Deep learning on curved surfaces: manifold-formulation of convolutional neural networks and its operations

Deep neural networks have recently been proven to be powerful tools for a broad range of problems from computer vision, natural-language processing and audio analysis. For the past decades, these tools have achieved great success on data with an underlying Euclidean or grid-like structure such as speech, images or video on 1D-, 2D-, 3D Euclidean domains, respectively, where the invariances of these structures are built into networks used to model them. However, in the recent years, more and more fields have to deal with data residing on geometric domains where trivial Euclidean structure is not available, such as computer-aided design (CAD) models, three-dimensional (3D) scans, polygonal meshes and graphs.

This thesis explores a few approaches to generalize deep learning techniques to arbitrary 3d shapes. Upon our approaches, we go beyond the original geometric data structure, extracting the numerical fingerprint of the geometry that correlated to any quantity of interest dependent on the task, which permit seamless applications of various mathematical tools (Deep neural networks). Several interesting and challenge problems defined on arbitrary 3d shapes are successfully addressed in an efficient manner via data driven approach, building upon our generalization of deep learning techniques. For example, two popular tasks in the community of geometry processing, dense correspondence matching and sematic segmentation on 3d shapes are tackled in this thesis. And another interesting engineering problem, wall stress estimation on cerebral aneurysm, which typically lacks an analytic solution, is addressed via data driven approach.