Neural Implicit Surface Evolution
Tiago Novello, Vinicius da Silva, Guilherme Schardong, Luiz Schirmer, Helio Lopes, Luiz Velho
TL;DR
The paper tackles evolving implicit surfaces under the level set equation by extending neural implicit representations to a space-time domain and learning a single smooth network to encode the entire surface animation. It enforces the LSE constraint via a PDE loss together with data terms for initial conditions, enabling continuous evolution without numerical LSE discretization. Key contributions include space-time neural implicit surfaces, an LSE-based training framework with initial-condition data, a novel initialization strategy, and demonstrations on vector-field, mean-curvature, and interpolation evolutions. The approach yields compact, differentiable surface evolutions suitable for graphics and geometry processing, with potential for faster training and robust handling of topological changes.
Abstract
This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time $\mathbb{R}^3\times \mathbb{R}$, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually $\mathbb{R}^3 \times \{0\}$. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.