Neural Implicit Surface Evolution

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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 \(R^3\!\!\times\! R\), 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.

Overview

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:

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

• 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 \(v=-(h-f)\).

Results

Paper

Neural Implicit Surface Evolution

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

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Citation

@InProceedings{Novello_2023_ICCV,
    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 = {https://openaccess.thecvf.com/content/ICCV2023/html/Novello_Neural_Implicit_Surface_Evolution_ICCV_2023_paper.html},
    pages = {14279-14289}
}

Acknowledgements