Implicit Geometric Regularization for Learning Shapes (2002.10099v2)

Abstract: Representing shapes as level sets of neural networks has been recently proved to be useful for different shape analysis and reconstruction tasks. So far, such representations were computed using either: (i) pre-computed implicit shape representations; or (ii) loss functions explicitly defined over the neural level sets. In this paper we offer a new paradigm for computing high fidelity implicit neural representations directly from raw data (i.e., point clouds, with or without normal information). We observe that a rather simple loss function, encouraging the neural network to vanish on the input point cloud and to have a unit norm gradient, possesses an implicit geometric regularization property that favors smooth and natural zero level set surfaces, avoiding bad zero-loss solutions. We provide a theoretical analysis of this property for the linear case, and show that, in practice, our method leads to state of the art implicit neural representations with higher level-of-details and fidelity compared to previous methods.

Implicit Geometric Regularization for Learning Shapes

The paper "Implicit Geometric Regularization for Learning Shapes" by Amos Gropp et al. presents a novel method for constructing high-fidelity implicit neural representations of shapes directly from raw data, such as point clouds with or without normal information. The authors identify and leverage the implicit geometric regularization property of a simple loss function, which drives the neural network to produce smooth and natural zero level set surfaces. This approach avoids unfavorable zero-loss solutions prevalent in previous methods for learning shapes from implicit representations.

Methodology

The core idea revolves around defining shapes as level sets of multi-layer perceptrons (MLPs). Unlike traditional methods that rely on pre-computed implicit representations or explicit loss functions over neural level sets, this method extracts shapes directly from raw data. The proposed loss function encourages the neural network to vanish on the input point cloud and to maintain a unit norm gradient. This loss function can be expressed as:

$$\ell(\theta) = \ell_{\mathcal{D}}(\theta) + \lambda \mathbb{E}_{X} \left( \|\nabla f(X; \theta)\|_2^2 - 1 \right)^2$$

Here, $\ell_{\mathcal{D}}(\theta)$ ensures the neural network outputs zero at the input points, while the Eikonal term encourages unit norm gradients. Theoretical analysis proves that even though the plane reproduction property holds true only locally for this simple linear case, the empirical evidence suggests this property extends well to more complex non-linear cases.

Key Results

The paper demonstrates the efficacy of this approach through multiple experiments:

• SDF Approximation: The model accurately learns signed distance functions (SDFs) for various shapes, achieving mean relative errors as low as 0.3% for planar data.
• Fidelity and Level of Details: Compared to regression-based methods such as DeepSDF, the implicit geometric regularization method generates significantly more detailed and high-fidelity reconstructions from raw point cloud data.
• Surface Reconstruction Benchmark: On the benchmark dataset consisting of complex 3D shapes, this method outperforms state-of-the-art deep learning chart-based surface reconstruction techniques across several metrics, including Chamfer and Hausdorff distances.

Practical and Theoretical Implications

The main advantage of this approach is its ability to construct implicit neural representations in a data-driven way, thereby eliminating the need for pre-computed implicit representations. The implications for 3D shape analysis and reconstruction are noteworthy:

• Higher Fidelity Representations: This method's ability to capture intricate shape details and produce smooth surfaces is significantly advantageous for applications requiring high precision, such as medical imaging and detailed CAD models.
• Implicit Regularization: The empirical observation that the method avoids bad local minima and produces plausible reconstructions without explicit geometric constraints suggests a powerful implicit regularization effect in neural networks.

Future Directions

Future work may focus on further theoretical validation of the implicit geometric regularization phenomenon in non-linear models. Given the success demonstrated in these various contexts:

• Learning Complex Shape Spaces: Extending this method to more diverse datasets, particularly those with noisy or incomplete data, will provide further insights and improvements in reconstructive modeling.
• Incorporation into Generative Models: Integrating this regularization technique with generative adversarial networks (GANs) or variational autoencoders (VAEs) could enhance the fidelity of generated synthetic data for training downstream tasks.
• Differentiable Rendering: Applying this loss function in differentiable rendering pipelines promises improved performance in single-view 3D reconstruction and image-based shape modeling tasks.

Conclusion

This paper provides strong numerical evidence supporting the effectiveness of implicit geometric regularization for learning high-fidelity neural shape representations from raw data. It stands out by achieving superior results over previous methods without the need for explicit shape supervision or regularization constraints, making a substantial contribution to the field of machine learning-based 3D shape reconstruction. The theoretical and practical implications outlined pave the way for further exploration and application of implicit geometric regularization in advanced neural network tasks.