nach oben

International Journal of Computer Vision

Erschienen in:

06.09.2019

Learning SO(3) Equivariant Representations with Spherical CNNs

verfasst von: Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia, Kostas Daniilidis

Erschienen in: International Journal of Computer Vision | Ausgabe 3/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations are independent of the underlying spherical resolution throughout the network. We show that networks with much lower capacity and without requiring data augmentation can exhibit performance comparable to the state of the art in standard 3D shape retrieval and classification benchmarks.

Vorheriger Artikel Robust Fitting in Computer Vision: Easy or Hard?

Nächster Artikel EKLT: Asynchronous Photometric Feature Tracking Using Events and Frames

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

The first version of this work was submitted to CVPR on 11/15/2017, shortly after we became aware of Cohen et al. (2018) ICLR submission on 10/27/2017.

In a CNN setting, f represents inputs/feature maps, and h the learned filters.

For the experiments in Table 6, one epoch for the WAP model in the first row takes 234 s, versus 132 s for the SP model in the third row, both on a Nvidia 1080 Ti.

Arfken, G. (1966). Mathematical methods for physicists. No. v. 2 in Mathematical methods for physicists. New York: Academic Press.

Bai, S., Bai, X., Zhou, Z., Zhang, Z., & Jan Latecki, L. (2016). Gift: A real-time and scalable 3d shape search engine. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 5023–5032).

Boscaini, D., Masci, J., Rodolà, E., & Bronstein, M. (2016). Learning shape correspondence with anisotropic convolutional neural networks. In Advances in neural information processing systems (pp. 3189–3197).

Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., & Vandergheynst, P. (2017). Geometric deep learning: Going beyond euclidean data. IEEE Signal Processing Magazine, 34(4), 18–42.CrossRef

Bruna, J., Szlam, A., & LeCun, Y. (2013a). Learning stable group invariant representations with convolutional networks. arXiv preprint arXiv:1301.3537.

Bruna, J., Zaremba, W., Szlam, A., & LeCun, Y. (2013b). Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:1312.6203.

Bruna, J., Zaremba, W., Szlam, A., & LeCun, Y. (2013c). Spectral networks and locally connected networks on graphs. CoRR arXiv:1312.6203v3.

Chang, A. X., Funkhouser, T., Guibas, L., Hanrahan, P., Huang, Q., Li, Z., et al. (2015). Shapenet: An information-rich 3d model repository. CoRR arXiv:1512.03012v1.

Cohen, T. S., & Welling, M. (2016). Group equivariant convolutional networks. arXiv preprint arXiv:1602.07576.

Cohen, T. S., Geiger, M., Köhler, J., & Welling, M. (2018). Spherical CNNs. In International conference on learning representations.

Defferrard, M., Bresson, X., & Vandergheynst, P. (2016). Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in neural information processing systems (pp. 3844–3852).

Dieleman, S., Willett, K. W., & Dambre, J. (2015). Rotation-invariant convolutional neural networks for galaxy morphology prediction. Monthly Notices of the Royal Astronomical Society, 450(2), 1441–1459.CrossRef

Driscoll, J. R., & Healy, D. M. (1994). Computing fourier transforms and convolutions on the 2-sphere. Advances in Applied Mathematics, 15(2), 202–250.MathSciNetCrossRef

Frome, A., Huber, D., Kolluri, R., Bülow, T., & Malik, J. (2004). Recognizing objects in range data using regional point descriptors. In T. Pajdla & J. Matas (Eds.), Computer Vision - ECCV 2004. ECCV 2004. Lecture notes in computer science (vol. 3023). Berlin, Heidelberg: Springer.

Furuya, T., & Ohbuchi, R. (2016). Deep aggregation of local 3d geometric features for 3d model retrieval. In BMVC (p. 121).

Gens, R., & Domingos, P. M. (2014). Deep symmetry networks. In Advances in neural information processing systems (pp. 2537–2545).

