📢New ICCV Paper: Neural Surface Evolution!📢

— Vinícius da Silva (@dsilvavinicius) July 14, 2023

Do you want to evolve your implicit surface using the Level Set equation without any supervision at the boundary conditions or using discretized methods such as FEM? Follow the 🧵 (rendering powered by @nvidiaomniverse). pic.twitter.com/sofIf7pqpI

[Jul 20th 2023] Page online.

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 \(R^3\!\!\times\! R\), 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 \(R^3\!\! \times\! \{0\}\). 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 when compared with the standard approach.

Let \(g:R^3\rightarrow R\) be a *neural implicit function* (coord-based neural network)
approximating the signed distance function of a given surface \(S\).

This work investigates the extension of the domain of \(g\) to the *space-time* \(R^3\times R\),
encoding the evolution of \(g\) as a higher-dimensional function \(f:R^3\times R \to R\).
The resulting animation is governed by the *level set equation* (LSE)
\(\frac{\partial f}{\partial t}=v|{\nabla f}|\),
which encodes the propagation of the level sets \(S_t\) of \(f(\cdot, t)\) towards their normals with speed \(v\).
The choice of the function \(v\) depends on the underlying geometric model.

We propose representing this implicit evolution by a (coord-based) neural network \(f_\theta:R^3\times R\to R\)
and define the following loss functional to train \(f_\theta\) to approximate a solution to the LSE problem:

\(L(\theta)\!=\!\!\!\!\! \int\limits_{R^3\times {0}}\!\!\!\left(f_\theta\!-\!g\right)^2 dx+\!\!\!\!\!\!\! \int\limits_{R^3\times (a,b)}\!\!\!\!\!\!\big(\frac{\partial f_\theta}{\partial t}\!-\!v|{\nabla f_\theta}|\big)^2 dx dt\).

We present three examples of neural implicit evolution using the LSE. For this, we only need to define the function \(v\) for each example:

- First, we consider moving the surface \(S\) towards a vector field \(V:R^3\to R^3\), in this case, \(v=\langle V,N\rangle\), where \(N\) is the normal field.
- Second, we move \(S\) by the mean curvature equation which is expressed as a LSE with \(v=div (N)\).
- Finally, we deform \(g\) into a second function \(h\) using a LSE with \(v=-(h-f)\).

**Neural Implicit Surface Evolution**

Tiago Novello, Vinícius da Silva, Guilherme Schardong, Luiz Schirmer, Hélio Lopes and Luiz Velho

Please send feedback and questions to Tiago Novello.

```
@article{novello2022neural,
title = {Neural Implicit Surface Evolution},
author = {Novello, Tiago and da Silva, Vin\'icius and Schardong, Guilherme and Schirmer,
Luiz and Lopes, H\'elio and Velho, Luiz},
booktitle={Proceedings of the IEEE/CVF international conference on computer vision},
year = {2023}
}
```

We would like to thank
Towaki Takikawa,
Joey Litalien,
Kangxue Yin,
Karsten Kreis,
Charles Loop,
Derek Nowrouzezahrai,
Alec Jacobson,
Morgan McGuire and
Sanja Fidler
for licensing the code of the paper Neural Geometric Level of Detail:
Real-time Rendering with Implicit 3D Surfaces and project page under the MIT License. This website is based on that page.

*We also thank the Stanford Computer Graphics Laboratory for the Bunny, Dragon, Armadillo, and Happy Buddha, acquired through the Stanford 3D scan repository. Finally, we thank Keenan Crane for the Spot and Bob models.
*