Geometric implicit neural representations for signed distance functions

TL;DR

Geometric implicit neural representations (INRs) extend neural surface modeling by enforcing differential-geometry constraints on signed distance functions, notably the Eikonal condition $||∇f||=1$ and normal alignment, to recover smooth surfaces from oriented point clouds and posed images. The framework combines data fidelity terms with geometry-aware regularizers, employing point-based and image-based losses, curvature-aware sampling, and efficient inference via sphere tracing. It surveys seminal methods (SIREN, IGR) and image-based approaches (IDR, NeuS, Neuralangelo), and discusses advances in multiresolution and dynamic INRs for scalable, time-evolving surfaces. The work highlights practical benefits for accurate, differentiable surface reconstruction and animation, enabling robust geometry processing in 3D vision and graphics pipelines.

Abstract

\textit{Implicit neural representations} (INRs) have emerged as a promising framework for representing signals in low-dimensional spaces. This survey reviews the existing literature on the specialized INR problem of approximating \textit{signed distance functions} (SDFs) for surface scenes, using either oriented point clouds or a set of posed images. We refer to neural SDFs that incorporate differential geometry tools, such as normals and curvatures, in their loss functions as \textit{geometric} INRs. The key idea behind this 3D reconstruction approach is to include additional \textit{regularization} terms in the loss function, ensuring that the INR satisfies certain global properties that the function should hold -- such as having unit gradient in the case of SDFs. We explore key methodological components, including the definition of INR, the construction of geometric loss functions, and sampling schemes from a differential geometry perspective. Our review highlights the significant advancements enabled by geometric INRs in surface reconstruction from oriented point clouds and posed images.

Figures (16)

• Figure 1: Geometric INR pipeline: The input data can be either an oriented point cloud $\{p_i, N_i\}$ or a set of posed images $\{\mathscr{I}_j\}$. A neural network $f$ is then defined to fit a solution to the Eikonal equation. To train $f$, we define a loss function consisting of two terms: data constraint and Eikonal constraint. For the point-based data, we simply enforce $f(p_i) = 0$ and $\nabla f(p_i) = N_i$. For the image-based data, we rely on volume rendering techniques. Finally, gradient descent is used to optimize the resulting loss function.
• Figure 2: Neural implicit surfaces approximating the Armadillo model. The columns indicate the zero-level sets after $29$, $52$, $76$, and $100$ epochs of training. Line $1$ shows the results using minibatches sampled uniformly. Line $2$ presents the results using the adapted sampling of minibatches with $10\%$ / $70\%$ / $20\%$ of points with low/medium/high features. Image from novello2022exploring.
• Figure 3: Sphere tracing of neural SDFs representing the Armadillo and Bunny models. Both INRs have the same architecture and were trained on the same data during $500$ epochs. Image adapted from novello2022exploring
• Figure 4: Visual comparison of the discrete and continuous mean curvatures of the Dragon model. The top row shows the discrete mean curvature, while the bottom row shows the mean curvatures computed from the neural SDF using PyTorch framework. Image from novello2022exploring
• Figure 5: Principal curvatures and directions of the Dragon. Maximal curvatures are shown on the left, while minimal curvatures are on the right. Notice how their directions align nicely with the mesh's ridges and valleys.