siamese” time delay neural network. In: Advances in Neural Information Processing Systems, pp. 737–744 (1994)

    Google Scholar 

  5. Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Numerical Geometry of Non-rigid Shapes. Springer, New York (2008)

    MATH  Google Scholar 

  6. Bruckstein, A.M., Netravali, A.N.: On differential invariants of planar curves and recognizing partially occluded planar shapes. Ann. Math. Artif. Intell. 13(3–4), 227–250 (1995)

    Article  MathSciNet  Google Scholar 

  7. Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Comput. Vis. 26(2), 107–135 (1998)

    Article  Google Scholar 

  8. Carlevaris-Bianco, N., Eustice, R.M.: Learning visual feature descriptors for dynamic lighting conditions. In: 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2769–2776. IEEE, Piscataway (2014)

    Google Scholar 

  9. Cartan, E.: Geometry of Riemannian Spaces: Lie Groups; History, Frontiers and Applications Series, vol. 13. Math Science Press, , Berkeley (1983)

    Google Scholar 

  10. Chopra, S., Hadsell, R., LeCun, Y.: Learning a similarity metric discriminatively, with application to face verification. In: 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol. 1, pp. 539–546. IEEE, Piscataway (2005)

    Google Scholar 

  11. Fidler, T., Grasmair, M., Scherzer, O.: Identifiability and reconstruction of shapes from integral invariants. Inverse Probl. Imag. 2(3), 341–354 (2008)

    Article  MathSciNet  Google Scholar 

  12. Hadsell, R., Chopra, S., LeCun, Y.: Dimensionality reduction by learning an invariant mapping. In: 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), vol. 2, pp. 1735–1742. IEEE, Piscataway (2006)

    Google Scholar 

  13. Hong, B.W., Soatto, S.: Shape matching using multiscale integral invariants. IEEE Trans. Pattern Anal. Mach. Intell. 37(1), 151–160 (2014)

    Article  Google Scholar 

  14. Hu, J., Lu, J., Tan, Y.P.: Discriminative deep metric learning for face verification in the wild. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1875–1882 (2014)

    Google Scholar 

  15. Hua, B.S., Tran, M.K., Yeung, S.K.: Pointwise convolutional neural networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 984–993 (2018)

    Google Scholar 

  16. Huang, Q., Wang, W., Neumann, U.: Recurrent slice networks for 3d segmentation of point clouds. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2626–2635 (2018)

    Google Scholar 

  17. Kanezaki, A., Matsushita, Y., Nishida, Y.: Rotationnet: joint object categorization and pose estimation using multiviews from unsupervised viewpoints. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5010–5019 (2018)

    Google Scholar 

  18. Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)

    Article  MathSciNet  Google Scholar 

  19. Kimmel, R.: Numerical Geometry of Images: Theory, Algorithms, and Applications. Springer, New York (2012)

    MATH  Google Scholar 

  20. Klokov, R., Lempitsky, V.: Escape from cells: deep kd-networks for the recognition of 3D point cloud models. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 863–872 (2017)

    Google Scholar 

  21. Manay, S., Hong, B.W., Yezzi, A.J., Soatto, S.: Integral invariant signatures. In: European Conference on Computer Vision, pp. 87–99. Springer, Berlin (2004)

    Google Scholar 

  22. Masci, J., Boscaini, D., Bronstein, M., Vandergheynst, P.: Geodesic convolutional neural networks on riemannian manifolds. In: Proceedings of the IEEE International Conference on Computer Vision Workshops, pp. 37–45 (2015)

    Google Scholar 

  23. Maturana, D., Scherer, S.: Voxnet: a 3D convolutional neural network for real-time object recognition. In: Intelligent Robots and Systems (IROS), 2015 IEEE/RSJ International Conference on, pp. 922–928. IEEE, Piscataway (2015)

    Google Scholar 

  24. Pai, G., Wetzler, A., Kimmel, R.: Learning invariant representations of planar curves. In: International Conference on Learning Representations (2017)

    Google Scholar 

  25. Pottmann, H., Wallner, J., Huang, Q.X., Yang, Y.L.: Integral invariants for robust geometry processing. Comput. Aided Geom. Des. 26(1), 37–60 (2009)

