Neural Implicit Surface Evolution

3University of Coimbra

Neural implicit surface evolutions using our method: interpolation between implicit surfaces, deformation driven by vector fields, and smoothing using the mean curvature equation.


description [Oct 2nd 2023] Code available.
description [Jul 14th 2023] Paper accepted to #ICCV23.
description [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:


Vector Field: Bob

Vector Field: Spot

Smoothing: Bunny

Smoothing: Dumbbell

Interpolation: Torus -> Bob

Interpolation: Spot -> Bob

Interpolation: Spot -> Bob

Interpolation: Falcon -> Witch


Neural Implicit Surface Evolution

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

description Paper preprint (PDF)
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Please send feedback and questions to Tiago Novello.


    author = {Novello, Tiago and da Silva, Vin\'icius and Schardong, Guilherme and Schirmer, Luiz and
            Lopes, H\'elio and Velho, Luiz},
    title = {Neural Implicit Surface Evolution},
    booktitle = {Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)},
    month = {October},
    year = {2023},
    url = {},
    pages = {14279-14289}


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.