Exploring Differential Geometry in Neural Implicits

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.

Figures (11)

• Figure 1: Gaussian and mean curvatures of the smooth Armadillo model.
• Figure 2: Discrete shape operator setting.
• Figure 3: Comparison between the ground truth Armadillo model (right) and its reconstructed neural implicit surface (left) after $1000$ epochs of training.
• Figure 4: Neural implicit surfaces trained to approximate the Armadillo. The columns indicate the neural surfaces after $100$, $200$, $300$, and $500$ epochs of training. Line $1$ shows the results using the SDF functional. Line $2$ also consider the alignment between the principal directions in the loss functional.
• Figure 5: Neural implicit surfaces approximating the Armadillo model. The columns indicate the zero-level sets of the neural implicit functions after $29$, $52$, $76$, and $100$ epochs of training. Line $1$ shows the results using minibatches sampled uniformly in $V$. Line $2$ presents the results using the adapted sampling of minibatches with $10\%$ / $70\%$ / $20\%$ of points with low/medium/high features.