    Article  MathSciNet  Google Scholar 

  26. Qi, C.R., Su, H., Mo, K., Guibas, L.J.: Pointnet: deep learning on point sets for 3D classification and segmentation. Proc. Comput. Vision Pattern Recogn. IEEE 1(2), 4 (2017)

    Google Scholar 

  27. Qi, C.R., Yi, L., Su, H., Guibas, L.J.: Pointnet+ +: Deep hierarchical feature learning on point sets in a metric space. In: Advances in Neural Information Processing Systems, pp. 5105–5114 (2017)

    Google Scholar 

  28. Riegler, G., Osman Ulusoy, A., Geiger, A.: Octnet: learning deep 3D representations at high resolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3577–3586 (2017)

    Google Scholar 

  29. Su, B.: Affine Differential Geometry. CRC Press, Boca Raton (1983)

    MATH  Google Scholar 

  30. Su, H., Maji, S., Kalogerakis, E., Learned-Miller, E.: Multi-view convolutional neural networks for 3D shape recognition. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 945–953 (2015)

    Google Scholar 

  31. Sun, Y., Wang, X., Tang, X.: Deep learning face representation from predicting 10,000 classes. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1891–1898 (2014)

    Google Scholar 

  32. Taigman, Y., Yang, M., Ranzato, M., Wolf, L.: Deepface: closing the gap to human-level performance in face verification. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1701–1708 (2014)

    Google Scholar 

  33. Wang, Y., Sun, Y., Liu, Z., Sarma, S.E., Bronstein, M.M., Solomon, J.M.: Dynamic graph CNN for learning on point clouds (2018). Preprint. arXiv: 1801.07829

    Google Scholar 

  34. Weingarten, J.W., Gruener, G., Siegwart, R.: A state-of-the-art 3D sensor for robot navigation. In: 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No. 04CH37566), vol. 3, pp. 2155–2160. IEEE, Piscataway (2004)

    Google Scholar 

  35. Weiss, I.: Noise-resistant invariants of curves. IEEE Trans. Pattern Anal. Mach. Intell. 15(9), 943–948 (1993)

    Article  Google Scholar 

  36. Wu, Z., Song, S., Khosla, A., Yu, F., Zhang, L., Tang, X., Xiao, J.: 3D shapenets: a deep representation for volumetric shapes. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1912–1920 (2015)

    Google Scholar 

  37. Xiang, Y., Choi, W., Lin, Y., Savarese, S.: Data-driven 3D voxel patterns for object category recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1903–1911 (2015)

    Google Scholar 

  38. Xie, J., Fang, Y., Zhu, F., Wong, E.: Deepshape: deep learned shape descriptor for 3D shape matching and retrieval. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1275–1283 (2015)

    Google Scholar 

  39. Yan, C., Xie, H., Yang, D., Yin, J., Zhang, Y., Dai, Q.: Supervised hash coding with deep neural network for environment perception of intelligent vehicles. IEEE Trans. Intell. Transp. Syst. 19(1), 284–295 (2017)

    Article  Google Scholar 

  40. Yarotsky, D.: Error bounds for approximations with deep relu networks. Neural Netw. 94, 103–114 (2017)

    Article  Google Scholar 

  41. Zhao, J., Xie, X., Xu, X., Sun, S.: Multi-view learning overview: recent progress and new challenges. Inf. Fusion 38, 43–54 (2017)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Computer Science, Technion, Haifa, Israel

    Gautam Pai & Ron Kimmel

  2. Faculty of Electrical Engineering, Technion, Haifa, Israel

    Mor Joseph-Rivlin

  3. Department of Applied Mathematics, Tel-Aviv University, Tel Aviv-Yafo, Israel

    Nir Sochen

Authors
  1. Gautam Pai
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  2. Mor Joseph-Rivlin
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  3. Ron Kimmel
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  4. Nir Sochen
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Corresponding author

Correspondence to Gautam Pai .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Pai, G., Joseph-Rivlin, M., Kimmel, R., Sochen, N. (2020). On Geometric Invariants, Learning, and Recognition of Shapes and Forms. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_16

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