Deep learning on curved surfaces
Zhiyu Sun, Joël St‐Aubin, Yong Chen, Xuan Song, Jia Lu, Shaoping Xiao, Stephen Baek · 2019
In many fields of science and engineering, geometric features play a key role in understanding certain quantity or phenomena. Thus, an ability to codify geometric features into a mathematical quantity and formulate an equation that describes a certain scientific quantity in terms of geometric features can be critical. Recently, artificial neural networks (ANNs), and in particular convolutional neural networks (CNNs) have demonstrated an exceptional capacity to discern visual patterns from digital images and signals. However, the application of such capable CNNs algorithm has been quite limited to mostly computer vision problems where visual information is inherently given on a grid-like structure. This unfortunately was not the case for many geometric pattern recognition problems defined on curved surfaces, or manifolds. A major technical challenge to this end is that, often times, a canonical tensor-like representation is not granted for these manifolds such that they are not compatible with the unprecedented advances of convolutional neural networks. Hence, many fields of science and engineering, where data points possess some manifold structure, cannot enjoy the full benefits of the recent advances in CNNs. The goal of this thesis is to address this issue by generalizing deep learning techniques, in particular, CNNs on curved surfaces. The key contribution of this work resides in concise but rigorous mathematical generalization of convolutional neural networks. The proposed approach permits the use of deep learning for various geometry processing applications without a loss of generality of the original convolutional neural networks. At the moment of publication, the proposed formulation demonstrates significantly better accuracy than other state-of-the-art methods on benchmark tests. The method also enjoys high scalability to different computational geometry problems, which will be demonstrated through a stress estimation problem on membrane surfaces.