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Online Learning of Continuous Signed Distance Fields Using Piecewise Polynomials

Ante Marić, Yiming Li, Sylvain Calinon

TL;DR

The paper tackles online learning of signed distance fields for robotics by representing SDFs with piecewise Bernstein polynomials, enabling a $C^1$ continuous distance field with analytic gradients while avoiding storage of training data for prediction. It introduces an incremental online least-squares framework that uses a Sherman–Morrison–Woodbury update to refine a compact set of weights, starting from a prior and enforcing continuity through inter-segment constraints. Across real YCB objects, the method achieves reconstruction accuracy comparable to neural-network SDFs and GPIS baselines but with far fewer parameters and real-time query/update capabilities, demonstrated further in a physical grasping task. The approach offers a memory-efficient, gradient-accessible alternative for online scene understanding in manipulation, with clear paths to scalability via adaptive grids or hierarchical representations.

Abstract

Reasoning about distance is indispensable for establishing or avoiding contact in manipulation tasks. To this end, we present an online approach for learning implicit representations of signed distance using piecewise polynomial basis functions. Starting from an arbitrary prior shape, our method incrementally constructs a continuous and smooth distance representation from incoming surface points, with analytical access to gradient information. The underlying model does not store training data for prediction, and its performance can be balanced through interpretable hyperparameters such as polynomial degree and number of segments. We assess the accuracy of the incrementally learned model on a set of household objects and compare it to neural network and Gaussian process counterparts. The utility of intermediate results and analytical gradients is further demonstrated in a physical experiment. For code and video, see https://sites.google.com/view/pp-sdf/.

Online Learning of Continuous Signed Distance Fields Using Piecewise Polynomials

TL;DR

The paper tackles online learning of signed distance fields for robotics by representing SDFs with piecewise Bernstein polynomials, enabling a continuous distance field with analytic gradients while avoiding storage of training data for prediction. It introduces an incremental online least-squares framework that uses a Sherman–Morrison–Woodbury update to refine a compact set of weights, starting from a prior and enforcing continuity through inter-segment constraints. Across real YCB objects, the method achieves reconstruction accuracy comparable to neural-network SDFs and GPIS baselines but with far fewer parameters and real-time query/update capabilities, demonstrated further in a physical grasping task. The approach offers a memory-efficient, gradient-accessible alternative for online scene understanding in manipulation, with clear paths to scalability via adaptive grids or hierarchical representations.

Abstract

Reasoning about distance is indispensable for establishing or avoiding contact in manipulation tasks. To this end, we present an online approach for learning implicit representations of signed distance using piecewise polynomial basis functions. Starting from an arbitrary prior shape, our method incrementally constructs a continuous and smooth distance representation from incoming surface points, with analytical access to gradient information. The underlying model does not store training data for prediction, and its performance can be balanced through interpretable hyperparameters such as polynomial degree and number of segments. We assess the accuracy of the incrementally learned model on a set of household objects and compare it to neural network and Gaussian process counterparts. The utility of intermediate results and analytical gradients is further demonstrated in a physical experiment. For code and video, see https://sites.google.com/view/pp-sdf/.
Paper Structure (15 sections, 20 equations, 7 figures, 3 tables, 1 algorithm)

This paper contains 15 sections, 20 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Implicit environment representations
  3. Implicit signed distance functions
  4. Gaussian process implicit surfaces
  5. Basis function representations
  6. Piecewise polynomial SDF
  7. Bernstein polynomial basis functions
  8. Computation of weights
  9. Incremental formulation
  10. Evaluation
  11. Reconstruction accuracy
  12. Computational requirements
  13. Physical experiment
  14. Discussion
  15. Conclusion

Figures (7)

  • Figure 1: Mesh and distance level sets of the Stanford Bunny reconstructed from piecewise polynomial basis functions. The model is learned incrementally from $1283$ randomly sampled surface points and corresponding normal vectors.
  • Figure 2: Incremental model updates used to model a 2D shape on a $4\times 4$ grid, starting from a circular prior. Sampled points and normals are shown in dark blue, and the reconstructed zero-level contour in black. The reconstructed SDF is visualized as a color map. The normal ray and regularization points of a single sample are displayed in the second image.
  • Figure 3: Mesh (gray) and level set (red) reconstruction results for objects from the YCB YCBds test set: (a) 003_cracker_box, (b) 006_mustard_bottle, (c) 011_banana, (d) 016_pear, (e) 019_pitcher_base, (f) 021_bleach_cleanser, (g) 048_hammer, (h) 035_power_drill, (i) 063-a_marbles, (j) 053_mini_soccer_ball. Models were learned from $800$ non-uniformly sampled points and normals using 6 segments per input dimension.
  • Figure 4: Distance reconstruction accuracy compared on varying amounts of training data.
  • Figure 5: Accuracy of reconstructed gradient fields with varying amounts of training data.
  • ...and 2 more figures