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Spherical Convolutional Neural Networks Empirical analysis of SCNNs LTS2 Prof. Pierre Vandergheynst Sup. Michal Defferrard Nathanal Perraudin Master Thesis - Frdrick Gusset 16.07.2019 Introduction CNNs are very powerful

  1. Spherical Convolutional Neural Networks Empirical analysis of SCNNs LTS2 Prof. Pierre Vandergheynst Sup. Michaël Defferrard Nathanaël Perraudin Master Thesis - Frédérick Gusset 16.07.2019

  2. Introduction ● CNNs are very powerful tools in Deep Learning Equivariance to translation ○ [1] 2

  3. Introduction ● Different symmetries such as rotations Use of sphere S 2 or SO(3) domain ○ Cosmological maps [2] 3D objects Omnidirectional imaging [3] 3

  4. Sphere representation HEALPix [4] Equiangular [2] Polyhedron [2] Iso-latitude ● Same area coverage ● Hierarchical ● 4

  5. Equivariance Example: Segmentation Rotation [5] 5

  6. Spherical CNNs ● 2D CNNs on planar projection ○ not desired rotation equivariance Planar projection [6] 6

  7. Spherical CNNs ● 2D CNNs on planar projection ○ not desired rotation equivariance Planar projection [6] Spherical Fourier Transform ● ○ computationally expensive Spherical Harmonics [7] 7

  8. Spherical CNNs ● 2D CNNs on planar projection ○ not desired rotation equivariance Planar projection [6] Spherical Fourier Transform ● ○ computationally expensive ● Graph CNN Spherical Harmonics [7] Graph of USA 8

  9. DeepSphere [2] Advantages ● Similar to standard CNN (computationally efficient) Can operate with any graph (flexible) ● 9

  10. DeepSphere [2] Advantages ● Similar to standard CNN (computationally efficient) Can operate with any graph (flexible) ● Differences Almost rotation equivariant (graph construction) ● Equivariant only on S 2 , but invariant to 3rd rotation of SO(3) ● 10

  11. Different tasks Shape retrieval and classification ● ○ SHREC17 and ModelNet40 11

  12. Different tasks Shape retrieval and classification ● ○ SHREC17 and ModelNet40 ● Global and Dense regression ○ GHCN-daily, planetarian data 12

  13. Different tasks Shape retrieval and classification ● ○ SHREC17 and ModelNet40 ● Global and Dense regression ○ GHCN-daily, planetarian data ● Segmentation ○ Climate Pattern Detection 13

  14. SHREC17 ● Shape retrieval contest 55 classes: [airplane, drawer, lamp, …] Spherical signal → All orientations in 3D ● Back Front Front Back Ray-casting on a sphere Distance feature 14

  15. SHREC17 - Results local filter 4 to 40 times faster 15

  16. Equiangular Tested on SHREC17 16

  17. SHREC17 - Time evaluation 17

  18. ModelNet40 ● Shape classification - similar to SHREC17 Accuracy 18

  19. ModelNet40 Logits evolution Confusion matrix 19

  20. GHCN-daily ● Non-uniform sampling → prove DeepSphere flexibility No specific task ● Temperature over the globe 20

  21. GHCN-daily Dense regression Find future temperature 21

  22. GHCN-daily Dense regression Global regression Find future temperature Find day in year 22

  23. Climate Pattern Detection ● Segmentation problem 23

  24. Climate Pattern Detection Results 24

  25. Conclusion ● Computationally 4 to 40 times faster ● Similar results to the other SCNNs Invariance to 3rd rotation is an unnecessary price to pay ○ ● Sufficiently equivariant to rotation ● Works on any sampling, as long as a graph is built and pooling operation adapted 25

  26. Thanks for your attention Questions? 26

  27. Equivariance to rotation N side = 32 27

  28. New graph ● Sampling density 28

  29. New graph ● Sampling density DeepSphere V2 ● 29

  30. Equiangular 30

  31. Overfit 31

  32. Bibliography 1. Li et al., 2018, Deeply Supervised Rotation Equivariant Network for Lesion Segmentation in Dermoscopy Images 2. Perraudin et al., 2018 3. http://cmp.felk.cvut.cz/cmp/demos/Omni/omni-ibr/, 14.07.2019 4. https://healpix.sourceforge.io/, 14.07.2019 5. https://www.machinelearningtutorial.net/2018/01/11/dynamic-routing-between-capsules-a-novel-archit ecture-for-convolutional-neural-networks/ 6. Cohen et al., 2018 7. Starry documentation (rodluger.github.io/starry) 32

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