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Structures for deep learning and topology optimization of functions on 3D shapes

Abstract : The field of geometry processing is following a similar path as image analysis with the explosion of publications dedicated to deep learning in recent years. An important research effort is being made to reproduce the successes of deep learning 2D computer vision in the context of 3D shape analysis. Unlike images shapes comes in various representations like meshes or point clouds which often lack canonical structure. This makes traditional deep learning algorithms like Convolutional Neural Networks (CNN) non straightforward to apply to 3D data. In this thesis we propose three main contributions:First, we introduce a method to compare functions on different domains without correspondences and to deform them to make the topology of their set of levels more alike. We apply our method to the classical problem of shape matching in the context of functional maps to produce smoother and more accurate correspondences. Furthermore, our method is based on the continuous optimization of a differentiable energy with respect to the compared functions and is applicable to deep learning. We make two direct contributions to deep learning on 3D data. We introduce a new convolution operator over triangles meshes based on local polar coordinates and apply it to deep learning on meshes. Unlike previous works our operator takes all choices of polar coordinates into account without loss of directional information. Lastly we introduce a new rotation invariant convolution layer over point clouds and show that CNNs based on this layer can outperform state of the art methods in standard tasks on un-alligned datasets even with data augmentation.
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Submitted on : Thursday, June 11, 2020 - 4:03:15 PM
Last modification on : Friday, June 12, 2020 - 3:52:03 AM

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Adrien Poulenard. Structures for deep learning and topology optimization of functions on 3D shapes. Computational Geometry [cs.CG]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX007⟩. ⟨tel-02865275⟩

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