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Convolutional Neural Networks on Surfaces: Seamless Toric Covers

This lecture by Jiacheng Cheng on Feb 21, 2018, explains defining and implementing CNNs on surfaces using seamless toric covers. It covers problem statements, translations, surfaces, mappings, layers, data generation, human body segmentation, and biological landmark detection. The lecture concludes with a discussion on limitations and future work.

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Convolutional Neural Networks on Surfaces: Seamless Toric Covers

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  1. Lecture 14: Convolutional Neural Networks on Surfaces via Seamless Toric Covers Jiacheng Cheng Feb, 21, 2018

  2. 1 Convolutional neural networkson surfacesviaseamlesstoriccovers HaggaiMaron,MeiravGalun,NoamAigerman,MiriTrope,NadavDym,ErsinYumer, Vladimir G. Kim, YaronLipman

  3. Problemstatement Easy Hard

  4. DeepLearning Geometric Deep Learning

  5. WhatisrequiredtodefineaCNN?

  6. Translations • Two dimensional,commutative Isometries ofR^2 • Convolution • Linear • Translationinvariant • Pooling • Non-linear(max) • Sub-translation invariant

  7. DefiningCNNsonsurfaces

  8. Translations onsurfaces? • Translationonsurface≝locallyEuclideantranslation • Flowalongnon-vanishingvectorfields

  9. Flat torus! • Translations “modulo1” • Full translation invariance on the flattorus !

  10. Only thetorus! Index of vectorfield Euler characteristic • Poincaré-Hopf:Foracompactorientablesurface • Index–ameasureofthecomplexitynearavanishingpoint • Non-vanishingvectorfieldimpliesgenus1-torus

  11. 15 CNNonflattorus Cyclic padding

  12. 16 Recap • CNNiswell-definedoverflat-torus • RoadblocksforCNNonsphere-typesurfaces • Topological:NolocallyEuclideantranslationsonspheres • Geometrical:Theflattorusisflatandoursurfaceisnot

  13. 17 Solution: Map the surface to a flattorus

  14. Torus4-cover ≅

  15. 19 MappingtheTorustotheflatTorus ! Aigerman and Lipman,2015

  16. The pullbacktranslation !

  17. 22 Pull-back Translations: pull-back Euclideantranslations Two dimensional,commutative Conformalmaps ! Pull-backconvolution Linear Theorem: Translationinvariance Pull-backpooling Non-linear(max) Sub-translationinvariant

  18. 24 Newlayers projection cyclicpadding

  19. Datageneration Inputimage Labels

  20. 26 Testphase • Aggregation from differenttriplets • “Magnifyingglass” • Scale factor asweights • + + + =

  21. 27 Human bodysegmentation Train: 370models FAUST, MIT, SCAPE,ADOBE Test: 18models SHREC07

  22. Easyfunctions Raw • Normals • Average geodesicdistance • Wave kernelsignature Complex

  23. Human bodysegmentation Train: 370models FAUST, MIT, SCAPE,ADOBE Test: 18models SHREC07

  24. CNNappliedtootherdata

  25. Biological landmarksdetection • Train: 73teeth from BOYER • Onlycurvatureandscalefactor Test: 8 teeth fromBOYER

  26. 32 Biologicallandmarks

  27. 35 Conclusion • CNN of sphere-typesurfaces • Wedefinedameaningfulconvolutiononsurfaces • Learns from rawfeatures • ReusingCNNsoftwareforimages • Limitationsandfuturework • Scope:Onlyspheretypesurfaces • Nocanonicalchoicefortriplets(andconvolutions) • Learn aggregationoperator

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