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$\nabla$-SDF: Learning Euclidean Signed Distance Functions Online with Gradient-Augmented Octree Interpolation and Neural Residual

Zhirui Dai, Qihao Qian, Tianxing Fan, Nikolay Atanasov

TL;DR

This work addresses online, large-scale Euclidean SDF reconstruction from streaming point clouds by merging an explicit gradient-augmented octree prior with an implicit neural residual. The method, $\nabla$-SDF, uses a semi-sparse octree to store $d_k$ and $\mathbf{g}_k$, a gradient-augmented interpolation to form $d_{ga}(\mathbf{x})$, and a multi-resolution hash-grid decoder to predict a residual $\delta_d(\mathbf{x})$, yielding $\hat{d}(\mathbf{x}) = d_{ga}(\mathbf{x}) + \delta_d(\mathbf{x})$. Training utilizes reconstruction, Eikonal, and projection losses to enforce surface fidelity and global consistency, enabling real-time performance. Experiments on Replica show that $\nabla$-SDF achieves higher mesh completion and strong SDF accuracy compared to Voxblox, H2-Mapping, PIN-SLAM, and HIO-SDF, while maintaining non-truncated, differentiable SDF estimates. The approach offers a scalable, differentiable representation with practical impact on localization, planning, and manipulation in robotics and computer vision.

Abstract

Estimation of signed distance functions (SDFs) from point cloud data has been shown to benefit many robot autonomy capabilities, including localization, mapping, motion planning, and control. Methods that support online and large-scale SDF reconstruction tend to rely on discrete volumetric data structures, which affect the continuity and differentiability of the SDF estimates. Recently, using implicit features, neural network methods have demonstrated high-fidelity and differentiable SDF reconstruction but they tend to be less efficient, can experience catastrophic forgetting and memory limitations in large environments, and are often restricted to truncated SDFs. This work proposes $\nabla$-SDF, a hybrid method that combines an explicit prior obtained from gradient-augmented octree interpolation with an implicit neural residual. Our method achieves non-truncated (Euclidean) SDF reconstruction with computational and memory efficiency comparable to volumetric methods and differentiability and accuracy comparable to neural network methods. Extensive experiments demonstrate that \methodname{} outperforms the state of the art in terms of accuracy and efficiency, providing a scalable solution for downstream tasks in robotics and computer vision.

$\nabla$-SDF: Learning Euclidean Signed Distance Functions Online with Gradient-Augmented Octree Interpolation and Neural Residual

TL;DR

This work addresses online, large-scale Euclidean SDF reconstruction from streaming point clouds by merging an explicit gradient-augmented octree prior with an implicit neural residual. The method, -SDF, uses a semi-sparse octree to store and , a gradient-augmented interpolation to form , and a multi-resolution hash-grid decoder to predict a residual , yielding . Training utilizes reconstruction, Eikonal, and projection losses to enforce surface fidelity and global consistency, enabling real-time performance. Experiments on Replica show that -SDF achieves higher mesh completion and strong SDF accuracy compared to Voxblox, H2-Mapping, PIN-SLAM, and HIO-SDF, while maintaining non-truncated, differentiable SDF estimates. The approach offers a scalable, differentiable representation with practical impact on localization, planning, and manipulation in robotics and computer vision.

Abstract

Estimation of signed distance functions (SDFs) from point cloud data has been shown to benefit many robot autonomy capabilities, including localization, mapping, motion planning, and control. Methods that support online and large-scale SDF reconstruction tend to rely on discrete volumetric data structures, which affect the continuity and differentiability of the SDF estimates. Recently, using implicit features, neural network methods have demonstrated high-fidelity and differentiable SDF reconstruction but they tend to be less efficient, can experience catastrophic forgetting and memory limitations in large environments, and are often restricted to truncated SDFs. This work proposes -SDF, a hybrid method that combines an explicit prior obtained from gradient-augmented octree interpolation with an implicit neural residual. Our method achieves non-truncated (Euclidean) SDF reconstruction with computational and memory efficiency comparable to volumetric methods and differentiability and accuracy comparable to neural network methods. Extensive experiments demonstrate that \methodname{} outperforms the state of the art in terms of accuracy and efficiency, providing a scalable solution for downstream tasks in robotics and computer vision.
Paper Structure (31 sections, 1 theorem, 15 equations, 6 figures, 4 tables)

This paper contains 31 sections, 1 theorem, 15 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Volumetric SDF Reconstruction
  4. Gaussian Process SDF Reconstruction
  5. Neural Network SDF Reconstruction
  6. Hybrid Methods for SDF Reconstruction
  7. Problem Statement
  8. $\nabla$-SDF: Efficient Network for Learning Globally Accurate SDF
  9. SDF Prior via Gradient-Augmented Octree Interpolation
  10. Semi-Sparse Octree
  11. Gradient-Augmented Interpolation
  12. SDF Residual via Neural Feature Decoding
  13. Key Frame Selection
  14. Dataset Generation
  15. Free-space Points
  16. ...and 16 more sections

Key Result

Proposition 1

Consider an octant $\mathcal{V} \subset \mathbb{R}^3$ of size $L$. Assume that the SDF $d(\mathbf{x})$ is twice differentiable and the spectral norm of its Hessian is bounded: Assume that each vertex $\mathbf{x}_k$ has ground-truth SDF value $d_k=d(\mathbf{x}_k)$ and gradient $\mathbf{g}_k=\nabla d(\mathbf{x}_k)$. Then, given arbitrary $\mathbf{x} \in \mathcal{V}$, the errors of gradient-augmente

Figures (6)

  • Figure 1: $\nabla$-SDF reconstructs an accurate Euclidean signed distance function online from streaming point cloud data.
  • Figure 2: Method Overview: a) We keep key frames with small overlap and those that maximize the surface coverage for training; b) with the selected key frames and the current frame, we generate three types of samples: surface points, perturbed points around the surface, and free-space points; c) to predict SDF, we first obtain an SDF prior $d_{ga}(\mathbf{x})$ with gradient-augmented interpolation in a semi-sparse octree, where each octant vertex has estimated SDF value and gradient; d) a multi-resolution hash grid with an MLP decoder is used to obtain an SDF residual correction $\delta_d(\mathbf{x})$; e) the SDF prior $d_{ga}(\mathbf{x})$ and the SDF residual $\delta_d(\mathbf{x})$ are combined as the final SDF prediction $\hat{d}(\mathbf{x}) = d_{ga}(\mathbf{x}) + \delta_d(\mathbf{x})$, and the parameters are trained with three loss functions: reconstruction loss, Eikonal loss and projection loss.
  • Figure 3: 2D visualization of SDF interpolation without gradient augmentation using (a) a sparse octree and (b) a semi-sparse with corresponding interpolation error shown in (c) and (d) respectively. The bottom-left red region is an obstacle containing one vertex.
  • Figure 4: 2D visualization of interpolation with and without gradient augmentation for one (red region, top row) and four obstacles (red regions, bottom row). Gradient-augmented interpolation produces a better SDF prior (b) with smaller error (d). Empirically, positions where the SDF gradient is not well defined (large Hessian spectral norm), as shown in (f), have small interpolation error with gradient augmentation as shown in (d).
  • Figure 5: Qualitative comparison of mesh reconstruction (top row) and z-plane slice of SDF reconstruction (bottom row) on Replica room 0 replica19arxiv. $\nabla$-SDF reconstructs a mesh with the highest completion ratio and accurate SDF both near and far from the surface. $\text{H}_2$-Mapping and PIN-SLAM only learn truncated SDF. HIO-SDF learns an over smooth result. Voxblox significantly under-estimates the SDF.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof