Neural Implicit Surfaces in Higher Dimension

1IMPA
2PUC-Rio
3University of Coimbra

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description [Jan 27th 2022] Page online.

Abstract

This work investigates the use of neural networks admitting high-order derivatives for modeling dynamic variations of smooth implicit surfaces. For this purpose, it extends the representation of differentiable neural implicit surfaces to higher dimensions, which opens up mechanisms that allow to exploit geometric transformations in many settings, from animation and surface evolution to shape morphing and design galleries.
The problem is modeled by a $$k-$$parameter family of surfaces $$S_c$$, specified as a neural network function $$f : R^3 \times R^k \rightarrow R$$, where $$S_c$$ is the zero-level set of the implicit function $$f(\cdot, c) : R^3 \rightarrow R$$, with $$c \in R^k$$, with variations induced by the control variable $$c$$. In that context, restricted to each coordinate of $$R^k$$, the underlying representation is a neural homotopy which is the solution of a general partial differential equation (PDE).

Overview

Let $$S$$ be a surface in $$R^3$$, and $$g:R^3\rightarrow R$$ be an unknown neural implicit function. To compute the parameter set of $$g$$ such that $$g^{-1}(0)$$ approximates $$S$$, it is common to consider the Eikonal problem.
We augment the domain $$R^3$$ by a product space $$R^3\times R^k$$, where $$p\in R^3$$ denotes a point in the space and $$c\in R^k$$ denotes a vector that control $$k$$ directions of possible animations of the initial implicit surface. In this work we focus on the space-time $$R^3\times R$$.
Specifically, let $$f:R^3\times R \rightarrow R$$ be a smooth neural function representing an one-parameter family of neural implicit functions $$f_t:R^3\rightarrow R$$ defined by $$f_t(p)=f(p,t)$$. Topologically, the family $$f_t$$ represents homotopies between any two functions $$f_{t_0},\,f_{t_1}:R^3\rightarrow R$$, with $$t_0, t_1\in R$$. Thus, we say that $$f$$ is a neural homotopy function. The underlying family of neural implicit surfaces $$S_t = f_t^{-1} (0)$$ is a neural animation of the initial surface $$S_0=S$$. Restricted to the interval $$[t_0,t_1]$$, the surfaces $$S_t$$ represent a neural morphing between the neural implicit surfaces $$S_{t_0}$$ and $$S_{t_1}$$.
In the same way that we considered a neural implicit function as a solution of the Eikonal problem, we can consider the neural homotopy $$f: R^3\times R\rightarrow R$$ to be a solution of a general PDE problem $$\mathcal{F}\big(\nabla^n f, \dots, \nabla^1 f, f, p, t\big)=0$$. Initial-boundary conditions can also be used. In this work, we consider several definitions of $$\mathcal{F}$$ to model problems such as animation, smoothing, morphing, keyframe interpolation, etc. The time $$t$$ allows a continuous navigation in the neural animation $$S_t=f_t^{-1}(0)$$.

Paper

Neural Implicit Surfaces in Higher Dimension

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

description Paper preprint (PDF, 6.0 MB)
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Citation

@article{novello2022neural,
title = {Neural Implicit Surfaces in Higher Dimension},
author = {Novello, Tiago and da Silva, Vin\'icius and Schardong, Guilherme and Schirmer,
Luiz and Lopes, H\'elio and Velho, Luiz},
journal = {arXiv:2201.09636},
year = {2022},
month = jan
}



Acknowledgements

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.