Górski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D., Hansen, F. K., Reinecke, M., et al. (2005). HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. The Astrophysical Journal, 622, 759–771. https://doi.org/10.1086/427976.CrossRef

Healy, D. M., Rockmore, D. N., Kostelec, P. J., & Moore, S. (2003). Ffts for the 2-sphere-improvements and variations. Journal of Fourier Analysis and Applications, 9(4), 341–385.MathSciNetCrossRef

Hel-Or, Y., & Teo, P. C. (1996). Canonical decomposition of steerable functions. In Computer vision and pattern recognition, 1996. Proceedings CVPR’96, 1996 IEEE computer society conference on (pp. 809–816). IEEE.

Jaderberg, M., Simonyan, K., & Zisserman, A., et al. (2015). Spatial transformer networks. In Advances in neural information processing systems (pp. 2017–2025).

Kanezaki, A., Matsushita, Y., & Nishida, Y. (2018). Rotationnet: Joint object categorization and pose estimation using multiviews from unsupervised viewpoints. In Proceedings of IEEE international conference on computer vision and pattern recognition (CVPR).

Kazhdan, M., & Funkhouser, T. (2002). Harmonic 3d shape matching. In ACM SIGGRAPH 2002 conference abstracts and applications (pp. 191–191). New York: ACM.

Kipf, T. N., & Welling, M. (2016). Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907.

Klokov, R., & Lempitsky, V. (2017). Escape from cells: Deep kd-networks for the recognition of 3d point cloud models. In International conference on compute vision (ICCV) (pp. 863–872).

Lebedev, N., & Silverman, R. (1972). Special functions and their applications. Dover Books on Mathematics, Dover Publications.

Lenc, K., & Vedaldi, A. (2015). Understanding image representations by measuring their equivariance and equivalence. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 991–999).

Li, J., Chen, B M., & Lee, G. H. (2018). SO-net: Self-organizing network for point cloud analysis. CoRR arXiv:1803.04249v4.

Makadia, A., & Daniilidis, K. (2010). Spherical correlation of visual representations for 3d model retrieval. International Journal of Computer Vision, 89(2), 193–210.CrossRef

Marcos, D., Volpi, M., Komodakis, N., & Tuia, D. (2016). Rotation equivariant vector field networks. CoRR arXiv:1612.09346.

Masci, J., Boscaini, D., Bronstein, M., & Vandergheynst, P. (2015). Geodesic convolutional neural networks on Riemannian manifolds. In Proceedings of the IEEE international conference on computer vision workshops (pp. 37–45).

Maturana, D., & Scherer, S. (2015). Voxnet: A 3d convolutional neural network for real-time object recognition. In 2015 IEEE/RSJ international conference on intelligent robots and systems, IROS 2015, Hamburg, Germany, September 28–October 2, 2015 (pp. 922–928). https://doi.org/10.1109/IROS.2015.7353481.

Monti, F., Boscaini, D., Masci, J., Rodolà, E., Svoboda, J., & Bronstein, M. M. (2016). Geometric deep learning on graphs and manifolds using mixture model CNNs. arXiv preprint arXiv:1611.08402.

Qi, C. R., Su, H., Nießner, M., Dai, A., Yan, M., & Guibas, LJ. (2016). Volumetric and multi-view CNNs for object classification on 3d data. In 2016 IEEE conference on computer vision and pattern recognition, CVPR 2016, Las Vegas, NV, USA, June 27–30, 2016 (pp. 5648–5656). https://doi.org/10.1109/CVPR.2016.609.

Qi, C. R., Su, H., Mo, K., & Guibas, L. J. (2017a). Pointnet: Deep learning on point sets for 3d classification and segmentation. In Processing computer vision and pattern recognition (CVPR) (Vol. 1(2), p. 4). IEEE.

