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Lecture 1 4 : Convolutional Neural Networks on Surfaces via Seamless Toric Covers

Lecture 1 4 : Convolutional Neural Networks on Surfaces via Seamless Toric Covers. Jiacheng Cheng Feb, 21, 2018. 1. Convolutional neural networks on surfaces via seamless toric covers.

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Lecture 1 4 : Convolutional Neural Networks on Surfaces via 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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