Exploring differential geometry in neural implicits. (November 2022) - Novello, Tiago; Schardong, Guilherme; Schirmer, Luiz; da Silva, Vinícius; Lopes, Hélio; Velho, Luiz
Abstract: We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions . To train a neural implicit function, we propose a loss functional that approximates a signed distance function, and allows terms with high-order derivatives, such as the alignment between the principal directions of curvature, to learn more geometric details. During training, we consider a non-uniform sampling strategy based on the curvatures of the point-sampled surface to prioritize points with more geometric details. This sampling implies faster learning while preserving geometric accuracy when compared with previous approaches. We also use the analytical derivatives of a neural implicit function to estimate the differential measures of the underlying point-sampled surface. Graphical abstract: Highlights: Introduction of a neural implicit framework to approximate the signed distance functions of surfaces by smooth neural networks using differential geometry. Definition of a loss function that allows the exploration of tools from differential geometry during the training of the network. During the training, we use the discrete differential geometry of the dataset to sample important regions. We also use the analytical derivatives of the neural network to estimate the curvatures of the underlying point-sampled surface.