Qi, C. R., Yi, L., Su, H., & Guibas, LJ. (2017b). Pointnet++: Deep hierarchical feature learning on point sets in a metric space. In Advances in neural information processing systems (pp. 5105–5114).

Rippel, O., Snoek, J., & Adams, R. P. (2015). Spectral representations for convolutional neural networks. CoRR arXiv:1506.03767

Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., et al. (2015). Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3), 211–252. https://doi.org/10.1007/s11263-015-0816-y.MathSciNetCrossRef

Savva, M., Yu, F., Su, H., Kanezaki, A., Furuya, T., et al. (2017). Shrec’17 track: Large-scale 3d shape retrieval from shapenet core55. In 10th Eurographics workshop on 3D object retrieval (pp. 1–11).

Schroff, F., Kalenichenko, D., & Philbin, J. (2015). Facenet: A unified embedding for face recognition and clustering. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 815–823).

Segman, J., Rubinstein, J., & Zeevi, Y. Y. (1992). The canonical coordinates method for pattern deformation: Theoretical and computational considerations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(12), 1171–1183.CrossRef

Su, H., Maji, S., Kalogerakis, E., & Learned-Miller, E. (2015). Multi-view convolutional neural networks for 3d shape recognition. In Proceedings of the IEEE international conference on computer vision (pp. 945–953).

Tatsuma, A., & Aono, M. (2009). Multi-fourier spectra descriptor and augmentation with spectral clustering for 3d shape retrieval. The Visual Computer, 25(8), 785–804.CrossRef

Thurston, W. P. (1997). Three-dimensional geometry and topology (Vol. 1). Princeton, NJ: Princeton University Press.CrossRef

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

Worrall, D. E., Garbin, S. J., Turmukhambetov, D., & Brostow, G. J. (2016). Harmonic networks: Deep translation and rotation equivariance. arXiv preprint arXiv:1612.04642.

Worrall, D. E., Garbin, S. J., Turmukhambetov, D., & Brostow, G. J. (2017). Harmonic networks: deep translation and rotation equivariance. In Proceedings IEEE conference on computer vision and pattern recognition (CVPR) (Vol. 2, pp. 5028–5037).

Wu, Z., Song, S., Khosla, A., Yu, F., Zhang, L., Tang, X., & Xiao, J. (2015). 3d shapenets: A deep representation for volumetric shapes. In IEEE conference on computer vision and pattern recognition, CVPR 2015, Boston, MA, USA, June 7–12, 2015 (pp. 1912–1920). https://doi.org/10.1109/CVPR.2015.7298801.

Yi, L., Su, H., Guo, X., & Guibas, L. (2016). SyncSpecCNN: Synchronized spectral CNN for 3d shape segmentation. arXiv preprint arXiv:1612.00606.

Zhang, R. (2019). Making convolutional networks shift-invariant again. In International conference on machine learning (ICML)

Zhou, Y., Ye, Q., Qiu, Q., & Jiao, J. (2017). Oriented response networks. In The IEEE conference on computer vision and pattern recognition (CVPR).

Titel: Learning SO(3) Equivariant Representations with Spherical CNNs
verfasst von: Carlos Esteves
Christine Allen-Blanchette
Ameesh Makadia
Kostas Daniilidis
Publikationsdatum: 06.09.2019
Verlag: Springer US
Erschienen in: International Journal of Computer Vision / Ausgabe 3/2020
Print ISSN: 0920-5691
Elektronische ISSN: 1573-1405
DOI: https://doi.org/10.1007/s11263-019-01220-1

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3/2020

Differential Scene Flow from Light Field Gradients

GANimation: One-Shot Anatomically Consistent Facial Animation

Jointly Discovering Visual Objects and Spoken Words from Raw Sensory Input

Robust Fitting in Computer Vision: Easy or Hard?

Correction to: EKLT: Asynchronous Photometric Feature Tracking Using Events and Frames

DeepIM: Deep Iterative Matching for 6D Pose Estimation

Premium Partner